:: Properties of Fields
:: by J\'ozef Bia{\l}as
::
:: Received June 20, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

func add_2 -> BinOp of 2 equals :: REALSET2:def 1
((((0,0) .--> 0) +* ((0,1) .--> 1)) +* ((1,0) .--> 1)) +* ((1,1) .--> 0);

coherence
((((0,0) .--> 0) +* ((0,1) .--> 1)) +* ((1,0) .--> 1)) +* ((1,1) .--> 0) is BinOp of 2

proof end;

func mult_2 -> BinOp of 2 equals :: REALSET2:def 2
((((0,0) .--> 0) +* ((0,1) .--> 0)) +* ((1,0) .--> 0)) +* ((1,1) .--> 1);

coherence
((((0,0) .--> 0) +* ((0,1) .--> 0)) +* ((1,0) .--> 0)) +* ((1,1) .--> 1) is BinOp of 2

proof end;

end;

:: deftheorem defines add_2 REALSET2:def 1 :
add_2 = ((((0,0) .--> 0) +* ((0,1) .--> 1)) +* ((1,0) .--> 1)) +* ((1,1) .--> 0);

:: deftheorem defines mult_2 REALSET2:def 2 :
mult_2 = ((((0,0) .--> 0) +* ((0,1) .--> 0)) +* ((1,0) .--> 0)) +* ((1,1) .--> 1);

set x = In (0,2);

set y = In (1,2);

Lm1: 1 in 2
by CARD_1:50, TARSKI:def 2;

Lm2: 0 in 2
by CARD_1:50, TARSKI:def 2;

then Lm3: In (0,2) = 0
by SUBSET_1:def 8;

Lm4: In (1,2) = 1
by Lm1, SUBSET_1:def 8;

set Z = 2;

reconsider A = 2 as non trivial set by Lm1, Lm2, ZFMISC_1:def 10;

reconsider nd = In (0,2) as Element of A ;

Lm5: dom ((1,1) .--> 0) = {[1,1]}
by FUNCOP_1:13;

Lm6: dom ((1,0) .--> 1) = {[1,0]}
by FUNCOP_1:13;

Lm7: dom ((0,1) .--> 1) = {[0,1]}
by FUNCOP_1:13;

Lm8: add_2 . ((In (0,2)),(In (0,2))) = In (0,2)

proof end;

Lm9: add_2 . ((In (0,2)),(In (1,2))) = In (1,2)

proof end;

Lm10: add_2 . ((In (1,2)),(In (0,2))) = In (1,2)

proof end;

Lm11: add_2 . ((In (1,2)),(In (1,2))) = In (0,2)

proof end;

Lm12: dom ((1,1) .--> 1) = {[1,1]}
by FUNCOP_1:13;

Lm13: dom ((1,0) .--> 0) = {[1,0]}
by FUNCOP_1:13;

Lm14: dom ((0,1) .--> 0) = {[0,1]}
by FUNCOP_1:13;

Lm15: mult_2 . ((In (0,2)),(In (0,2))) = In (0,2)

proof end;

Lm16: mult_2 . ((In (0,2)),(In (1,2))) = In (0,2)

proof end;

Lm17: mult_2 . ((In (1,2)),(In (0,2))) = In (0,2)

proof end;

Lm18: mult_2 . ((In (1,2)),(In (1,2))) = In (1,2)

proof end;

set om = mult_2 ;

Lm19: A \ {(In (0,2))} = {(In (1,2))}
by Lm3, Lm4, CARD_1:50, ZFMISC_1:17;

then Lm20: [:(A \ {(In (0,2))}),(A \ {(In (0,2))}):] = {[(In (1,2)),(In (1,2))]}
by ZFMISC_1:29;

Lm21: for t being set st t in [:(A \ {(In (0,2))}),(A \ {(In (0,2))}):] holds
mult_2 . t in A \ {(In (0,2))}

proof end;

reconsider nm = In (1,2) as Element of A \ {nd} by Lm19, TARSKI:def 1;

reconsider od0 = add_2 as BinOp of A ;

reconsider om0 = mult_2 as BinOp of A ;

Lm22: for a, b, d being Element of A holds add_2 . ((add_2 . (a,b)),d) = add_2 . (a,(add_2 . (b,d)))

proof end;

reconsider dL = doubleLoopStr(# A,od0,om0,nm,nd #) as non empty doubleLoopStr ;

Lm23: for a being Element of dL holds a + (0. dL) = a

proof end;

Lm24: for a, b, c being Element of dL holds (a + b) + c = a + (b + c)
by Lm22;

Lm25: for a, b being Element of dL holds a + b = b + a

proof end;

reconsider om1 = mult_2 as A \ {nd} -subsetpreserving BinOp of A by Lm21, REALSET1:def 4;

