:: The Product of the Families of the Groups
:: by Artur Korni{\l}owicz
::
:: Received June 10, 1998
:: Copyright (c) 1998-2021 Association of Mizar Users

definition

let R be Relation;

attr R is multMagma-yielding means :Def1: :: GROUP_7:def 1
for y being set st y in rng R holds
y is non empty multMagma ;

end;

:: deftheorem Def1 defines multMagma-yielding GROUP_7:def 1 :
for R being Relation holds
( R is multMagma-yielding iff for y being set st y in rng R holds
y is non empty multMagma );

registration

cluster Relation-like Function-like multMagma-yielding -> 1-sorted-yielding for set ;

coherence
for b₁ being Function st b₁ is multMagma-yielding holds
b₁ is 1-sorted-yielding

proof end;

end;

registration

let I be set ;

cluster Relation-like I -defined Function-like total multMagma-yielding for set ;

existence
ex b₁ being ManySortedSet of I st b₁ is multMagma-yielding

proof end;

end;

registration

cluster Relation-like Function-like multMagma-yielding for set ;

existence
ex b₁ being Function st b₁ is multMagma-yielding

proof end;

end;

definition

let I be set ;
mode multMagma-Family of I is multMagma-yielding ManySortedSet of I;

end;

definition

let I be non empty set ;
let F be multMagma-Family of I;
let i be Element of I;
:: original: .
redefine func F . i -> non empty multMagma ;
coherence
F . i is non empty multMagma

proof end;

end;

registration

let I be set ;
let F be multMagma-Family of I;

cluster Carrier F -> non-empty ;

coherence
Carrier F is non-empty

proof end;

end;

Lm1: now :: thesis: for I being non empty set
for i being Element of I
for F being multMagma-yielding ManySortedSet of I
for f being Element of product (Carrier F) holds f . i in the carrier of (F . i)

let I be non empty set ; :: thesis: for i being Element of I
for F being multMagma-yielding ManySortedSet of I
for f being Element of product (Carrier F) holds f . i in the carrier of (F . i)
let i be Element of I; :: thesis: for F being multMagma-yielding ManySortedSet of I
for f being Element of product (Carrier F) holds f . i in the carrier of (F . i)
let F be multMagma-yielding ManySortedSet of I; :: thesis: for f being Element of product (Carrier F) holds f . i in the carrier of (F . i)
let f be Element of product (Carrier F); :: thesis: f . i in the carrier of (F . i)
A1: dom (Carrier F) = I by PARTFUN1:def 2;
ex R being 1-sorted st
( R = F . i & (Carrier F) . i = the carrier of R ) by PRALG_1:def 15;
hence f . i in the carrier of (F . i) by A1, CARD_3:9; :: thesis: verum

end;

definition

let I be set ;
let F be multMagma-Family of I;

func product F -> strict multMagma means :Def2: :: GROUP_7:def 2
( the carrier of it = product (Carrier F) & ( for f, g being Element of product (Carrier F)
for i being set st i in I holds
ex Fi being non empty multMagma ex h being Function st
( Fi = F . i & h = the multF of it . (f,g) & h . i = the multF of Fi . ((f . i),(g . i)) ) ) );

existence
ex b₁ being strict multMagma st
( the carrier of b₁ = product (Carrier F) & ( for f, g being Element of product (Carrier F)
for i being set st i in I holds
ex Fi being non empty multMagma ex h being Function st
( Fi = F . i & h = the multF of b₁ . (f,g) & h . i = the multF of Fi . ((f . i),(g . i)) ) ) )

proof end;

