:: Monoids
:: by Grzegorz Bancerek
::
:: Received December 29, 1992
:: Copyright (c) 1992 Association of Mizar Users

begin

deffunc H₁( 1-sorted ) -> set = the carrier of $1;

deffunc H₂( multMagma ) -> Element of bool [:[: the carrier of $1, the carrier of $1:], the carrier of $1:] = the multF of $1;

definition

let G be 1-sorted ;
mode BinOp of G is BinOp of the carrier of G;

end;

definition

let IT be 1-sorted ;
attr IT is constituted-Functions means :Def1: :: MONOID_0:def 1
for a being Element of IT holds a is Function;
attr IT is constituted-FinSeqs means :Def2: :: MONOID_0:def 2
for a being Element of IT holds a is FinSequence;

end;

:: deftheorem Def1 defines constituted-Functions MONOID_0:def 1 :
for IT being 1-sorted holds
( IT is constituted-Functions iff for a being Element of IT holds a is Function );

:: deftheorem Def2 defines constituted-FinSeqs MONOID_0:def 2 :
for IT being 1-sorted holds
( IT is constituted-FinSeqs iff for a being Element of IT holds a is FinSequence );

registration

cluster constituted-Functions 1-sorted ;
existence
ex b₁ being 1-sorted st b₁ is constituted-Functions

proof end;

cluster constituted-FinSeqs 1-sorted ;
existence
ex b₁ being 1-sorted st b₁ is constituted-FinSeqs

proof end;

end;

registration

let X be constituted-Functions 1-sorted ;
cluster -> Relation-like Function-like Element of the carrier of X;
coherence
for b₁ being Element of X holds
( b₁ is Function-like & b₁ is Relation-like ) by Def1;

end;

registration

cluster constituted-FinSeqs -> constituted-Functions 1-sorted ;
coherence
for b₁ being 1-sorted st b₁ is constituted-FinSeqs holds
b₁ is constituted-Functions

proof end;

end;

registration

let X be constituted-FinSeqs 1-sorted ;
cluster -> FinSequence-like Element of the carrier of X;
coherence
for b₁ being Element of X holds b₁ is FinSequence-like by Def2;

end;

definition

let D be set ;
let p, q be FinSequence of D;
:: original: ^
redefine func p ^ q -> Element of D * ;
coherence
p ^ q is Element of D *

proof end;

end;

notation

let g, f be Function;
synonym f (*) g for f * g;

end;

definition

let D be non empty set ;
let IT be BinOp of D;
attr IT is left-invertible means :: MONOID_0:def 3
for a, b being Element of D ex l being Element of D st IT . (l,a) = b;
attr IT is right-invertible means :: MONOID_0:def 4
for a, b being Element of D ex r being Element of D st IT . (a,r) = b;
attr IT is invertible means :Def5: :: MONOID_0:def 5
for a, b being Element of D ex r, l being Element of D st
( IT . (a,r) = b & IT . (l,a) = b );
attr IT is left-cancelable means :: MONOID_0:def 6
for a, b, c being Element of D st IT . (a,b) = IT . (a,c) holds
b = c;
attr IT is right-cancelable means :: MONOID_0:def 7
for a, b, c being Element of D st IT . (b,a) = IT . (c,a) holds
b = c;
attr IT is cancelable means :: MONOID_0:def 8
for a, b, c being Element of D st ( IT . (a,b) = IT . (a,c) or IT . (b,a) = IT . (c,a) ) holds
b = c;
attr IT is uniquely-decomposable means :: MONOID_0:def 9
( IT is having_a_unity & ( for a, b being Element of D st IT . (a,b) = the_unity_wrt IT holds
( a = b & b = the_unity_wrt IT ) ) );

end;

:: deftheorem defines left-invertible MONOID_0:def 3 :
for D being non empty set
for IT being BinOp of D holds
( IT is left-invertible iff for a, b being Element of D ex l being Element of D st IT . (l,a) = b );

:: deftheorem defines right-invertible MONOID_0:def 4 :
for D being non empty set
for IT being BinOp of D holds
( IT is right-invertible iff for a, b being Element of D ex r being Element of D st IT . (a,r) = b );

:: deftheorem Def5 defines invertible MONOID_0:def 5 :
for D being non empty set
for IT being BinOp of D holds
( IT is invertible iff for a, b being Element of D ex r, l being Element of D st
( IT . (a,r) = b & IT . (l,a) = b ) );

:: deftheorem defines left-cancelable MONOID_0:def 6 :
for D being non empty set
for IT being BinOp of D holds
( IT is left-cancelable iff for a, b, c being Element of D st IT . (a,b) = IT . (a,c) holds
b = c );

:: deftheorem defines right-cancelable MONOID_0:def 7 :
for D being non empty set
for IT being BinOp of D holds
( IT is right-cancelable iff for a, b, c being Element of D st IT . (b,a) = IT . (c,a) holds
b = c );

:: deftheorem defines cancelable MONOID_0:def 8 :
for D being non empty set
for IT being BinOp of D holds
( IT is cancelable iff for a, b, c being Element of D st ( IT . (a,b) = IT . (a,c) or IT . (b,a) = IT . (c,a) ) holds
b = c );

:: deftheorem defines uniquely-decomposable MONOID_0:def 9 :
for D being non empty set
for IT being BinOp of D holds
( IT is uniquely-decomposable iff ( IT is having_a_unity & ( for a, b being Element of D st IT . (a,b) = the_unity_wrt IT holds
( a = b & b = the_unity_wrt IT ) ) ) );

theorem Th1: :: MONOID_0:1

for D being non empty set
for f being BinOp of D holds
( f is invertible iff ( f is left-invertible & f is right-invertible ) )

proof end;

theorem Th2: :: MONOID_0:2

for D being non empty set
for f being BinOp of D holds
( f is cancelable iff ( f is left-cancelable & f is right-cancelable ) )

proof end;

theorem Th3: :: MONOID_0:3

for x being set
for f being BinOp of {x} holds
( f = (x,x) .--> x & f is having_a_unity & f is commutative & f is associative & f is idempotent & f is invertible & f is cancelable & f is uniquely-decomposable )

proof end;

begin

definition

let IT be non empty multMagma ;
redefine attr IT is unital means :Def10: :: MONOID_0:def 10
the multF of IT is having_a_unity ;
compatibility
( IT is unital iff the multF of IT is having_a_unity )

proof end;

end;

:: deftheorem Def10 defines unital MONOID_0:def 10 :
for IT being non empty multMagma holds
( IT is unital iff the multF of IT is having_a_unity );

definition

let G be non empty multMagma ;
redefine attr G is commutative means :Def11: :: MONOID_0:def 11
the multF of G is commutative ;
compatibility
( G is commutative iff the multF of G is commutative )

proof end;

redefine attr G is associative means :Def12: :: MONOID_0:def 12
the multF of G is associative ;
compatibility
( G is associative iff the multF of G is associative )

proof end;

end;

