:: GROUP_7 semantic presentation

theorem Th1: :: GROUP_7:1

for a, b being set st <*a*> = <*b*> holds
a = b

proof end;

theorem :: GROUP_7:2

for a, b, c, d being set st <*a,b*> = <*c,d*> holds
( a = c & b = d )

proof end;

theorem :: GROUP_7:3

for a, b, c, d, e, f being set st <*a,b,c*> = <*d,e,f*> holds
( a = d & b = e & c = f )

proof end;

definition

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

end;

:: deftheorem Def1 defines HGrStr-yielding GROUP_7:def 1 :

registration

cluster HGrStr-yielding -> 1-sorted-yielding set ;
coherence

proof end;

end;

registration

let I be set ;
cluster 1-sorted-yielding HGrStr-yielding ManySortedSet of I;
existence

proof end;

end;

registration

cluster 1-sorted-yielding HGrStr-yielding set ;
existence

proof end;

end;

definition

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

end;

definition

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

proof end;

end;

registration

let I be set ;
let F be HGrStr-Family of I;
cluster Carrier F -> V2 ;
coherence

proof end;

end;

Lm1: now

let I be non empty set ; :: thesis:
let i be Element of I; :: thesis:
let F be HGrStr-yielding ManySortedSet of I; :: thesis:
let f be Element of product (Carrier F); :: thesis:
A1: ex R being 1-sorted st
( R = F . i & (Carrier F) . i = the carrier of R ) by PRALG_1:def 13;
dom (Carrier F) = I by PBOOLE:def 3;
hence f . i in the carrier of (F . i) by A1, CARD_3:18; :: thesis:

end;

definition

let I be set ;
let F be HGrStr-Family of I;
func product F -> strict HGrStr 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 HGrStr ex h being Function st
( Fi = F . i & h = the mult of it . f,g & h . i = the mult of Fi . (f . i),(g . i) ) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines product GROUP_7:def 2 :

registration

let I be set ;
let F be HGrStr-Family of I;
cluster product F -> non empty strict constituted-Functions ;
coherence

proof end;

end;

theorem Th4: :: GROUP_7:4

for I, i being set
for f, g, h being Function
for F being HGrStr-Family of I
for G being non empty HGrStr
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 HGrStr-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 HGrStr 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 HGrStr 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 HGrStr st Fi = F . i;

end;

:: deftheorem Def3 defines Group-like GROUP_7:def 3 :

:: deftheorem Def4 defines associative GROUP_7:def 4 :

:: deftheorem Def5 defines commutative GROUP_7:def 5 :

definition

let I be non empty set ;
let F be HGrStr-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

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

proof end;

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

proof end;

end;

:: deftheorem Def6 defines Group-like GROUP_7:def 6 :

:: deftheorem Def7 defines associative GROUP_7:def 7 :

:: deftheorem defines commutative GROUP_7:def 8 :

registration

let I be set ;
cluster Group-like associative commutative ManySortedSet of I;
existence

proof end;

end;

registration

let I be set ;
let F be Group-like HGrStr-Family of I;
cluster product F -> non empty strict Group-like constituted-Functions ;
coherence

proof end;

end;

registration

let I be set ;
let F be associative HGrStr-Family of I;
cluster product F -> non empty strict associative constituted-Functions ;
coherence

proof end;

end;

registration

let I be set ;
let F be commutative HGrStr-Family of I;
cluster product F -> non empty strict commutative constituted-Functions ;
coherence

proof end;

end;

theorem :: GROUP_7:5

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

proof end;

theorem :: GROUP_7:6

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

proof end;

theorem :: GROUP_7:7

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

proof end;

theorem Th8: :: GROUP_7:8

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

proof end;

theorem Th9: :: GROUP_7:9

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

proof end;

theorem Th10: :: GROUP_7:10

for I being set
for g being Function
for s being ManySortedSet of I
for F being Group-like associative HGrStr-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 Th11: :: GROUP_7:11

for I, i being set
for f, g being Function
for F being Group-like associative HGrStr-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 HGrStr-Family of I;
func sum F -> strict Subgroup of product F means :Def9: :: GROUP_7:def 9
for x being set 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 HGrStr st
( G = F . j & f . j in the carrier of G & f . j <> 1. G ) ) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def9 defines sum GROUP_7:def 9 :

registration

let I be set ;
let F be Group-like associative HGrStr-Family of I;
let f, g be Element of (sum F);
cluster the mult of (sum F) . f,g -> Relation-like Function-like ;
coherence

proof end;

end;

theorem :: GROUP_7:12

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

proof end;

theorem Th13: :: GROUP_7:13

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

proof end;

definition

let G1 be non empty HGrStr ;
:: original: <*
redefine func <*G1*> -> HGrStr-Family of {1};
coherence by Th13;

end;

theorem Th14: :: GROUP_7:14

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

proof end;

definition

let G1 be non empty Group-like HGrStr ;
:: original: <*
redefine func <*G1*> -> Group-like HGrStr-Family of {1};
coherence by Th14;

end;

theorem Th15: :: GROUP_7:15

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

proof end;

definition

let G1 be non empty associative HGrStr ;
:: original: <*
redefine func <*G1*> -> associative HGrStr-Family of {1};
coherence by Th15;

end;

theorem Th16: :: GROUP_7:16

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

proof end;

definition

let G1 be non empty commutative HGrStr ;
:: original: <*
redefine func <*G1*> -> commutative HGrStr-Family of {1};
coherence by Th16;