uniqueness
for b₁, b₂ being strict multMagma st the carrier of b₁ = product (Carrier F) & ( for f, g being Element of product (Carrier F)
for i being set st i in I holds
ex Fi being non empty multMagma ex h being Function st
( Fi = F . i & h = the multF of b₁ . (f,g) & h . i = the multF of Fi . ((f . i),(g . i)) ) ) & the carrier of b₂ = product (Carrier F) & ( for f, g being Element of product (Carrier F)
for i being set st i in I holds
ex Fi being non empty multMagma ex h being Function st
( Fi = F . i & h = the multF of b₂ . (f,g) & h . i = the multF of Fi . ((f . i),(g . i)) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines product GROUP_7:def 2 :
for I being set
for F being multMagma-Family of I
for b₃ being strict multMagma holds
( b₃ = product F iff ( the carrier of b₃ = product (Carrier F) & ( for f, g being Element of product (Carrier F)
for i being set st i in I holds
ex Fi being non empty multMagma ex h being Function st
( Fi = F . i & h = the multF of b₃ . (f,g) & h . i = the multF of Fi . ((f . i),(g . i)) ) ) ) );

registration

let I be set ;
let F be multMagma-Family of I;

cluster product F -> non empty strict constituted-Functions ;

coherence
( not product F is empty & product F is constituted-Functions )

proof end;

end;

theorem Th1: :: GROUP_7:1

for i, I being set
for f, g, h being Function
for F being multMagma-Family of I
for G being non empty multMagma
for p, q being Element of (product F)
for x, y being Element of G st i in I & G = F . i & f = p & g = q & h = p * q & f . i = x & g . i = y holds
x * y = h . i

proof end;

definition

let I be set ;
let F be multMagma-Family of I;

attr F is Group-like means :Def3: :: GROUP_7:def 3
for i being set st i in I holds
ex Fi being non empty Group-like multMagma st Fi = F . i;

attr F is associative means :Def4: :: GROUP_7:def 4
for i being set st i in I holds
ex Fi being non empty associative multMagma st Fi = F . i;

attr F is commutative means :Def5: :: GROUP_7:def 5
for i being set st i in I holds
ex Fi being non empty commutative multMagma st Fi = F . i;

end;

:: deftheorem Def3 defines Group-like GROUP_7:def 3 :
for I being set
for F being multMagma-Family of I holds
( F is Group-like iff for i being set st i in I holds
ex Fi being non empty Group-like multMagma st Fi = F . i );

:: deftheorem Def4 defines associative GROUP_7:def 4 :
for I being set
for F being multMagma-Family of I holds
( F is associative iff for i being set st i in I holds
ex Fi being non empty associative multMagma st Fi = F . i );

:: deftheorem Def5 defines commutative GROUP_7:def 5 :
for I being set
for F being multMagma-Family of I holds
( F is commutative iff for i being set st i in I holds
ex Fi being non empty commutative multMagma st Fi = F . i );

definition

let I be non empty set ;
let F be multMagma-Family of I;

redefine attr F is Group-like means :Def6: :: GROUP_7:def 6
for i being Element of I holds F . i is Group-like ;

compatibility
( F is Group-like iff for i being Element of I holds F . i is Group-like )

proof end;

redefine attr F is associative means :Def7: :: GROUP_7:def 7
for i being Element of I holds F . i is associative ;

compatibility
( F is associative iff for i being Element of I holds F . i is associative )

proof end;

redefine attr F is commutative means :: GROUP_7:def 8
for i being Element of I holds F . i is commutative ;

compatibility
( F is commutative iff for i being Element of I holds F . i is commutative )

proof end;

end;

:: deftheorem Def6 defines Group-like GROUP_7:def 6 :
for I being non empty set
for F being multMagma-Family of I holds
( F is Group-like iff for i being Element of I holds F . i is Group-like );

:: deftheorem Def7 defines associative GROUP_7:def 7 :
for I being non empty set
for F being multMagma-Family of I holds
( F is associative iff for i being Element of I holds F . i is associative );