:: deftheorem Def11 defines commutative MONOID_0:def 11 :
for G being non empty multMagma holds
( G is commutative iff the multF of G is commutative );

:: deftheorem Def12 defines associative MONOID_0:def 12 :
for G being non empty multMagma holds
( G is associative iff the multF of G is associative );

definition

let IT be non empty multMagma ;
attr IT is idempotent means :: MONOID_0:def 13
the multF of IT is idempotent ;
attr IT is left-invertible means :: MONOID_0:def 14
the multF of IT is left-invertible ;
attr IT is right-invertible means :: MONOID_0:def 15
the multF of IT is right-invertible ;
attr IT is invertible means :Def16: :: MONOID_0:def 16
the multF of IT is invertible ;
attr IT is left-cancelable means :: MONOID_0:def 17
the multF of IT is left-cancelable ;
attr IT is right-cancelable means :: MONOID_0:def 18
the multF of IT is right-cancelable ;
attr IT is cancelable means :: MONOID_0:def 19
the multF of IT is cancelable ;
attr IT is uniquely-decomposable means :Def20: :: MONOID_0:def 20
the multF of IT is uniquely-decomposable ;

end;

:: deftheorem defines idempotent MONOID_0:def 13 :
for IT being non empty multMagma holds
( IT is idempotent iff the multF of IT is idempotent );

:: deftheorem defines left-invertible MONOID_0:def 14 :
for IT being non empty multMagma holds
( IT is left-invertible iff the multF of IT is left-invertible );

:: deftheorem defines right-invertible MONOID_0:def 15 :
for IT being non empty multMagma holds
( IT is right-invertible iff the multF of IT is right-invertible );

:: deftheorem Def16 defines invertible MONOID_0:def 16 :
for IT being non empty multMagma holds
( IT is invertible iff the multF of IT is invertible );

:: deftheorem defines left-cancelable MONOID_0:def 17 :
for IT being non empty multMagma holds
( IT is left-cancelable iff the multF of IT is left-cancelable );

:: deftheorem defines right-cancelable MONOID_0:def 18 :
for IT being non empty multMagma holds
( IT is right-cancelable iff the multF of IT is right-cancelable );

:: deftheorem defines cancelable MONOID_0:def 19 :
for IT being non empty multMagma holds
( IT is cancelable iff the multF of IT is cancelable );

:: deftheorem Def20 defines uniquely-decomposable MONOID_0:def 20 :
for IT being non empty multMagma holds
( IT is uniquely-decomposable iff the multF of IT is uniquely-decomposable );

registration

cluster non empty strict unital associative commutative constituted-Functions constituted-FinSeqs idempotent invertible cancelable uniquely-decomposable multMagma ;
existence
ex b₁ being non empty multMagma st
( b₁ is unital & b₁ is commutative & b₁ is associative & b₁ is cancelable & b₁ is idempotent & b₁ is invertible & b₁ is uniquely-decomposable & b₁ is constituted-Functions & b₁ is constituted-FinSeqs & b₁ is strict )

proof end;

end;

theorem Th4: :: MONOID_0:4

for G being non empty multMagma st G is unital holds
the_unity_wrt the multF of G is_a_unity_wrt the multF of G

proof end;

theorem :: MONOID_0:5

for G being non empty multMagma holds
( G is unital iff for a being Element of G holds
( (the_unity_wrt the multF of G) * a = a & a * (the_unity_wrt the multF of G) = a ) )

proof end;

theorem Th6: :: MONOID_0:6

for G being non empty multMagma holds
( G is unital iff ex a being Element of G st
for b being Element of G holds
( a * b = b & b * a = b ) )

proof end;

Lm1: for G being non empty multMagma holds
( G is commutative iff for a, b being Element of G holds a * b = b * a )

proof end;

Lm2: for G being non empty multMagma holds
( G is associative iff for a, b, c being Element of G holds (a * b) * c = a * (b * c) )

proof end;

theorem :: MONOID_0:7

canceled;

theorem :: MONOID_0:8

canceled;

theorem Th9: :: MONOID_0:9

for G being non empty multMagma holds
( G is idempotent iff for a being Element of G holds a * a = a )

proof end;

theorem :: MONOID_0:10

for G being non empty multMagma holds
( G is left-invertible iff for a, b being Element of G ex l being Element of G st l * a = b )

proof end;

theorem :: MONOID_0:11

for G being non empty multMagma holds
( G is right-invertible iff for a, b being Element of G ex r being Element of G st a * r = b )

proof end;

theorem Th12: :: MONOID_0:12

for G being non empty multMagma holds
( G is invertible iff for a, b being Element of G ex r, l being Element of G st
( a * r = b & l * a = b ) )

proof end;

theorem :: MONOID_0:13

for G being non empty multMagma holds
( G is left-cancelable iff for a, b, c being Element of G st a * b = a * c holds
b = c )

proof end;

theorem :: MONOID_0:14

for G being non empty multMagma holds
( G is right-cancelable iff for a, b, c being Element of G st b * a = c * a holds
b = c )

proof end;

theorem Th15: :: MONOID_0:15

for G being non empty multMagma holds
( G is cancelable iff for a, b, c being Element of G st ( a * b = a * c or b * a = c * a ) holds
b = c )

proof end;

theorem Th16: :: MONOID_0:16

for G being non empty multMagma holds
( G is uniquely-decomposable iff ( the multF of G is having_a_unity & ( for a, b being Element of G st a * b = the_unity_wrt the multF of G holds
( a = b & b = the_unity_wrt the multF of G ) ) ) )

proof end;

theorem Th17: :: MONOID_0:17

for G being non empty multMagma st G is associative holds
( G is invertible iff ( G is unital & the multF of G is having_an_inverseOp ) )

proof end;

Lm3: for G being non empty multMagma holds
( G is invertible iff ( G is left-invertible & G is right-invertible ) )

proof end;

Lm4: for G being non empty multMagma holds
( G is cancelable iff ( G is left-cancelable & G is right-cancelable ) )

proof end;

