Theorem idghm 13628

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 2205	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
4	2, 3	grpcl 13373	. . . . . . 7 |- ( ( G e. Grp /\ a e. B /\ b e. B ) -> ( a ( +g ` G ) b ) e. B )
5	4	3expb 1207	. . . . . 6 |- ( ( G e. Grp /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` G ) b ) e. B )
6		fvresi 5779	. . . . . 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 5779	. . . . . . 7 |- ( a e. B -> ( ( _I |` B ) ` a ) = a )
9		fvresi 5779	. . . . . . 7 |- ( b e. B -> ( ( _I |` B ) ` b ) = b )
10	8, 9	oveqan12d 5965	. . . . . 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 2241	. . . 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 2588	. . 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 5562	. . . 4 |- ( _I |` B ) : B -1-1-onto-> B
15		f1of 5524	. . . 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 13612	. 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 1185	1 |- ( G e. Grp -> ( _I |` B ) e. ( G GrpHom G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   A.wral 2484   _I cid 4336   |` cres 4678   -->wf 5268   -1-1-onto->wf1o 5271   ` cfv 5272  (class class class)co 5946   Basecbs 12865   +g cplusg 12942   Grpcgrp 13365   GrpHom cghm 13609

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1re 8021  ax-addrcl 8024

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-inn 9039  df-2 9097  df-ndx 12868  df-slot 12869  df-base 12871  df-plusg 12955  df-mgm 13221  df-sgrp 13267  df-mnd 13282  df-grp 13368  df-ghm 13610

This theorem is referenced by: (None)