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.

Step	Hyp	Ref	Expression
1		id 22	. . 3 |- ( G e. Grp -> G e. Grp )
2	1	ancli 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 )
5	3, 4	grpcl 15864	. . . . . . 7 |- ( ( G e. Grp /\ a e. B /\ b e. B ) -> ( a ( +g ` G ) b ) e. B )
6	5	3expb 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 ) )
8	6, 7	syl 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 )
11	9, 10	oveqan12d 6301	. . . . . 6 |- ( ( a e. B /\ b e. B ) -> ( ( ( _I |` B ) ` a ) ( +g ` G ) ( ( _I |` B ) ` b ) ) = ( a ( +g ` G ) b ) )
12	11	adantl 466	. . . . 5 |- ( ( G e. Grp /\ ( a e. B /\ b e. B ) ) -> ( ( ( _I |` B ) ` a ) ( +g ` G ) ( ( _I |` B ) ` b ) ) = ( a ( +g ` G ) b ) )
13	8, 12	eqtr4d 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 ) ) )
14	13	ralrimivva 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 )
17	15, 16	ax-mp 5	. . 3 |- ( _I |` B ) : B --> B
18	14, 17	jctil 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 ) ) ) )
19	3, 3, 4, 4	isghm 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 ) ) ) ) )
20	2, 18, 19	sylanbrc 664	1 |- ( G e. Grp -> ( _I |` B ) e. ( G GrpHom G ) )

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