Lm5: for G being non empty multMagma st G is associative & G is invertible holds
G is Group-like

proof end;

registration

cluster non empty Group-like associative -> non empty invertible multMagma ;
coherence
for b₁ being non empty multMagma st b₁ is associative & b₁ is Group-like holds
b₁ is invertible

proof end;

cluster non empty associative invertible -> non empty Group-like multMagma ;
coherence
for b₁ being non empty multMagma st b₁ is associative & b₁ is invertible holds
b₁ is Group-like by Lm5;

end;

registration

cluster non empty invertible -> non empty left-invertible right-invertible multMagma ;
coherence
for b₁ being non empty multMagma st b₁ is invertible holds
( b₁ is left-invertible & b₁ is right-invertible ) by Lm3;
cluster non empty left-invertible right-invertible -> non empty invertible multMagma ;
coherence
for b₁ being non empty multMagma st b₁ is left-invertible & b₁ is right-invertible holds
b₁ is invertible by Lm3;
cluster non empty cancelable -> non empty left-cancelable right-cancelable multMagma ;
coherence
for b₁ being non empty multMagma st b₁ is cancelable holds
( b₁ is left-cancelable & b₁ is right-cancelable ) by Lm4;
cluster non empty left-cancelable right-cancelable -> non empty cancelable multMagma ;
coherence
for b₁ being non empty multMagma st b₁ is left-cancelable & b₁ is right-cancelable holds
b₁ is cancelable by Lm4;
cluster non empty associative invertible -> non empty unital cancelable multMagma ;
coherence
for b₁ being non empty multMagma st b₁ is associative & b₁ is invertible holds
( b₁ is unital & b₁ is cancelable )

proof end;

end;

begin

deffunc H₃( multLoopStr ) -> Element of the carrier of $1 = 1. $1;

definition

let IT be non empty multLoopStr ;
redefine attr IT is well-unital means :Def21: :: MONOID_0:def 21
1. IT is_a_unity_wrt the multF of IT;
compatibility
( IT is well-unital iff 1. IT is_a_unity_wrt the multF of IT )

proof end;

end;

:: deftheorem Def21 defines well-unital MONOID_0:def 21 :
for IT being non empty multLoopStr holds
( IT is well-unital iff 1. IT is_a_unity_wrt the multF of IT );

theorem Th18: :: MONOID_0:18

for M being non empty multLoopStr holds
( M is well-unital iff for a being Element of M holds
( (1. M) * a = a & a * (1. M) = a ) )

proof end;

theorem Th19: :: MONOID_0:19

for M being non empty multLoopStr st M is well-unital holds
1. M = the_unity_wrt the multF of M

proof end;

registration

cluster non empty strict unital associative commutative well-unital constituted-Functions constituted-FinSeqs idempotent invertible cancelable uniquely-decomposable multLoopStr ;
existence
ex b₁ being non empty multLoopStr st
( b₁ is well-unital & b₁ is commutative & b₁ is associative & b₁ is cancelable & b₁ is idempotent & b₁ is invertible & b₁ is uniquely-decomposable & b₁ is unital & b₁ is constituted-Functions & b₁ is constituted-FinSeqs & b₁ is strict )

proof end;

end;

definition

mode Monoid is non empty associative well-unital multLoopStr ;

end;

definition

let G be multMagma ;
mode MonoidalExtension of G -> multLoopStr means :Def22: :: MONOID_0:def 22
multMagma(# the carrier of it, the multF of it #) = multMagma(# the carrier of G, the multF of G #);
existence
ex b₁ being multLoopStr st multMagma(# the carrier of b₁, the multF of b₁ #) = multMagma(# the carrier of G, the multF of G #)

proof end;

end;

:: deftheorem Def22 defines MonoidalExtension MONOID_0:def 22 :
for G being multMagma
for b₂ being multLoopStr holds
( b₂ is MonoidalExtension of G iff multMagma(# the carrier of b₂, the multF of b₂ #) = multMagma(# the carrier of G, the multF of G #) );

registration

let G be non empty multMagma ;
cluster -> non empty MonoidalExtension of G;
coherence
for b₁ being MonoidalExtension of G holds not b₁ is empty

proof end;

end;

theorem Th20: :: MONOID_0:20

for G being non empty multMagma
for M being MonoidalExtension of G holds
( the carrier of M = the carrier of G & the multF of M = the multF of G & ( for a, b being Element of M
for a9, b9 being Element of G st a = a9 & b = b9 holds
a * b = a9 * b9 ) )

proof end;

registration

let G be multMagma ;
cluster strict MonoidalExtension of G;
existence
ex b₁ being MonoidalExtension of G st b₁ is strict

proof end;

end;

theorem Th21: :: MONOID_0:21

for G being non empty multMagma
for M being MonoidalExtension of G holds
( ( G is unital implies M is unital ) & ( G is commutative implies M is commutative ) & ( G is associative implies M is associative ) & ( G is invertible implies M is invertible ) & ( G is uniquely-decomposable implies M is uniquely-decomposable ) & ( G is cancelable implies M is cancelable ) )

proof end;

registration

let G be constituted-Functions multMagma ;
cluster -> constituted-Functions MonoidalExtension of G;
coherence
for b₁ being MonoidalExtension of G holds b₁ is constituted-Functions

proof end;

end;

registration

let G be constituted-FinSeqs multMagma ;
cluster -> constituted-FinSeqs MonoidalExtension of G;
coherence
for b₁ being MonoidalExtension of G holds b₁ is constituted-FinSeqs

proof end;

end;

registration

let G be non empty unital multMagma ;
cluster -> unital MonoidalExtension of G;
coherence
for b₁ being MonoidalExtension of G holds b₁ is unital by Th21;

end;

registration

let G be non empty associative multMagma ;
cluster -> associative MonoidalExtension of G;
coherence
for b₁ being MonoidalExtension of G holds b₁ is associative by Th21;

end;

registration

let G be non empty commutative multMagma ;
cluster -> commutative MonoidalExtension of G;
coherence
for b₁ being MonoidalExtension of G holds b₁ is commutative by Th21;

end;

registration

let G be non empty invertible multMagma ;
cluster -> invertible MonoidalExtension of G;
coherence
for b₁ being MonoidalExtension of G holds b₁ is invertible by Th21;

end;

registration

let G be non empty cancelable multMagma ;
cluster -> cancelable MonoidalExtension of G;
coherence
for b₁ being MonoidalExtension of G holds b₁ is cancelable by Th21;

end;

registration

let G be non empty uniquely-decomposable multMagma ;
cluster -> uniquely-decomposable MonoidalExtension of G;
coherence
for b₁ being MonoidalExtension of G holds b₁ is uniquely-decomposable by Th21;

end;

registration

let G be non empty unital multMagma ;
cluster non empty strict unital well-unital MonoidalExtension of G;
existence
ex b₁ being MonoidalExtension of G st
( b₁ is well-unital & b₁ is strict )

proof end;

end;

theorem Th22: :: MONOID_0:22

for G being non empty unital multMagma
for M1, M2 being strict well-unital MonoidalExtension of G holds M1 = M2

proof end;

begin

definition

let G be multMagma ;
mode SubStr of G -> multMagma means :Def23: :: MONOID_0:def 23
the multF of it c= the multF of G;
existence
ex b₁ being multMagma st the multF of b₁ c= the multF of G ;

end;

