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Theorem idghm 15873
Description: The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Hypothesis
Ref Expression
idghm.b  |-  B  =  ( Base `  G
)
Assertion
Ref Expression
idghm  |-  ( G  e.  Grp  ->  (  _I  |`  B )  e.  ( G  GrpHom  G ) )

Proof of Theorem idghm
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3  |-  ( G  e.  Grp  ->  G  e.  Grp )
21ancli 551 . 2  |-  ( G  e.  Grp  ->  ( G  e.  Grp  /\  G  e.  Grp ) )
3 idghm.b . . . . . . . 8  |-  B  =  ( Base `  G
)
4 eqid 2451 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
53, 4grpcl 15662 . . . . . . 7  |-  ( ( G  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  G ) b )  e.  B )
653expb 1189 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  G
) b )  e.  B )
7 fvresi 6006 . . . . . 6  |-  ( ( a ( +g  `  G
) b )  e.  B  ->  ( (  _I  |`  B ) `  ( a ( +g  `  G ) b ) )  =  ( a ( +g  `  G
) b ) )
86, 7syl 16 . . . . 5  |-  ( ( G  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
(  _I  |`  B ) `
 ( a ( +g  `  G ) b ) )  =  ( a ( +g  `  G ) b ) )
9 fvresi 6006 . . . . . . 7  |-  ( a  e.  B  ->  (
(  _I  |`  B ) `
 a )  =  a )
10 fvresi 6006 . . . . . . 7  |-  ( b  e.  B  ->  (
(  _I  |`  B ) `
 b )  =  b )
119, 10oveqan12d 6212 . . . . . 6  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ( (  _I  |`  B ) `  a
) ( +g  `  G
) ( (  _I  |`  B ) `  b
) )  =  ( a ( +g  `  G
) b ) )
1211adantl 466 . . . . 5  |-  ( ( G  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( (  _I  |`  B ) `
 a ) ( +g  `  G ) ( (  _I  |`  B ) `
 b ) )  =  ( a ( +g  `  G ) b ) )
138, 12eqtr4d 2495 . . . 4  |-  ( ( G  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
(  _I  |`  B ) `
 ( a ( +g  `  G ) b ) )  =  ( ( (  _I  |`  B ) `  a
) ( +g  `  G
) ( (  _I  |`  B ) `  b
) ) )
1413ralrimivva 2907 . . 3  |-  ( G  e.  Grp  ->  A. a  e.  B  A. b  e.  B  ( (  _I  |`  B ) `  ( a ( +g  `  G ) b ) )  =  ( ( (  _I  |`  B ) `
 a ) ( +g  `  G ) ( (  _I  |`  B ) `
 b ) ) )
15 f1oi 5777 . . . 4  |-  (  _I  |`  B ) : B -1-1-onto-> B
16 f1of 5742 . . . 4  |-  ( (  _I  |`  B ) : B -1-1-onto-> B  ->  (  _I  |`  B ) : B --> B )
1715, 16ax-mp 5 . . 3  |-  (  _I  |`  B ) : B --> B
1814, 17jctil 537 . 2  |-  ( G  e.  Grp  ->  (
(  _I  |`  B ) : B --> B  /\  A. a  e.  B  A. b  e.  B  (
(  _I  |`  B ) `
 ( a ( +g  `  G ) b ) )  =  ( ( (  _I  |`  B ) `  a
) ( +g  `  G
) ( (  _I  |`  B ) `  b
) ) ) )
193, 3, 4, 4isghm 15858 . 2  |-  ( (  _I  |`  B )  e.  ( G  GrpHom  G )  <-> 
( ( G  e. 
Grp  /\  G  e.  Grp )  /\  (
(  _I  |`  B ) : B --> B  /\  A. a  e.  B  A. b  e.  B  (
(  _I  |`  B ) `
 ( a ( +g  `  G ) b ) )  =  ( ( (  _I  |`  B ) `  a
) ( +g  `  G
) ( (  _I  |`  B ) `  b
) ) ) ) )
202, 18, 19sylanbrc 664 1  |-  ( G  e.  Grp  ->  (  _I  |`  B )  e.  ( G  GrpHom  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795    _I cid 4732    |` cres 4943   -->wf 5515   -1-1-onto->wf1o 5518   ` cfv 5519  (class class class)co 6193   Basecbs 14285   +g cplusg 14349   Grpcgrp 15521    GrpHom cghm 15855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-mnd 15526  df-grp 15656  df-ghm 15856
This theorem is referenced by:  gicref  15910  symgga  16022  0frgp  16389  idlmhm  17237  frgpcyg  18124  nmoid  20446  idnghm  20447
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