Lm26: for B being non empty set
for P being BinOp of B
for e being Element of B st B = A \ {nd} & e = nm holds
addLoopStr(# B,P,e #) is AbGroup

proof end;

Lm27: for a, b, d being Element of dL holds
( a * (b + d) = (a * b) + (a * d) & (b + d) * a = (b * a) + (d * a) )

proof end;

definition

func dL-Z_2 -> doubleLoopStr equals :: REALSET2:def 3
doubleLoopStr(# 2,add_2,mult_2,(In (1,2)),(In (0,2)) #);

coherence
doubleLoopStr(# 2,add_2,mult_2,(In (1,2)),(In (0,2)) #) is doubleLoopStr ;

end;

:: deftheorem defines dL-Z_2 REALSET2:def 3 :
dL-Z_2 = doubleLoopStr(# 2,add_2,mult_2,(In (1,2)),(In (0,2)) #);

registration

cluster dL-Z_2 -> non empty non degenerated strict ;

coherence
( dL-Z_2 is strict & not dL-Z_2 is empty & not dL-Z_2 is degenerated )

proof end;

end;

registration

cluster dL-Z_2 -> add-associative distributive ;

coherence
( dL-Z_2 is add-associative & dL-Z_2 is distributive )

proof end;

end;

registration

cluster non empty non trivial strict add-associative for doubleLoopStr ;

existence
ex b₁ being non trivial strict doubleLoopStr st b₁ is add-associative

proof end;

end;

registration

cluster the carrier of dL-Z_2 -> natural-membered ;

coherence
the carrier of dL-Z_2 is natural-membered by CARD_1:50, TARSKI:def 2;

end;

registration

cluster empty for Element of the carrier of dL-Z_2;

existence
ex b₁ being Element of dL-Z_2 st b₁ is empty

proof end;

cluster non empty for Element of the carrier of dL-Z_2;

existence
not for b₁ being Element of dL-Z_2 holds b₁ is empty

proof end;

end;

definition

let L be doubleLoopStr ;

attr L is Field-like means :Def4: :: REALSET2:def 4
( the multF of L is the carrier of L \ {(0. L)} -subsetpreserving & ( for B being non empty set
for P being BinOp of B
for e being Element of B st B = NonZero L & e = 1. L & P = the multF of L || (NonZero L) holds
addLoopStr(# B,P,e #) is AbGroup ) );

end;

:: deftheorem Def4 defines Field-like REALSET2:def 4 :
for L being doubleLoopStr holds
( L is Field-like iff ( the multF of L is the carrier of L \ {(0. L)} -subsetpreserving & ( for B being non empty set
for P being BinOp of B
for e being Element of B st B = NonZero L & e = 1. L & P = the multF of L || (NonZero L) holds
addLoopStr(# B,P,e #) is AbGroup ) ) );

registration

let A be non empty set ;
let od be BinOp of A;
let nd be Element of A;
let om be BinOp of A;
let nm be Element of A;

cluster doubleLoopStr(# A,od,om,nm,nd #) -> non empty ;

coherence
not doubleLoopStr(# A,od,om,nm,nd #) is empty ;

end;

Lm28: for B being non empty set
for P being BinOp of B
for e being Element of B st B = A \ {nd} & e = nm & P = om1 ! (A,nd) holds
addLoopStr(# B,P,e #) is AbGroup
by Lm26;

registration

cluster non empty non degenerated non trivial right_complementable strict Abelian add-associative right_zeroed distributive Field-like for doubleLoopStr ;

existence
ex b₁ being non degenerated doubleLoopStr st
( b₁ is strict & b₁ is Field-like & b₁ is Abelian & b₁ is distributive & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable )

proof end;

end;

registration

cluster non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed well-unital distributive associative commutative -> non degenerated Field-like for doubleLoopStr ;

coherence
for b₁ being non degenerated doubleLoopStr st b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is Abelian & b₁ is commutative & b₁ is associative & b₁ is well-unital & b₁ is distributive & b₁ is almost_left_invertible holds
b₁ is Field-like

proof end;

end;

deffunc H₁( non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr ) -> set = the carrier of $1;

definition

let F be non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr ;

func omf F -> the carrier of F \ {(0. F)} -subsetpreserving BinOp of F equals :: REALSET2:def 5
the multF of F;

coherence
the multF of F is the carrier of F \ {(0. F)} -subsetpreserving BinOp of F by Def4;

end;