:: deftheorem Def23 defines SubStr MONOID_0:def 23 :
for G, b₂ being multMagma holds
( b₂ is SubStr of G iff the multF of b₂ c= the multF of G );

registration

let G be multMagma ;
cluster strict SubStr of G;
existence
ex b₁ being SubStr of G st b₁ is strict

proof end;

end;

registration

let G be non empty multMagma ;
cluster non empty strict SubStr of G;
existence
ex b₁ being SubStr of G st
( b₁ is strict & not b₁ is empty )

proof end;

end;

registration

let G be non empty unital multMagma ;
cluster non empty strict unital associative commutative idempotent invertible cancelable uniquely-decomposable SubStr of G;
existence
ex b₁ being non empty SubStr of G st
( b₁ is unital & b₁ is associative & b₁ is commutative & b₁ is cancelable & b₁ is idempotent & b₁ is invertible & b₁ is uniquely-decomposable & b₁ is strict )

proof end;

end;

definition

let G be multMagma ;
mode MonoidalSubStr of G -> multLoopStr means :Def24: :: MONOID_0:def 24
( the multF of it c= the multF of G & ( for M being multLoopStr st G = M holds
1. it = 1. M ) );
existence
ex b₁ being multLoopStr st
( the multF of b₁ c= the multF of G & ( for M being multLoopStr st G = M holds
1. b₁ = 1. M ) )

proof end;

end;

:: deftheorem Def24 defines MonoidalSubStr MONOID_0:def 24 :
for G being multMagma
for b₂ being multLoopStr holds
( b₂ is MonoidalSubStr of G iff ( the multF of b₂ c= the multF of G & ( for M being multLoopStr st G = M holds
1. b₂ = 1. M ) ) );

registration

let G be multMagma ;
cluster strict MonoidalSubStr of G;
existence
ex b₁ being MonoidalSubStr of G st b₁ is strict

proof end;

end;

registration

let G be non empty multMagma ;
cluster non empty strict MonoidalSubStr of G;
existence
ex b₁ being MonoidalSubStr of G st
( b₁ is strict & not b₁ is empty )

proof end;

end;

definition

let M be multLoopStr ;
redefine mode MonoidalSubStr of M means :Def25: :: MONOID_0:def 25
( the multF of it c= the multF of M & 1. it = 1. M );
compatibility
for b₁ being multLoopStr holds
( b₁ is MonoidalSubStr of M iff ( the multF of b₁ c= the multF of M & 1. b₁ = 1. M ) )

proof end;

end;

:: deftheorem Def25 defines MonoidalSubStr MONOID_0:def 25 :
for M, b₂ being multLoopStr holds
( b₂ is MonoidalSubStr of M iff ( the multF of b₂ c= the multF of M & 1. b₂ = 1. M ) );

registration

let G be non empty well-unital multLoopStr ;
cluster non empty strict associative commutative well-unital idempotent invertible cancelable uniquely-decomposable MonoidalSubStr of G;
existence
ex b₁ being non empty MonoidalSubStr of G st
( b₁ is well-unital & b₁ is associative & b₁ is commutative & b₁ is cancelable & b₁ is idempotent & b₁ is invertible & b₁ is uniquely-decomposable & b₁ is strict )

proof end;

end;

theorem Th23: :: MONOID_0:23

for G being multMagma
for M being MonoidalSubStr of G holds M is SubStr of G

proof end;

definition

let G be multMagma ;
let M be MonoidalExtension of G;
:: original: SubStr
redefine mode SubStr of M -> SubStr of G;
coherence
for b₁ being SubStr of M holds b₁ is SubStr of G

proof end;

end;

definition

let G1 be multMagma ;
let G2 be SubStr of G1;
:: original: SubStr
redefine mode SubStr of G2 -> SubStr of G1;
coherence
for b₁ being SubStr of G2 holds b₁ is SubStr of G1

proof end;

end;

definition

let G1 be multMagma ;
let G2 be MonoidalSubStr of G1;
:: original: SubStr
redefine mode SubStr of G2 -> SubStr of G1;
coherence
for b₁ being SubStr of G2 holds b₁ is SubStr of G1

proof end;

end;

definition

let G be multMagma ;
let M be MonoidalSubStr of G;
:: original: MonoidalSubStr
redefine mode MonoidalSubStr of M -> MonoidalSubStr of G;
coherence
for b₁ being MonoidalSubStr of M holds b₁ is MonoidalSubStr of G

proof end;

end;

theorem :: MONOID_0:24

for G being non empty multMagma
for M being non empty multLoopStr holds
( G is SubStr of G & M is MonoidalSubStr of M )

proof end;

theorem Th25: :: MONOID_0:25

for G being non empty multMagma
for H being non empty SubStr of G
for N being non empty MonoidalSubStr of G holds
( the carrier of H c= the carrier of G & the carrier of N c= the carrier of G )

proof end;

theorem Th26: :: MONOID_0:26

for G being non empty multMagma
for H being non empty SubStr of G holds the multF of H = the multF of G || the carrier of H

proof end;

theorem Th27: :: MONOID_0:27

for G being non empty multMagma
for H being non empty SubStr of G
for a, b being Element of H
for a9, b9 being Element of G st a = a9 & b = b9 holds
a * b = a9 * b9

proof end;

theorem Th28: :: MONOID_0:28

for G being non empty multMagma
for H1, H2 being non empty SubStr of G st the carrier of H1 = the carrier of H2 holds
multMagma(# the carrier of H1, the multF of H1 #) = multMagma(# the carrier of H2, the multF of H2 #)

proof end;

theorem :: MONOID_0:29

for M being non empty multLoopStr
for H1, H2 being non empty MonoidalSubStr of M st the carrier of H1 = the carrier of H2 holds
multLoopStr(# the carrier of H1, the multF of H1, the OneF of H1 #) = multLoopStr(# the carrier of H2, the multF of H2, the OneF of H2 #)

proof end;

theorem Th30: :: MONOID_0:30

for G being non empty multMagma
for H1, H2 being non empty SubStr of G st the carrier of H1 c= the carrier of H2 holds
H1 is SubStr of H2

proof end;

theorem :: MONOID_0:31

for M being non empty multLoopStr
for H1, H2 being non empty MonoidalSubStr of M st the carrier of H1 c= the carrier of H2 holds
H1 is MonoidalSubStr of H2

proof end;

theorem Th32: :: MONOID_0:32

for G being non empty multMagma
for H being non empty SubStr of G st G is unital & the_unity_wrt the multF of G in the carrier of H holds
( H is unital & the_unity_wrt the multF of G = the_unity_wrt the multF of H )

proof end;

theorem Th33: :: MONOID_0:33

for M being non empty well-unital multLoopStr
for N being non empty MonoidalSubStr of M holds N is well-unital

proof end;

theorem Th34: :: MONOID_0:34

for G being non empty multMagma
for H being non empty SubStr of G st G is commutative holds
H is commutative

proof end;

theorem Th35: :: MONOID_0:35

for G being non empty multMagma
for H being non empty SubStr of G st G is associative holds
H is associative

