Theorem ghminv 13320

Description: A homomorphism of groups preserves inverses. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Hypotheses

Ref	Expression
ghminv.b	|- B = ( Base ` S )
ghminv.y	|- M = ( invg ` S )
ghminv.z	|- N = ( invg ` T )

Assertion

Ref	Expression
ghminv	|- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( F ` ( M ` X ) ) = ( N ` ( F ` X ) ) )

Proof of Theorem ghminv

Step	Hyp	Ref	Expression
1		ghmgrp1 13315	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2		ghminv.b	. . . . . . 7 |- B = ( Base ` S )
3		eqid 2193	. . . . . . 7 |- ( +g ` S ) = ( +g ` S )
4		eqid 2193	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
5		ghminv.y	. . . . . . 7 |- M = ( invg ` S )
6	2, 3, 4, 5	grprinv 13123	. . . . . 6 |- ( ( S e. Grp /\ X e. B ) -> ( X ( +g ` S ) ( M ` X ) ) = ( 0g ` S ) )
7	1, 6	sylan 283	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( X ( +g ` S ) ( M ` X ) ) = ( 0g ` S ) )
8	7	fveq2d 5558	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( F ` ( X ( +g ` S ) ( M ` X ) ) ) = ( F ` ( 0g ` S ) ) )
9	2, 5	grpinvcl 13120	. . . . . 6 |- ( ( S e. Grp /\ X e. B ) -> ( M ` X ) e. B )
10	1, 9	sylan 283	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( M ` X ) e. B )
11		eqid 2193	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
12	2, 3, 11	ghmlin 13318	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ X e. B /\ ( M ` X ) e. B ) -> ( F ` ( X ( +g ` S ) ( M ` X ) ) ) = ( ( F ` X ) ( +g ` T ) ( F ` ( M ` X ) ) ) )
13	10, 12	mpd3an3 1349	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( F ` ( X ( +g ` S ) ( M ` X ) ) ) = ( ( F ` X ) ( +g ` T ) ( F ` ( M ` X ) ) ) )
14		eqid 2193	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
15	4, 14	ghmid 13319	. . . . 5 |- ( F e. ( S GrpHom T ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
16	15	adantr 276	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
17	8, 13, 16	3eqtr3d 2234	. . 3 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( ( F ` X ) ( +g ` T ) ( F ` ( M ` X ) ) ) = ( 0g ` T ) )
18		ghmgrp2 13316	. . . . 5 |- ( F e. ( S GrpHom T ) -> T e. Grp )
19	18	adantr 276	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> T e. Grp )
20		eqid 2193	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
21	2, 20	ghmf 13317	. . . . 5 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
22	21	ffvelcdmda 5693	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( F ` X ) e. ( Base ` T ) )
23	21	adantr 276	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> F : B --> ( Base ` T ) )
24	23, 10	ffvelcdmd 5694	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( F ` ( M ` X ) ) e. ( Base ` T ) )
25		ghminv.z	. . . . 5 |- N = ( invg ` T )
26	20, 11, 14, 25	grpinvid1 13124	. . . 4 |- ( ( T e. Grp /\ ( F ` X ) e. ( Base ` T ) /\ ( F ` ( M ` X ) ) e. ( Base ` T ) ) -> ( ( N ` ( F ` X ) ) = ( F ` ( M ` X ) ) <-> ( ( F ` X ) ( +g ` T ) ( F ` ( M ` X ) ) ) = ( 0g ` T ) ) )
27	19, 22, 24, 26	syl3anc 1249	. . 3 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( ( N ` ( F ` X ) ) = ( F ` ( M ` X ) ) <-> ( ( F ` X ) ( +g ` T ) ( F ` ( M ` X ) ) ) = ( 0g ` T ) ) )
28	17, 27	mpbird 167	. 2 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( N ` ( F ` X ) ) = ( F ` ( M ` X ) ) )
29	28	eqcomd 2199	1 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( F ` ( M ` X ) ) = ( N ` ( F ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   -->wf 5250   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   0gc0g 12867   Grpcgrp 13072   invgcminusg 13073   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-rmo 2480  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-riota 5873  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-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-ghm 13311

This theorem is referenced by:  ghmsub  13321  ghmmulg  13326  ghmrn  13327  ghmpreima  13336  ghmeql  13337