:: deftheorem defines commutative GROUP_7:def 8 :
for I being non empty set
for F being multMagma-Family of I holds
( F is commutative iff for i being Element of I holds F . i is commutative );

registration

let I be set ;

cluster Relation-like I -defined Function-like total 1-sorted-yielding multMagma-yielding Group-like associative commutative for set ;

existence
ex b₁ being multMagma-Family of I st
( b₁ is Group-like & b₁ is associative & b₁ is commutative )

proof end;

end;

registration

let I be set ;
let F be Group-like multMagma-Family of I;

cluster product F -> strict Group-like ;

coherence
product F is Group-like

proof end;

end;

registration

let I be set ;
let F be associative multMagma-Family of I;

cluster product F -> strict associative ;

coherence
product F is associative

proof end;

end;

registration

let I be set ;
let F be commutative multMagma-Family of I;

cluster product F -> strict commutative ;

coherence
product F is commutative

proof end;

end;

theorem :: GROUP_7:2

for i, I being set
for F being multMagma-Family of I
for G being non empty multMagma st i in I & G = F . i & product F is Group-like holds
G is Group-like

proof end;

theorem :: GROUP_7:3

for i, I being set
for F being multMagma-Family of I
for G being non empty multMagma st i in I & G = F . i & product F is associative holds
G is associative

proof end;

theorem :: GROUP_7:4

for i, I being set
for F being multMagma-Family of I
for G being non empty multMagma st i in I & G = F . i & product F is commutative holds
G is commutative

proof end;

theorem Th5: :: GROUP_7:5

for I being set
for s being ManySortedSet of I
for F being Group-like multMagma-Family of I st ( for i being set st i in I holds
ex G being non empty Group-like multMagma st
( G = F . i & s . i = 1_ G ) ) holds
s = 1_ (product F)

proof end;

theorem Th6: :: GROUP_7:6

for i, I being set
for f being Function
for F being Group-like multMagma-Family of I
for G being non empty Group-like multMagma st i in I & G = F . i & f = 1_ (product F) holds
f . i = 1_ G

proof end;

theorem Th7: :: GROUP_7:7

for I being set
for g being Function
for s being ManySortedSet of I
for F being Group-like associative multMagma-Family of I
for x being Element of (product F) st x = g & ( for i being set st i in I holds
ex G being Group ex y being Element of G st
( G = F . i & s . i = y " & y = g . i ) ) holds
s = x "

proof end;

theorem Th8: :: GROUP_7:8

for i, I being set
for f, g being Function
for F being Group-like associative multMagma-Family of I
for x being Element of (product F)
for G being Group
for y being Element of G st i in I & G = F . i & f = x & g = x " & f . i = y holds
g . i = y "

proof end;

definition

let I be set ;
let F be Group-like associative multMagma-Family of I;

func sum F -> strict Subgroup of product F means :Def9: :: GROUP_7:def 9
for x being object holds
( x in the carrier of it iff ex g being Element of product (Carrier F) ex J being finite Subset of I ex f being ManySortedSet of J st
( g = 1_ (product F) & x = g +* f & ( for j being set st j in J holds
ex G being non empty Group-like multMagma st
( G = F . j & f . j in the carrier of G & f . j <> 1_ G ) ) ) );

existence
ex b₁ being strict Subgroup of product F st
for x being object holds
( x in the carrier of b₁ iff ex g being Element of product (Carrier F) ex J being finite Subset of I ex f being ManySortedSet of J st
( g = 1_ (product F) & x = g +* f & ( for j being set st j in J holds
ex G being non empty Group-like multMagma st
( G = F . j & f . j in the carrier of G & f . j <> 1_ G ) ) ) )

proof end;

uniqueness
for b₁, b₂ being strict Subgroup of product F st ( for x being object holds
( x in the carrier of b₁ iff ex g being Element of product (Carrier F) ex J being finite Subset of I ex f being ManySortedSet of J st
( g = 1_ (product F) & x = g +* f & ( for j being set st j in J holds
ex G being non empty Group-like multMagma st
( G = F . j & f . j in the carrier of G & f . j <> 1_ G ) ) ) ) ) & ( for x being object holds
( x in the carrier of b₂ iff ex g being Element of product (Carrier F) ex J being finite Subset of I ex f being ManySortedSet of J st
( g = 1_ (product F) & x = g +* f & ( for j being set st j in J holds
ex G being non empty Group-like multMagma st
( G = F . j & f . j in the carrier of G & f . j <> 1_ G ) ) ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def9 defines sum GROUP_7:def 9 :
for I being set
for F being Group-like associative multMagma-Family of I
for b₃ being strict Subgroup of product F holds
( b₃ = sum F iff for x being object holds
( x in the carrier of b₃ iff ex g being Element of product (Carrier F) ex J being finite Subset of I ex f being ManySortedSet of J st
( g = 1_ (product F) & x = g +* f & ( for j being set st j in J holds
ex G being non empty Group-like multMagma st
( G = F . j & f . j in the carrier of G & f . j <> 1_ G ) ) ) ) );