proof end;

theorem Th36: :: MONOID_0:36

for G being non empty multMagma
for H being non empty SubStr of G st G is idempotent holds
H is idempotent

proof end;

theorem Th37: :: MONOID_0:37

for G being non empty multMagma
for H being non empty SubStr of G st G is cancelable holds
H is cancelable

proof end;

theorem Th38: :: MONOID_0:38

for G being non empty multMagma
for H being non empty SubStr of G st the_unity_wrt the multF of G in the carrier of H & G is uniquely-decomposable holds
H is uniquely-decomposable

proof end;

theorem Th39: :: MONOID_0:39

for M being non empty well-unital uniquely-decomposable multLoopStr
for N being non empty MonoidalSubStr of M holds N is uniquely-decomposable

proof end;

registration

let G be non empty constituted-Functions multMagma ;
cluster non empty -> non empty constituted-Functions SubStr of G;
coherence
for b₁ being non empty SubStr of G holds b₁ is constituted-Functions

proof end;

cluster non empty -> non empty constituted-Functions MonoidalSubStr of G;
coherence
for b₁ being non empty MonoidalSubStr of G holds b₁ is constituted-Functions

proof end;

end;

registration

let G be non empty constituted-FinSeqs multMagma ;
cluster non empty -> non empty constituted-FinSeqs SubStr of G;
coherence
for b₁ being non empty SubStr of G holds b₁ is constituted-FinSeqs

proof end;

cluster non empty -> non empty constituted-FinSeqs MonoidalSubStr of G;
coherence
for b₁ being non empty MonoidalSubStr of G holds b₁ is constituted-FinSeqs

proof end;

end;

registration

let M be non empty well-unital multLoopStr ;
cluster non empty -> non empty well-unital MonoidalSubStr of M;
coherence
for b₁ being non empty MonoidalSubStr of M holds b₁ is well-unital by Th33;

end;

registration

let G be non empty commutative multMagma ;
cluster non empty -> non empty commutative SubStr of G;
coherence
for b₁ being non empty SubStr of G holds b₁ is commutative by Th34;
cluster non empty -> non empty commutative MonoidalSubStr of G;
coherence
for b₁ being non empty MonoidalSubStr of G holds b₁ is commutative

proof end;

end;

registration

let G be non empty associative multMagma ;
cluster non empty -> non empty associative SubStr of G;
coherence
for b₁ being non empty SubStr of G holds b₁ is associative by Th35;
cluster non empty -> non empty associative MonoidalSubStr of G;
coherence
for b₁ being non empty MonoidalSubStr of G holds b₁ is associative

proof end;

end;

registration

let G be non empty idempotent multMagma ;
cluster non empty -> non empty idempotent SubStr of G;
coherence
for b₁ being non empty SubStr of G holds b₁ is idempotent by Th36;
cluster non empty -> non empty idempotent MonoidalSubStr of G;
coherence
for b₁ being non empty MonoidalSubStr of G holds b₁ is idempotent

proof end;

end;

registration

let G be non empty cancelable multMagma ;
cluster non empty -> non empty cancelable SubStr of G;
coherence
for b₁ being non empty SubStr of G holds b₁ is cancelable by Th37;
cluster non empty -> non empty cancelable MonoidalSubStr of G;
coherence
for b₁ being non empty MonoidalSubStr of G holds b₁ is cancelable

proof end;

end;

registration

let M be non empty well-unital uniquely-decomposable multLoopStr ;
cluster non empty -> non empty uniquely-decomposable MonoidalSubStr of M;
coherence
for b₁ being non empty MonoidalSubStr of M holds b₁ is uniquely-decomposable by Th39;

end;

Lm6: now

let G be non empty multMagma ; :: thesis: for D being non empty Subset of G st ( for x, y being Element of D holds x * y in D ) holds
ex H being non empty strict SubStr of G st the carrier of H = D
let D be non empty Subset of G; :: thesis: ( ( for x, y being Element of D holds x * y in D ) implies ex H being non empty strict SubStr of G st the carrier of H = D )
assume A1: for x, y being Element of D holds x * y in D ; :: thesis: ex H being non empty strict SubStr of G st the carrier of H = D
thus ex H being non empty strict SubStr of G st the carrier of H = D :: thesis: verum

proof

set op = the multF of G;
set carr = the carrier of G;
A2: dom the multF of G = [: the carrier of G, the carrier of G:] by FUNCT_2:def 1;
A3: rng ( the multF of G || D) c= D

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng ( the multF of G || D) or x in D )
assume x in rng ( the multF of G || D) ; :: thesis: x in D
then consider y being set such that
A4: y in dom ( the multF of G || D) and
A5: x = ( the multF of G || D) . y by FUNCT_1:def 5;
reconsider y = y as Element of [:D,D:] by A2, A4, RELAT_1:91;
reconsider y1 = y `1 , y2 = y `2 as Element of D ;
y = [y1,y2] by MCART_1:23;
then x = y1 * y2 by A4, A5, FUNCT_1:70;
hence x in D by A1; :: thesis: verum

end;

dom ( the multF of G || D) = [:D,D:] by A2, RELAT_1:91;
then reconsider f = the multF of G || D as BinOp of D by A3, FUNCT_2:def 1, RELSET_1:11;
f c= the multF of G by RELAT_1:88;
then reconsider H = multMagma(# D,f #) as non empty strict SubStr of G by Def23;
take H ; :: thesis: the carrier of H = D
thus the carrier of H = D ; :: thesis: verum

end;

end;

scheme :: MONOID_0:sch 1
SubStrEx2{ F₁() -> non empty multMagma , P₁[ set ] } :

ex H being non empty strict SubStr of F₁() st
for x being Element of F₁() holds
( x in the carrier of H iff P₁[x] )

provided

A1: for x, y being Element of F₁() st P₁[x] & P₁[y] holds
P₁[x * y] and
A2: ex x being Element of F₁() st P₁[x]

proof end;

