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Theorem idghm 14698
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 19 . . 3  |-  ( G  e.  Grp  ->  G  e.  Grp )
21ancli 534 . 2  |-  ( G  e.  Grp  ->  ( G  e.  Grp  /\  G  e.  Grp ) )
3 idghm.b . . . . . . . 8  |-  B  =  ( Base `  G
)
4 eqid 2283 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
53, 4grpcl 14495 . . . . . . 7  |-  ( ( G  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  G ) b )  e.  B )
653expb 1152 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  G
) b )  e.  B )
7 fvresi 5711 . . . . . 6  |-  ( ( a ( +g  `  G
) b )  e.  B  ->  ( (  _I  |`  B ) `  ( a ( +g  `  G ) b ) )  =  ( a ( +g  `  G
) b ) )
86, 7syl 15 . . . . 5  |-  ( ( G  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
(  _I  |`  B ) `
 ( a ( +g  `  G ) b ) )  =  ( a ( +g  `  G ) b ) )
9 fvresi 5711 . . . . . . 7  |-  ( a  e.  B  ->  (
(  _I  |`  B ) `
 a )  =  a )
10 fvresi 5711 . . . . . . 7  |-  ( b  e.  B  ->  (
(  _I  |`  B ) `
 b )  =  b )
119, 10oveqan12d 5877 . . . . . 6  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ( (  _I  |`  B ) `  a
) ( +g  `  G
) ( (  _I  |`  B ) `  b
) )  =  ( a ( +g  `  G
) b ) )
1211adantl 452 . . . . 5  |-  ( ( G  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( (  _I  |`  B ) `
 a ) ( +g  `  G ) ( (  _I  |`  B ) `
 b ) )  =  ( a ( +g  `  G ) b ) )
138, 12eqtr4d 2318 . . . 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 2635 . . 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 5511 . . . 4  |-  (  _I  |`  B ) : B -1-1-onto-> B
16 f1of 5472 . . . 4  |-  ( (  _I  |`  B ) : B -1-1-onto-> B  ->  (  _I  |`  B ) : B --> B )
1715, 16ax-mp 8 . . 3  |-  (  _I  |`  B ) : B --> B
1814, 17jctil 523 . 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 14683 . 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 645 1  |-  ( G  e.  Grp  ->  (  _I  |`  B )  e.  ( G  GrpHom  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    _I cid 4304    |` cres 4691   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362    GrpHom cghm 14680
This theorem is referenced by:  gicref  14735  symgga  14786  0frgp  15088  idlmhm  15798  frgpcyg  16527  nmoid  18251  idnghm  18252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-mnd 14367  df-grp 14489  df-ghm 14681
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