registration

let I be set ;
let F be Group-like associative multMagma-Family of I;
let f, g be Element of (sum F);

cluster the multF of (sum F) . (f,g) -> Relation-like Function-like ;

coherence
( the multF of (sum F) . (f,g) is Function-like & the multF of (sum F) . (f,g) is Relation-like )

proof end;

end;

theorem :: GROUP_7:9

for I being finite set
for F being Group-like associative multMagma-Family of I holds product F = sum F

proof end;

theorem Th10: :: GROUP_7:10

for G1 being non empty multMagma holds <*G1*> is multMagma-Family of {1}

proof end;

registration

let G1 be non empty multMagma ;

cluster <*G1*> -> {1} -defined ;

coherence
<*G1*> is {1} -defined by Th10;

end;

registration

let G1 be non empty multMagma ;

cluster <*G1*> -> total multMagma-yielding ;

coherence
( <*G1*> is total & <*G1*> is multMagma-yielding ) by Th10;

end;

theorem Th11: :: GROUP_7:11

for G1 being non empty Group-like multMagma holds <*G1*> is Group-like multMagma-Family of {1}

proof end;

registration

let G1 be non empty Group-like multMagma ;

cluster <*G1*> -> Group-like ;

coherence
<*G1*> is Group-like by Th11;

end;

theorem Th12: :: GROUP_7:12

for G1 being non empty associative multMagma holds <*G1*> is associative multMagma-Family of {1}

proof end;

registration

let G1 be non empty associative multMagma ;

cluster <*G1*> -> associative ;

coherence
<*G1*> is associative by Th12;

end;

theorem Th13: :: GROUP_7:13

for G1 being non empty commutative multMagma holds <*G1*> is commutative multMagma-Family of {1}

proof end;

registration

let G1 be non empty commutative multMagma ;

cluster <*G1*> -> commutative ;

coherence
<*G1*> is commutative by Th13;

end;

theorem :: GROUP_7:14

for G1 being Group holds <*G1*> is Group-like associative multMagma-Family of {1} ;

theorem :: GROUP_7:15

for G1 being commutative Group holds <*G1*> is Group-like associative commutative multMagma-Family of {1} ;

registration

let G1 be non empty multMagma ;

cluster -> FinSequence-like for Element of product (Carrier <*G1*>);

coherence
for b₁ being Element of product (Carrier <*G1*>) holds b₁ is FinSequence-like by FINSEQ_1:2, PARTFUN1:def 2;

end;

registration

let G1 be non empty multMagma ;

cluster -> FinSequence-like for Element of the carrier of (product <*G1*>);

coherence
for b₁ being Element of (product <*G1*>) holds b₁ is FinSequence-like

proof end;

end;

definition

let G1 be non empty multMagma ;
let x be Element of G1;
:: original: <*
redefine func <*x*> -> Element of (product <*G1*>);
coherence
<*x*> is Element of (product <*G1*>)

proof end;

end;

theorem Th16: :: GROUP_7:16

for G1, G2 being non empty multMagma holds <*G1,G2*> is multMagma-Family of {1,2}

proof end;

registration

let G1, G2 be non empty multMagma ;

cluster <*G1,G2*> -> {1,2} -defined ;

coherence
<*G1,G2*> is {1,2} -defined by Th16;

end;

registration

let G1, G2 be non empty multMagma ;

cluster <*G1,G2*> -> total multMagma-yielding ;