:: deftheorem defines omf REALSET2:def 5 :
for F being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr holds omf F = the multF of F;

theorem :: REALSET2:1

for F being Field holds addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #) is AbGroup

proof end;

theorem :: REALSET2:2

for F being Field
for a being Element of F holds
( a + (0. F) = a & (0. F) + a = a ) ;

theorem Th3: :: REALSET2:3

for F being AbGroup
for a being Element of F ex b being Element of F st
( a + b = 0. F & b + a = 0. F )

proof end;

theorem Th4: :: REALSET2:4

for F being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr
for a, b, c being Element of NonZero F holds (a * b) * c = a * (b * c)

proof end;

theorem Th5: :: REALSET2:5

for F being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr
for a, b being Element of NonZero F holds a * b = b * a

proof end;

theorem Th6: :: REALSET2:6

for F being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr
for a being Element of NonZero F holds
( a * (1. F) = a & (1. F) * a = a )

proof end;

theorem Th7: :: REALSET2:7

for F being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr
for a being Element of NonZero F ex b being Element of NonZero F st
( a * b = 1. F & b * a = 1. F )

proof end;

theorem :: REALSET2:8

for F being Field
for x, y being Element of F st x + y = 0. F holds
y = (comp F) . x

proof end;

theorem :: REALSET2:9

for F being Field
for x being Element of F holds x = (comp F) . ((comp F) . x)

proof end;

theorem :: REALSET2:10

for F being Field
for a, b being Element of the carrier of F holds
( the addF of F . (a,b) is Element of the carrier of F & (omf F) . (a,b) is Element of the carrier of F & (comp F) . a is Element of the carrier of F ) ;

theorem :: REALSET2:11

for F being Field
for a, b, c being Element of F holds a * (b - c) = (a * b) - (a * c) by VECTSP_1:11;

theorem :: REALSET2:12

for F being Field
for a, b, c being Element of F holds (a - b) * c = (a * c) - (b * c) by VECTSP_1:13;

theorem :: REALSET2:13

for F being Field
for a being Element of F holds a * (0. F) = 0. F ;

theorem :: REALSET2:14

for F being Field
for a being Element of F holds (0. F) * a = 0. F ;

theorem :: REALSET2:15

for F being Field
for a, b being Element of F holds - (a * b) = a * (- b) by VECTSP_1:8;

theorem :: REALSET2:16

for F being Field holds (1. F) * (0. F) = 0. F ;

theorem :: REALSET2:17

for F being Field holds (0. F) * (1. F) = 0. F ;

theorem :: REALSET2:18

for F being Field
for a, b being Element of the carrier of F holds (omf F) . (a,b) is Element of the carrier of F ;

theorem Th19: :: REALSET2:19

for F being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr
for a, b, c being Element of F holds (a * b) * c = a * (b * c)

proof end;

theorem Th20: :: REALSET2:20

for F being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr
for a, b being Element of F holds a * b = b * a

proof end;

theorem Th21: :: REALSET2:21

for F being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr
for a being Element of F holds
( a * (1. F) = a & (1. F) * a = a )

proof end;

definition

let F be Field;

func revf F -> UnOp of (NonZero F) means :Def6: :: REALSET2:def 6
for x being Element of NonZero F holds (omf F) . (x,(it . x)) = 1. F;

existence
ex b₁ being UnOp of (NonZero F) st
for x being Element of NonZero F holds (omf F) . (x,(b₁ . x)) = 1. F

proof end;

uniqueness
for b₁, b₂ being UnOp of (NonZero F) st ( for x being Element of NonZero F holds (omf F) . (x,(b₁ . x)) = 1. F ) & ( for x being Element of NonZero F holds (omf F) . (x,(b₂ . x)) = 1. F ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines revf REALSET2:def 6 :
for F being Field
for b₂ being UnOp of (NonZero F) holds
( b₂ = revf F iff for x being Element of NonZero F holds (omf F) . (x,(b₂ . x)) = 1. F );

theorem :: REALSET2:22

for F being Field
for x, y being Element of NonZero F st x * y = 1. F holds
y = (revf F) . x

proof end;

theorem :: REALSET2:23

for F being Field
for x being Element of NonZero F holds x = (revf F) . ((revf F) . x)

proof end;

theorem :: REALSET2:24

for F being Field
for a, b being Element of NonZero F holds
( (omf F) . (a,b) is Element of NonZero F & (revf F) . a is Element of NonZero F )

proof end;

theorem :: REALSET2:25

for F being Field
for a, b, c being Element of F st a + b = a + c holds
b = c by RLVECT_1:8;

theorem :: REALSET2:26

for F being Field
for a being Element of NonZero F
for b, c being Element of F st a * b = a * c holds
b = c

proof end;

registration

cluster non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like -> non degenerated almost_left_invertible well-unital associative commutative for doubleLoopStr ;

coherence
for b₁ being non degenerated doubleLoopStr st b₁ is Field-like & b₁ is Abelian & b₁ is distributive & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable holds
( b₁ is commutative & b₁ is associative & b₁ is well-unital & b₁ is almost_left_invertible )

proof end;

end;