Theorem idghm 13329

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 2193	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
4	2, 3	grpcl 13080	. . . . . . 7 |- ( ( G e. Grp /\ a e. B /\ b e. B ) -> ( a ( +g ` G ) b ) e. B )
5	4	3expb 1206	. . . . . 6 |- ( ( G e. Grp /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` G ) b ) e. B )
6		fvresi 5751	. . . . . 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 5751	. . . . . . 7 |- ( a e. B -> ( ( _I |` B ) ` a ) = a )
9		fvresi 5751	. . . . . . 7 |- ( b e. B -> ( ( _I |` B ) ` b ) = b )
10	8, 9	oveqan12d 5937	. . . . . 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 2229	. . . 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 2576	. . 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 5538	. . . 4 |- ( _I |` B ) : B -1-1-onto-> B
15		f1of 5500	. . . 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 13313	. 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 1184	1 |- ( G e. Grp -> ( _I |` B ) e. ( G GrpHom G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   _I cid 4319   |` cres 4661   -->wf 5250   -1-1-onto->wf1o 5253   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   Grpcgrp 13072   GrpHom cghm 13310

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-ghm 13311

This theorem is referenced by: (None)