Lm7: now

let G be non empty multLoopStr ; :: thesis: for D being non empty Subset of G st ( for x, y being Element of D holds x * y in D ) & 1. G in D holds
ex H being non empty strict MonoidalSubStr of G st the carrier of H = D
let D be non empty Subset of G; :: thesis: ( ( for x, y being Element of D holds x * y in D ) & 1. G in D implies ex H being non empty strict MonoidalSubStr of G st the carrier of H = D )
assume that
A1: for x, y being Element of D holds x * y in D and
A2: 1. G in D ; :: thesis: ex H being non empty strict MonoidalSubStr of G st the carrier of H = D
thus ex H being non empty strict MonoidalSubStr of G st the carrier of H = D :: thesis: verum

proof

A3: for x, y being Element of D holds x * y in D by A1;
consider H being non empty strict SubStr of G such that
A4: the carrier of H = D by Lm6, A3;
reconsider e = 1. G as Element of H by A2, A4;
set N = multLoopStr(# the carrier of H, the multF of H,e #);
( H₂( multLoopStr(# the carrier of H, the multF of H,e #)) c= H₂(G) & ( for M being multLoopStr st G = M holds
1. multLoopStr(# the carrier of H, the multF of H,e #) = 1. M ) ) by Def23;
then reconsider N = multLoopStr(# the carrier of H, the multF of H,e #) as non empty strict MonoidalSubStr of G by Def25;
take N ; :: thesis: the carrier of N = D
thus the carrier of N = D by A4; :: thesis: verum

end;

end;

scheme :: MONOID_0:sch 2
MonoidalSubStrEx2{ F₁() -> non empty multLoopStr , P₁[ set ] } :

ex M being non empty strict MonoidalSubStr of F₁() st
for x being Element of F₁() holds
( x in the carrier of M iff P₁[x] )

provided

A1: for x, y being Element of F₁() st P₁[x] & P₁[y] holds
P₁[x * y] and
A2: P₁[ 1. F₁()]

proof end;

notation

let G be non empty multMagma ;
let a, b be Element of G;
synonym a [*] b for a * b;

end;

begin

definition

func <REAL,+> -> non empty strict unital associative commutative invertible cancelable multMagma equals :: MONOID_0:def 26
multMagma(# REAL,addreal #);
coherence
multMagma(# REAL,addreal #) is non empty strict unital associative commutative invertible cancelable multMagma by GROUP_1:92;

end;

:: deftheorem defines <REAL,+> MONOID_0:def 26 :
<REAL,+> = multMagma(# REAL,addreal #);

theorem :: MONOID_0:40

canceled;

theorem :: MONOID_0:41

for x being set holds
( x is Element of <REAL,+> iff x is Real ) ;

theorem :: MONOID_0:42

canceled;

theorem Th43: :: MONOID_0:43

for N being non empty SubStr of <REAL,+>
for a, b being Element of N
for x, y being Real st a = x & b = y holds
a * b = x + y

proof end;

theorem Th44: :: MONOID_0:44

for N being non empty unital SubStr of <REAL,+> holds the_unity_wrt the multF of N = 0

proof end;

registration

let G be non empty unital multMagma ;
cluster non empty associative invertible -> non empty unital Group-like cancelable SubStr of G;
coherence
for b₁ being non empty SubStr of G st b₁ is associative & b₁ is invertible holds
( b₁ is unital & b₁ is cancelable & b₁ is Group-like ) ;

end;

definition

:: original: INT.Group
redefine func INT.Group -> non empty strict unital invertible SubStr of <REAL,+> ;
coherence
INT.Group is non empty strict unital invertible SubStr of <REAL,+>

proof end;

end;

theorem :: MONOID_0:45

canceled;

theorem :: MONOID_0:46

for G being non empty strict SubStr of <REAL,+> holds
( G = INT.Group iff the carrier of G = INT ) by Th28, GR_CY_1:def 4;

theorem :: MONOID_0:47

for x being set holds
( x is Element of INT.Group iff x is Integer ) by GR_CY_1:def 4, INT_1:def 2;

definition

func <NAT,+> -> non empty strict unital uniquely-decomposable SubStr of INT.Group means :Def27: :: MONOID_0:def 27
the carrier of it = NAT ;
existence
ex b₁ being non empty strict unital uniquely-decomposable SubStr of INT.Group st the carrier of b₁ = NAT

proof end;

uniqueness
for b₁, b₂ being non empty strict unital uniquely-decomposable SubStr of INT.Group st the carrier of b₁ = NAT & the carrier of b₂ = NAT holds
b₁ = b₂ by Th28;

end;

:: deftheorem Def27 defines <NAT,+> MONOID_0:def 27 :
for b₁ being non empty strict unital uniquely-decomposable SubStr of INT.Group holds
( b₁ = <NAT,+> iff the carrier of b₁ = NAT );

definition

func <NAT,+,0> -> non empty strict well-unital MonoidalExtension of <NAT,+> means :: MONOID_0:def 28
verum;
existence
ex b₁ being non empty strict well-unital MonoidalExtension of <NAT,+> st verum ;
uniqueness
for b₁, b₂ being non empty strict well-unital MonoidalExtension of <NAT,+> holds b₁ = b₂ by Th22;

end;

:: deftheorem defines <NAT,+,0> MONOID_0:def 28 :
for b₁ being non empty strict well-unital MonoidalExtension of <NAT,+> holds
( b₁ = <NAT,+,0> iff verum );

definition

redefine func addnat equals :: MONOID_0:def 29
the multF of <NAT,+>;
compatibility
for b₁ being Element of bool [:[:NAT,NAT:],NAT:] holds
( b₁ = addnat iff b₁ = the multF of <NAT,+> )

proof end;

end;

:: deftheorem defines addnat MONOID_0:def 29 :
addnat = the multF of <NAT,+>;

theorem :: MONOID_0:48

canceled;

theorem Th49: :: MONOID_0:49

<NAT,+> = multMagma(# NAT,addnat #) by Def27;

theorem :: MONOID_0:50

for x being set holds
( x is Element of <NAT,+,0> iff x is Element of NAT )

proof end;

theorem :: MONOID_0:51

for n1, n2 being Element of NAT
for m1, m2 being Element of <NAT,+,0> st n1 = m1 & n2 = m2 holds
m1 * m2 = n1 + n2

proof end;

theorem :: MONOID_0:52

<NAT,+,0> = multLoopStr(# NAT,addnat,0 #)

proof end;

theorem :: MONOID_0:53

( addnat = addreal || NAT & addnat = addint || NAT )

proof end;

theorem :: MONOID_0:54

( 0 is_a_unity_wrt addnat & addnat is uniquely-decomposable )

proof end;

definition

func <REAL,*> -> non empty strict unital associative commutative multMagma equals :: MONOID_0:def 30
multMagma(# REAL,multreal #);
coherence
multMagma(# REAL,multreal #) is non empty strict unital associative commutative multMagma by Def10, Def11, Def12;

end;

:: deftheorem defines <REAL,*> MONOID_0:def 30 :
<REAL,*> = multMagma(# REAL,multreal #);

theorem :: MONOID_0:55

canceled;

theorem :: MONOID_0:56

for x being set holds
( x is Element of <REAL,*> iff x is Real ) ;

theorem :: MONOID_0:57

canceled;

theorem Th58: :: MONOID_0:58

for N being non empty SubStr of <REAL,*>
for a, b being Element of N
for x, y being Real st a = x & b = y holds
a * b = x * y

proof end;

theorem :: MONOID_0:59

canceled;

theorem :: MONOID_0:60

for N being non empty unital SubStr of <REAL,*> holds
( the_unity_wrt the multF of N = 0 or the_unity_wrt the multF of N = 1 )

proof end;

definition

func <NAT,*> -> non empty strict unital uniquely-decomposable SubStr of <REAL,*> means :Def31: :: MONOID_0:def 31
the carrier of it = NAT ;
existence
ex b₁ being non empty strict unital uniquely-decomposable SubStr of <REAL,*> st the carrier of b₁ = NAT

proof end;

uniqueness
for b₁, b₂ being non empty strict unital uniquely-decomposable SubStr of <REAL,*> st the carrier of b₁ = NAT & the carrier of b₂ = NAT holds
b₁ = b₂ by Th28;

end;