end;

theorem Th17: :: GROUP_7:17

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

proof end;

definition

let G1 be Group;
:: original: <*
redefine func <*G1*> -> Group-like associative HGrStr-Family of {1};
coherence by Th17;

end;

theorem Th18: :: GROUP_7:18

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

proof end;

definition

let G1 be commutative Group;
:: original: <*
redefine func <*G1*> -> Group-like associative commutative HGrStr-Family of {1};
coherence by Th18;

end;

registration

let G1 be non empty HGrStr ;
cluster -> FinSequence-like Element of product (Carrier <*G1*>);
coherence

proof end;

end;

registration

let G1 be non empty HGrStr ;
cluster -> FinSequence-like Element of the carrier of (product <*G1*>);
coherence

proof end;

end;

definition

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

proof end;

end;

theorem Th19: :: GROUP_7:19

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

proof end;

definition

let G1, G2 be non empty HGrStr ;
:: original: <*
redefine func <*G1,G2*> -> HGrStr-Family of {1,2};
coherence by Th19;

end;

theorem Th20: :: GROUP_7:20

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

proof end;

definition

let G1, G2 be non empty Group-like HGrStr ;
:: original: <*
redefine func <*G1,G2*> -> Group-like HGrStr-Family of {1,2};
coherence by Th20;

end;

theorem Th21: :: GROUP_7:21

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

proof end;

definition

let G1, G2 be non empty associative HGrStr ;
:: original: <*
redefine func <*G1,G2*> -> associative HGrStr-Family of {1,2};
coherence by Th21;

end;

theorem Th22: :: GROUP_7:22

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

proof end;

definition

let G1, G2 be non empty commutative HGrStr ;
:: original: <*
redefine func <*G1,G2*> -> commutative HGrStr-Family of {1,2};
coherence by Th22;

end;

theorem Th23: :: GROUP_7:23

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

proof end;

definition

let G1, G2 be Group;
:: original: <*
redefine func <*G1,G2*> -> Group-like associative HGrStr-Family of {1,2};
coherence by Th23;

end;

theorem Th24: :: GROUP_7:24

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

proof end;

definition

let G1, G2 be commutative Group;
:: original: <*
redefine func <*G1,G2*> -> Group-like associative commutative HGrStr-Family of {1,2};
coherence by Th24;

end;

registration

let G1, G2 be non empty HGrStr ;
cluster -> FinSequence-like Element of product (Carrier <*G1,G2*>);
coherence

proof end;

end;

registration

let G1, G2 be non empty HGrStr ;
cluster -> FinSequence-like Element of the carrier of (product <*G1,G2*>);
coherence

proof end;

end;

definition

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

proof end;

end;

theorem Th25: :: GROUP_7:25

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

proof end;

definition

let G1, G2, G3 be non empty HGrStr ;
:: original: <*
redefine func <*G1,G2,G3*> -> HGrStr-Family of {1,2,3};
coherence by Th25;

end;

theorem Th26: :: GROUP_7:26

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

proof end;

definition

let G1, G2, G3 be non empty Group-like HGrStr ;
:: original: <*
redefine func <*G1,G2,G3*> -> Group-like HGrStr-Family of {1,2,3};
coherence by Th26;

end;

theorem Th27: :: GROUP_7:27

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

proof end;

definition

let G1, G2, G3 be non empty associative HGrStr ;
:: original: <*
redefine func <*G1,G2,G3*> -> associative HGrStr-Family of {1,2,3};
coherence by Th27;

end;

theorem Th28: :: GROUP_7:28

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

proof end;

definition

let G1, G2, G3 be non empty commutative HGrStr ;
:: original: <*
redefine func <*G1,G2,G3*> -> commutative HGrStr-Family of {1,2,3};
coherence by Th28;

end;

theorem Th29: :: GROUP_7:29

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

proof end;

definition

let G1, G2, G3 be Group;
:: original: <*
redefine func <*G1,G2,G3*> -> Group-like associative HGrStr-Family of {1,2,3};
coherence by Th29;

end;

theorem Th30: :: GROUP_7:30

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

proof end;

definition

let G1, G2, G3 be commutative Group;
:: original: <*
redefine func <*G1,G2,G3*> -> Group-like associative commutative HGrStr-Family of {1,2,3};
coherence by Th30;

end;

registration

let G1, G2, G3 be non empty HGrStr ;
cluster -> FinSequence-like Element of product (Carrier <*G1,G2,G3*>);
coherence

proof end;

end;

registration

let G1, G2, G3 be non empty HGrStr ;
cluster -> FinSequence-like Element of the carrier of (product <*G1,G2,G3*>);
coherence

proof end;

end;

definition

let G1, G2, G3 be non empty HGrStr ;
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

proof end;

end;

theorem Th31: :: GROUP_7:31

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

proof end;

theorem :: GROUP_7:32

for G1, G2 being non empty HGrStr
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_7:33

for G1, G2, G3 being non empty HGrStr
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_7:34

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

proof end;

theorem :: GROUP_7:35

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

proof end;

theorem :: GROUP_7:36

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

proof end;

theorem :: GROUP_7:37

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

proof end;

theorem :: GROUP_7:38

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_7:39

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 Th40: :: GROUP_7:40

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 Th41: :: GROUP_7:41

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_isomorphism

proof end;

theorem :: GROUP_7:42

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

proof end;