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Theorem idghm 16608
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 23 . . 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 2404 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
53, 4grpcl 16389 . . . . . . 7  |-  ( ( G  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  G ) b )  e.  B )
653expb 1200 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  G
) b )  e.  B )
7 fvresi 6079 . . . . . 6  |-  ( ( a ( +g  `  G
) b )  e.  B  ->  ( (  _I  |`  B ) `  ( a ( +g  `  G ) b ) )  =  ( a ( +g  `  G
) b ) )
86, 7syl 17 . . . . 5  |-  ( ( G  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
(  _I  |`  B ) `
 ( a ( +g  `  G ) b ) )  =  ( a ( +g  `  G ) b ) )
9 fvresi 6079 . . . . . . 7  |-  ( a  e.  B  ->  (
(  _I  |`  B ) `
 a )  =  a )
10 fvresi 6079 . . . . . . 7  |-  ( b  e.  B  ->  (
(  _I  |`  B ) `
 b )  =  b )
119, 10oveqan12d 6299 . . . . . 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 2448 . . . 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 2827 . . 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 5836 . . . 4  |-  (  _I  |`  B ) : B -1-1-onto-> B
16 f1of 5801 . . . 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 16593 . 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 1407    e. wcel 1844   A.wral 2756    _I cid 4735    |` cres 4827   -->wf 5567   -1-1-onto->wf1o 5570   ` cfv 5571  (class class class)co 6280   Basecbs 14843   +g cplusg 14911   Grpcgrp 16379    GrpHom cghm 16590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-grp 16383  df-ghm 16591
This theorem is referenced by:  gicref  16645  symgga  16757  0frgp  17123  idrhm  17702  idlmhm  18009  frgpcyg  18912  nmoid  21543  idnghm  21544  idrnghm  38238
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