coherence
( <*G1,G2*> is total & <*G1,G2*> is multMagma-yielding ) by Th16;

end;

theorem Th17: :: GROUP_7:17

for G1, G2 being non empty Group-like multMagma holds <*G1,G2*> is Group-like multMagma-Family of {1,2}

proof end;

registration

let G1, G2 be non empty Group-like multMagma ;

cluster <*G1,G2*> -> Group-like ;

coherence
<*G1,G2*> is Group-like by Th17;

end;

theorem Th18: :: GROUP_7:18

for G1, G2 being non empty associative multMagma holds <*G1,G2*> is associative multMagma-Family of {1,2}

proof end;

registration

let G1, G2 be non empty associative multMagma ;

cluster <*G1,G2*> -> associative ;

coherence
<*G1,G2*> is associative by Th18;

end;

theorem Th19: :: GROUP_7:19

for G1, G2 being non empty commutative multMagma holds <*G1,G2*> is commutative multMagma-Family of {1,2}

proof end;

registration

let G1, G2 be non empty commutative multMagma ;

cluster <*G1,G2*> -> commutative ;

coherence
<*G1,G2*> is commutative by Th19;

end;

theorem :: GROUP_7:20

for G1, G2 being Group holds <*G1,G2*> is Group-like associative multMagma-Family of {1,2} ;

theorem :: GROUP_7:21

for G1, G2 being commutative Group holds <*G1,G2*> is Group-like associative commutative multMagma-Family of {1,2} ;

registration

let G1, G2 be non empty multMagma ;

cluster -> FinSequence-like for Element of product (Carrier <*G1,G2*>);

coherence
for b₁ being Element of product (Carrier <*G1,G2*>) holds b₁ is FinSequence-like by FINSEQ_1:2, PARTFUN1:def 2;

end;

registration

let G1, G2 be non empty multMagma ;

cluster -> FinSequence-like for Element of the carrier of (product <*G1,G2*>);

coherence
for b₁ being Element of (product <*G1,G2*>) holds b₁ is FinSequence-like

proof end;

end;

definition

let G1, G2 be non empty multMagma ;
let x be Element of G1;
let y be Element of G2;
:: original: <*
redefine func <*x,y*> -> Element of (product <*G1,G2*>);
coherence
<*x,y*> is Element of (product <*G1,G2*>)

proof end;

end;

theorem Th22: :: GROUP_7:22

for G1, G2, G3 being non empty multMagma holds <*G1,G2,G3*> is multMagma-Family of {1,2,3}

proof end;

registration

let G1, G2, G3 be non empty multMagma ;

cluster <*G1,G2,G3*> -> {1,2,3} -defined ;

coherence
<*G1,G2,G3*> is {1,2,3} -defined by Th22;

end;

registration

let G1, G2, G3 be non empty multMagma ;

cluster <*G1,G2,G3*> -> total multMagma-yielding ;

coherence
( <*G1,G2,G3*> is total & <*G1,G2,G3*> is multMagma-yielding ) by Th22;

end;

theorem Th23: :: GROUP_7:23

for G1, G2, G3 being non empty Group-like multMagma holds <*G1,G2,G3*> is Group-like multMagma-Family of {1,2,3}

proof end;

registration

let G1, G2, G3 be non empty Group-like multMagma ;

cluster <*G1,G2,G3*> -> Group-like ;

coherence
<*G1,G2,G3*> is Group-like by Th23;

end;

theorem Th24: :: GROUP_7:24

for G1, G2, G3 being non empty associative multMagma holds <*G1,G2,G3*> is associative multMagma-Family of {1,2,3}

proof end;

registration

let G1, G2, G3 be non empty associative multMagma ;

cluster <*G1,G2,G3*> -> associative ;

coherence
<*G1,G2,G3*> is associative by Th24;

end;

theorem Th25: :: GROUP_7:25

for G1, G2, G3 being non empty commutative multMagma holds <*G1,G2,G3*> is commutative multMagma-Family of {1,2,3}

proof end;

