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Theorem idghm 16077
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 2467 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
53, 4grpcl 15864 . . . . . . 7  |-  ( ( G  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  G ) b )  e.  B )
653expb 1197 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  G
) b )  e.  B )
7 fvresi 6085 . . . . . 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 6085 . . . . . . 7  |-  ( a  e.  B  ->  (
(  _I  |`  B ) `
 a )  =  a )
10 fvresi 6085 . . . . . . 7  |-  ( b  e.  B  ->  (
(  _I  |`  B ) `
 b )  =  b )
119, 10oveqan12d 6301 . . . . . 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 2511 . . . 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 2885 . . 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 5849 . . . 4  |-  (  _I  |`  B ) : B -1-1-onto-> B
16 f1of 5814 . . . 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 16062 . 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 1379    e. wcel 1767   A.wral 2814    _I cid 4790    |` cres 5001   -->wf 5582   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   Grpcgrp 15723    GrpHom cghm 16059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-mnd 15728  df-grp 15858  df-ghm 16060
This theorem is referenced by:  gicref  16114  symgga  16226  0frgp  16593  idlmhm  17470  frgpcyg  18379  nmoid  20984  idnghm  20985
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