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Theorem idghm 15011
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 20 . . 3  |-  ( G  e.  Grp  ->  G  e.  Grp )
21ancli 535 . 2  |-  ( G  e.  Grp  ->  ( G  e.  Grp  /\  G  e.  Grp ) )
3 idghm.b . . . . . . . 8  |-  B  =  ( Base `  G
)
4 eqid 2435 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
53, 4grpcl 14808 . . . . . . 7  |-  ( ( G  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  G ) b )  e.  B )
653expb 1154 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  G
) b )  e.  B )
7 fvresi 5916 . . . . . 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 5916 . . . . . . 7  |-  ( a  e.  B  ->  (
(  _I  |`  B ) `
 a )  =  a )
10 fvresi 5916 . . . . . . 7  |-  ( b  e.  B  ->  (
(  _I  |`  B ) `
 b )  =  b )
119, 10oveqan12d 6092 . . . . . 6  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ( (  _I  |`  B ) `  a
) ( +g  `  G
) ( (  _I  |`  B ) `  b
) )  =  ( a ( +g  `  G
) b ) )
1211adantl 453 . . . . 5  |-  ( ( G  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( (  _I  |`  B ) `
 a ) ( +g  `  G ) ( (  _I  |`  B ) `
 b ) )  =  ( a ( +g  `  G ) b ) )
138, 12eqtr4d 2470 . . . 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 2790 . . 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 5705 . . . 4  |-  (  _I  |`  B ) : B -1-1-onto-> B
16 f1of 5666 . . . 4  |-  ( (  _I  |`  B ) : B -1-1-onto-> B  ->  (  _I  |`  B ) : B --> B )
1715, 16ax-mp 8 . . 3  |-  (  _I  |`  B ) : B --> B
1814, 17jctil 524 . 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 14996 . 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 646 1  |-  ( G  e.  Grp  ->  (  _I  |`  B )  e.  ( G  GrpHom  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    _I cid 4485    |` cres 4872   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   Basecbs 13459   +g cplusg 13519   Grpcgrp 14675    GrpHom cghm 14993
This theorem is referenced by:  gicref  15048  symgga  15099  0frgp  15401  idlmhm  16107  frgpcyg  16844  nmoid  18766  idnghm  18767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-mnd 14680  df-grp 14802  df-ghm 14994
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