:: deftheorem Def31 defines <NAT,*> MONOID_0:def 31 :
for b₁ being non empty strict unital uniquely-decomposable SubStr of <REAL,*> holds
( b₁ = <NAT,*> iff the carrier of b₁ = NAT );

definition

func <NAT,*,1> -> non empty strict well-unital MonoidalExtension of <NAT,*> means :: MONOID_0:def 32
verum;
uniqueness
for b₁, b₂ being non empty strict well-unital MonoidalExtension of <NAT,*> holds b₁ = b₂ by Th22;
existence
ex b₁ being non empty strict well-unital MonoidalExtension of <NAT,*> st verum ;

end;

:: deftheorem defines <NAT,*,1> MONOID_0:def 32 :
for b₁ being non empty strict well-unital MonoidalExtension of <NAT,*> holds
( b₁ = <NAT,*,1> iff verum );

definition

redefine func multnat equals :: MONOID_0:def 33
the multF of <NAT,*>;
compatibility
for b₁ being Element of bool [:[:NAT,NAT:],NAT:] holds
( b₁ = multnat iff b₁ = the multF of <NAT,*> )

proof end;

end;

:: deftheorem defines multnat MONOID_0:def 33 :
multnat = the multF of <NAT,*>;

theorem Th61: :: MONOID_0:61

<NAT,*> = multMagma(# NAT,multnat #) by Def31;

theorem :: MONOID_0:62

for n1, n2 being Element of NAT
for m1, m2 being Element of <NAT,*> st n1 = m1 & n2 = m2 holds
m1 * m2 = n1 * n2 by Th58;

theorem Th63: :: MONOID_0:63

the_unity_wrt the multF of <NAT,*> = 1

proof end;

theorem :: MONOID_0:64

for n1, n2 being Element of NAT
for m1, m2 being Element of <NAT,*,1> st n1 = m1 & n2 = m2 holds
m1 * m2 = n1 * n2

proof end;

theorem :: MONOID_0:65

<NAT,*,1> = multLoopStr(# NAT,multnat,1 #)

proof end;

theorem :: MONOID_0:66

multnat = multreal || NAT by Th26, Th61;

theorem :: MONOID_0:67

( 1 is_a_unity_wrt multnat & multnat is uniquely-decomposable )

proof end;

begin

definition

let D be non empty set ;
func D *+^ -> non empty strict unital associative constituted-FinSeqs cancelable uniquely-decomposable multMagma means :Def34: :: MONOID_0:def 34
( the carrier of it = D * & ( for p, q being Element of it holds p [*] q = p ^ q ) );
existence
ex b₁ being non empty strict unital associative constituted-FinSeqs cancelable uniquely-decomposable multMagma st
( the carrier of b₁ = D * & ( for p, q being Element of b₁ holds p [*] q = p ^ q ) )

proof end;

uniqueness
for b₁, b₂ being non empty strict unital associative constituted-FinSeqs cancelable uniquely-decomposable multMagma st the carrier of b₁ = D * & ( for p, q being Element of b₁ holds p [*] q = p ^ q ) & the carrier of b₂ = D * & ( for p, q being Element of b₂ holds p [*] q = p ^ q ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def34 defines *+^ MONOID_0:def 34 :
for D being non empty set
for b₂ being non empty strict unital associative constituted-FinSeqs cancelable uniquely-decomposable multMagma holds
( b₂ = D *+^ iff ( the carrier of b₂ = D * & ( for p, q being Element of b₂ holds p [*] q = p ^ q ) ) );

definition

let D be non empty set ;
func D *+^+<0> -> non empty strict well-unital MonoidalExtension of D *+^ means :: MONOID_0:def 35
verum;
correctness
existence
ex b₁ being non empty strict well-unital MonoidalExtension of D *+^ st verum;
uniqueness
for b₁, b₂ being non empty strict well-unital MonoidalExtension of D *+^ holds b₁ = b₂;
by Th22;
func D -concatenation -> BinOp of (D *) equals :: MONOID_0:def 36
the multF of (D *+^);
correctness
coherence
the multF of (D *+^) is BinOp of (D *);

proof end;

end;

:: deftheorem defines *+^+<0> MONOID_0:def 35 :
for D being non empty set
for b₂ being non empty strict well-unital MonoidalExtension of D *+^ holds
( b₂ = D *+^+<0> iff verum );

:: deftheorem defines -concatenation MONOID_0:def 36 :
for D being non empty set holds D -concatenation = the multF of (D *+^);

theorem :: MONOID_0:68

for D being non empty set holds D *+^ = multMagma(# (D *),(D -concatenation) #) by Def34;

theorem Th69: :: MONOID_0:69

for D being non empty set holds the_unity_wrt the multF of (D *+^) = {}

proof end;

theorem :: MONOID_0:70

for D being non empty set holds
( the carrier of (D *+^+<0>) = D * & the multF of (D *+^+<0>) = D -concatenation & 1. (D *+^+<0>) = {} )

proof end;

theorem :: MONOID_0:71

for D being non empty set
for a, b being Element of (D *+^+<0>) holds a [*] b = a ^ b

proof end;

theorem Th72: :: MONOID_0:72

for D being non empty set
for F being non empty SubStr of D *+^
for p, q being Element of F holds p [*] q = p ^ q

proof end;

theorem :: MONOID_0:73

for D being non empty set
for F being non empty unital SubStr of D *+^ holds the_unity_wrt the multF of F = {}

proof end;

theorem :: MONOID_0:74

for D being non empty set
for F being non empty SubStr of D *+^ st {} is Element of F holds
( F is unital & the_unity_wrt the multF of F = {} )

proof end;

theorem :: MONOID_0:75

for A, B being non empty set st A c= B holds
A *+^ is SubStr of B *+^

proof end;

theorem :: MONOID_0:76

for D being non empty set holds
( D -concatenation is having_a_unity & the_unity_wrt (D -concatenation) = {} & D -concatenation is associative )

proof end;

begin

definition

let X be set ;
func GPFuncs X -> non empty strict unital associative constituted-Functions multMagma means :Def37: :: MONOID_0:def 37
( the carrier of it = PFuncs (X,X) & ( for f, g being Element of it holds f [*] g = f (*) g ) );
existence
ex b₁ being non empty strict unital associative constituted-Functions multMagma st
( the carrier of b₁ = PFuncs (X,X) & ( for f, g being Element of b₁ holds f [*] g = f (*) g ) )

