Theorem ghminv 13323

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 13318	. . . . . 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 13126	. . . . . 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 5559	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( F ` ( X ( +g ` S ) ( M ` X ) ) ) = ( F ` ( 0g ` S ) ) )
9	2, 5	grpinvcl 13123	. . . . . 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 13321	. . . . 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 13322	. . . . 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 13319	. . . . 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 13320	. . . . 5 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
22	21	ffvelcdmda 5694	. . . 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 5695	. . . 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 13127	. . . 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 5251   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   0gc0g 12870   Grpcgrp 13075   invgcminusg 13076   GrpHom cghm 13313

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 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

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 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-ghm 13314

This theorem is referenced by:  ghmsub  13324  ghmmulg  13329  ghmrn  13330  ghmpreima  13339  ghmeql  13340