Theorem ghminv 13380

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 13375	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2		ghminv.b	. . . . . . 7 |- B = ( Base ` S )
3		eqid 2196	. . . . . . 7 |- ( +g ` S ) = ( +g ` S )
4		eqid 2196	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
5		ghminv.y	. . . . . . 7 |- M = ( invg ` S )
6	2, 3, 4, 5	grprinv 13183	. . . . . 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 5562	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( F ` ( X ( +g ` S ) ( M ` X ) ) ) = ( F ` ( 0g ` S ) ) )
9	2, 5	grpinvcl 13180	. . . . . 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 2196	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
12	2, 3, 11	ghmlin 13378	. . . . 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 2196	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
15	4, 14	ghmid 13379	. . . . 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 2237	. . 3 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( ( F ` X ) ( +g ` T ) ( F ` ( M ` X ) ) ) = ( 0g ` T ) )
18		ghmgrp2 13376	. . . . 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 2196	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
21	2, 20	ghmf 13377	. . . . 5 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
22	21	ffvelcdmda 5697	. . . 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 5698	. . . 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 13184	. . . 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 2202	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 2167   -->wf 5254   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927   Grpcgrp 13132   invgcminusg 13133   GrpHom cghm 13370

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-ghm 13371

This theorem is referenced by:  ghmsub  13381  ghmmulg  13386  ghmrn  13387  ghmpreima  13396  ghmeql  13397