proof end;

uniqueness
for b₁, b₂ being non empty strict unital associative constituted-Functions multMagma st the carrier of b₁ = PFuncs (X,X) & ( for f, g being Element of b₁ holds f [*] g = f (*) g ) & the carrier of b₂ = PFuncs (X,X) & ( for f, g being Element of b₂ holds f [*] g = f (*) g ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def37 defines GPFuncs MONOID_0:def 37 :
for X being set
for b₂ being non empty strict unital associative constituted-Functions multMagma holds
( b₂ = GPFuncs X iff ( the carrier of b₂ = PFuncs (X,X) & ( for f, g being Element of b₂ holds f [*] g = f (*) g ) ) );

definition

let X be set ;
func MPFuncs X -> non empty strict well-unital MonoidalExtension of GPFuncs X means :: MONOID_0:def 38
verum;
existence
ex b₁ being non empty strict well-unital MonoidalExtension of GPFuncs X st verum ;
uniqueness
for b₁, b₂ being non empty strict well-unital MonoidalExtension of GPFuncs X holds b₁ = b₂ by Th22;
func X -composition -> BinOp of (PFuncs (X,X)) equals :: MONOID_0:def 39
the multF of (GPFuncs X);
correctness
coherence
the multF of (GPFuncs X) is BinOp of (PFuncs (X,X));

proof end;

end;

:: deftheorem defines MPFuncs MONOID_0:def 38 :
for X being set
for b₂ being non empty strict well-unital MonoidalExtension of GPFuncs X holds
( b₂ = MPFuncs X iff verum );

:: deftheorem defines -composition MONOID_0:def 39 :
for X being set holds X -composition = the multF of (GPFuncs X);

theorem :: MONOID_0:77

for x, X being set holds
( x is Element of (GPFuncs X) iff x is PartFunc of X,X )

proof end;

theorem Th78: :: MONOID_0:78

for X being set holds the_unity_wrt the multF of (GPFuncs X) = id X

proof end;

theorem Th79: :: MONOID_0:79

for X being set
for F being non empty SubStr of GPFuncs X
for f, g being Element of F holds f [*] g = f (*) g

proof end;

theorem Th80: :: MONOID_0:80

for X being set
for F being non empty SubStr of GPFuncs X st id X is Element of F holds
( F is unital & the_unity_wrt the multF of F = id X )

proof end;

theorem :: MONOID_0:81

for Y, X being set st Y c= X holds
GPFuncs Y is SubStr of GPFuncs X

proof end;

Lm8: for x, X being set st x in Funcs (X,X) holds
x is Function of X,X

proof end;

definition

let X be set ;
func GFuncs X -> non empty strict unital SubStr of GPFuncs X means :Def40: :: MONOID_0:def 40
the carrier of it = Funcs (X,X);
existence
ex b₁ being non empty strict unital SubStr of GPFuncs X st the carrier of b₁ = Funcs (X,X)

proof end;

uniqueness
for b₁, b₂ being non empty strict unital SubStr of GPFuncs X st the carrier of b₁ = Funcs (X,X) & the carrier of b₂ = Funcs (X,X) holds
b₁ = b₂ by Th28;

end;

:: deftheorem Def40 defines GFuncs MONOID_0:def 40 :
for X being set
for b₂ being non empty strict unital SubStr of GPFuncs X holds
( b₂ = GFuncs X iff the carrier of b₂ = Funcs (X,X) );

definition

let X be set ;
func MFuncs X -> strict well-unital MonoidalExtension of GFuncs X means :: MONOID_0:def 41
verum;
correctness
existence
ex b₁ being strict well-unital MonoidalExtension of GFuncs X st verum;
uniqueness
for b₁, b₂ being strict well-unital MonoidalExtension of GFuncs X holds b₁ = b₂;
by Th22;

end;

:: deftheorem defines MFuncs MONOID_0:def 41 :
for X being set
for b₂ being strict well-unital MonoidalExtension of GFuncs X holds
( b₂ = MFuncs X iff verum );

theorem :: MONOID_0:82

for x, X being set holds
( x is Element of (GFuncs X) iff x is Function of X,X )

proof end;

theorem Th83: :: MONOID_0:83

for X being set holds the multF of (GFuncs X) = (X -composition) || (Funcs (X,X))

proof end;

theorem Th84: :: MONOID_0:84

for X being set holds the_unity_wrt the multF of (GFuncs X) = id X

proof end;

theorem :: MONOID_0:85

for X being set holds
( the carrier of (MFuncs X) = Funcs (X,X) & the multF of (MFuncs X) = (X -composition) || (Funcs (X,X)) & 1. (MFuncs X) = id X )

proof end;

definition

let X be set ;
func GPerms X -> non empty strict unital invertible SubStr of GFuncs X means :Def42: :: MONOID_0:def 42
for f being Element of (GFuncs X) holds
( f in the carrier of it iff f is Permutation of X );
existence
ex b₁ being non empty strict unital invertible SubStr of GFuncs X st
for f being Element of (GFuncs X) holds
( f in the carrier of b₁ iff f is Permutation of X )

proof end;

uniqueness
for b₁, b₂ being non empty strict unital invertible SubStr of GFuncs X st ( for f being Element of (GFuncs X) holds
( f in the carrier of b₁ iff f is Permutation of X ) ) & ( for f being Element of (GFuncs X) holds
( f in the carrier of b₂ iff f is Permutation of X ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def42 defines GPerms MONOID_0:def 42 :
for X being set
for b₂ being non empty strict unital invertible SubStr of GFuncs X holds
( b₂ = GPerms X iff for f being Element of (GFuncs X) holds
( f in the carrier of b₂ iff f is Permutation of X ) );

theorem Th86: :: MONOID_0:86

for x, X being set holds
( x is Element of (GPerms X) iff x is Permutation of X )

proof end;

theorem Th87: :: MONOID_0:87

for X being set holds
( the_unity_wrt the multF of (GPerms X) = id X & 1_ (GPerms X) = id X )

proof end;

theorem :: MONOID_0:88

for X being set
for f being Element of (GPerms X) holds f " = f "

proof end;

theorem :: MONOID_0:89

for S being 1-sorted st the carrier of S is functional holds
S is constituted-Functions

proof end;

theorem :: MONOID_0:90

for G being non empty multMagma
for D being non empty Subset of G st ( for x, y being Element of D holds x * y in D ) holds
ex H being non empty strict SubStr of G st the carrier of H = D by Lm6;

theorem :: MONOID_0:91

for G being non empty multLoopStr
for D being non empty Subset of G st ( for x, y being Element of D holds x * y in D ) & 1. G in D holds
ex H being non empty strict MonoidalSubStr of G st the carrier of H = D by Lm7;