Theorem idghm 15873

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 2451	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
5	3, 4	grpcl 15662	. . . . . . 7 |- ( ( G e. Grp /\ a e. B /\ b e. B ) -> ( a ( +g ` G ) b ) e. B )
6	5	3expb 1189	. . . . . 6 |- ( ( G e. Grp /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` G ) b ) e. B )
7		fvresi 6006	. . . . . 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 6006	. . . . . . 7 |- ( a e. B -> ( ( _I |` B ) ` a ) = a )
10		fvresi 6006	. . . . . . 7 |- ( b e. B -> ( ( _I |` B ) ` b ) = b )
11	9, 10	oveqan12d 6212	. . . . . 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 2495	. . . 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 2907	. . 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 5777	. . . 4 |- ( _I |` B ) : B -1-1-onto-> B
16		f1of 5742	. . . 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 15858	. 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 1370   e. wcel 1758   A.wral 2795   _I cid 4732   |` cres 4943   -->wf 5515   -1-1-onto->wf1o 5518   ` cfv 5519  (class class class)co 6193   Basecbs 14285   +g cplusg 14349   Grpcgrp 15521   GrpHom cghm 15855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-mnd 15526  df-grp 15656  df-ghm 15856

This theorem is referenced by:  gicref  15910  symgga  16022  0frgp  16389  idlmhm  17237  frgpcyg  18124  nmoid  20446  idnghm  20447