registration

let G1, G2, G3 be non empty commutative multMagma ;

cluster <*G1,G2,G3*> -> commutative ;

coherence
<*G1,G2,G3*> is commutative by Th25;

end;

theorem :: GROUP_7:26

for G1, G2, G3 being Group holds <*G1,G2,G3*> is Group-like associative multMagma-Family of {1,2,3} ;

theorem :: GROUP_7:27

for G1, G2, G3 being commutative Group holds <*G1,G2,G3*> is Group-like associative commutative multMagma-Family of {1,2,3} ;

registration

let G1, G2, G3 be non empty multMagma ;

cluster -> FinSequence-like for Element of product (Carrier <*G1,G2,G3*>);

coherence
for b₁ being Element of product (Carrier <*G1,G2,G3*>) holds b₁ is FinSequence-like by FINSEQ_3:1, PARTFUN1:def 2;

end;

registration

let G1, G2, G3 be non empty multMagma ;

cluster -> FinSequence-like for Element of the carrier of (product <*G1,G2,G3*>);

coherence
for b₁ being Element of (product <*G1,G2,G3*>) holds b₁ is FinSequence-like

proof end;

end;

definition

let G1, G2, G3 be non empty multMagma ;
let x be Element of G1;
let y be Element of G2;
let z be Element of G3;
:: original: <*
redefine func <*x,y,z*> -> Element of (product <*G1,G2,G3*>);
coherence
<*x,y,z*> is Element of (product <*G1,G2,G3*>)

proof end;

end;

theorem Th28: :: GROUP_7:28

for G1 being non empty multMagma
for x1, x2 being Element of G1 holds <*x1*> * <*x2*> = <*(x1 * x2)*>

proof end;

theorem :: GROUP_7:29

for G1, G2 being non empty multMagma
for x1, x2 being Element of G1
for y1, y2 being Element of G2 holds <*x1,y1*> * <*x2,y2*> = <*(x1 * x2),(y1 * y2)*>

proof end;

theorem :: GROUP_7:30

for G1, G2, G3 being non empty multMagma
for x1, x2 being Element of G1
for y1, y2 being Element of G2
for z1, z2 being Element of G3 holds <*x1,y1,z1*> * <*x2,y2,z2*> = <*(x1 * x2),(y1 * y2),(z1 * z2)*>

proof end;

theorem :: GROUP_7:31

for G1 being non empty Group-like multMagma holds 1_ (product <*G1*>) = <*(1_ G1)*>

proof end;

theorem :: GROUP_7:32

for G1, G2 being non empty Group-like multMagma holds 1_ (product <*G1,G2*>) = <*(1_ G1),(1_ G2)*>

proof end;

theorem :: GROUP_7:33

for G1, G2, G3 being non empty Group-like multMagma holds 1_ (product <*G1,G2,G3*>) = <*(1_ G1),(1_ G2),(1_ G3)*>

proof end;

theorem :: GROUP_7:34

for G1 being Group
for x being Element of G1 holds <*x*> " = <*(x ")*>

proof end;

theorem :: GROUP_7:35

for G1, G2 being Group
for x being Element of G1
for y being Element of G2 holds <*x,y*> " = <*(x "),(y ")*>

proof end;

theorem :: GROUP_7:36

for G1, G2, G3 being Group
for x being Element of G1
for y being Element of G2
for z being Element of G3 holds <*x,y,z*> " = <*(x "),(y "),(z ")*>

proof end;

theorem Th37: :: GROUP_7:37

for G1 being Group
for f being Function of the carrier of G1, the carrier of (product <*G1*>) st ( for x being Element of G1 holds f . x = <*x*> ) holds
f is Homomorphism of G1,(product <*G1*>)

proof end;

theorem Th38: :: GROUP_7:38

for G1 being Group
for f being Homomorphism of G1,(product <*G1*>) st ( for x being Element of G1 holds f . x = <*x*> ) holds
f is bijective

proof end;

theorem :: GROUP_7:39

for G1 being Group holds G1, product <*G1*> are_isomorphic

proof end;