Theorem idghm 14012

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 19	. 2 |- ( G e. Grp -> G e. Grp )
2		idghm.b	. . . . . . . 8 |- B = ( Base ` G )
3		eqid 2234	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
4	2, 3	grpcl 13763	. . . . . . 7 |- ( ( G e. Grp /\ a e. B /\ b e. B ) -> ( a ( +g ` G ) b ) e. B )
5	4	3expb 1231	. . . . . 6 |- ( ( G e. Grp /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` G ) b ) e. B )
6		fvresi 5882	. . . . . 6 |- ( ( a ( +g ` G ) b ) e. B -> ( ( _I |` B ) ` ( a ( +g ` G ) b ) ) = ( a ( +g ` G ) b ) )
7	5, 6	syl 14	. . . . 5 |- ( ( G e. Grp /\ ( a e. B /\ b e. B ) ) -> ( ( _I |` B ) ` ( a ( +g ` G ) b ) ) = ( a ( +g ` G ) b ) )
8		fvresi 5882	. . . . . . 7 |- ( a e. B -> ( ( _I |` B ) ` a ) = a )
9		fvresi 5882	. . . . . . 7 |- ( b e. B -> ( ( _I |` B ) ` b ) = b )
10	8, 9	oveqan12d 6077	. . . . . 6 |- ( ( a e. B /\ b e. B ) -> ( ( ( _I |` B ) ` a ) ( +g ` G ) ( ( _I |` B ) ` b ) ) = ( a ( +g ` G ) b ) )
11	10	adantl 277	. . . . 5 |- ( ( G e. Grp /\ ( a e. B /\ b e. B ) ) -> ( ( ( _I |` B ) ` a ) ( +g ` G ) ( ( _I |` B ) ` b ) ) = ( a ( +g ` G ) b ) )
12	7, 11	eqtr4d 2270	. . . 4 |- ( ( G e. Grp /\ ( a e. B /\ b e. B ) ) -> ( ( _I |` B ) ` ( a ( +g ` G ) b ) ) = ( ( ( _I |` B ) ` a ) ( +g ` G ) ( ( _I |` B ) ` b ) ) )
13	12	ralrimivva 2626	. . 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 ) ) )
14		f1oi 5659	. . . 4 |- ( _I |` B ) : B -1-1-onto-> B
15		f1of 5619	. . . 4 |- ( ( _I |` B ) : B -1-1-onto-> B -> ( _I |` B ) : B --> B )
16	14, 15	ax-mp 5	. . 3 |- ( _I |` B ) : B --> B
17	13, 16	jctil 312	. 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 ) ) ) )
18	2, 2, 3, 3	isghm 13996	. 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 ) ) ) ) )
19	1, 1, 17, 18	syl21anbrc 1209	1 |- ( G e. Grp -> ( _I |` B ) e. ( G GrpHom G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   _I cid 4414   |` cres 4756   -->wf 5353   -1-1-onto->wf1o 5356   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   Grpcgrp 13755   GrpHom cghm 13993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-ghm 13994

This theorem is referenced by: (None)