Theorem ghminv 13214

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 13209	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2		ghminv.b	. . . . . . 7 |- B = ( Base ` S )
3		eqid 2189	. . . . . . 7 |- ( +g ` S ) = ( +g ` S )
4		eqid 2189	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
5		ghminv.y	. . . . . . 7 |- M = ( invg ` S )
6	2, 3, 4, 5	grprinv 13018	. . . . . 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 5541	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( F ` ( X ( +g ` S ) ( M ` X ) ) ) = ( F ` ( 0g ` S ) ) )
9	2, 5	grpinvcl 13015	. . . . . 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 2189	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
12	2, 3, 11	ghmlin 13212	. . . . 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 2189	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
15	4, 14	ghmid 13213	. . . . 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 2230	. . 3 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( ( F ` X ) ( +g ` T ) ( F ` ( M ` X ) ) ) = ( 0g ` T ) )
18		ghmgrp2 13210	. . . . 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 2189	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
21	2, 20	ghmf 13211	. . . . 5 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
22	21	ffvelcdmda 5675	. . . 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 5676	. . . 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 13019	. . . 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 2195	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 2160   -->wf 5234   ` cfv 5238  (class class class)co 5900   Basecbs 12523   +g cplusg 12600   0gc0g 12772   Grpcgrp 12968   invgcminusg 12969   GrpHom cghm 13204

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1re 7940  ax-addrcl 7943

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-inn 8955  df-2 9013  df-ndx 12526  df-slot 12527  df-base 12529  df-plusg 12613  df-0g 12774  df-mgm 12843  df-sgrp 12888  df-mnd 12901  df-grp 12971  df-minusg 12972  df-ghm 13205

This theorem is referenced by:  ghmsub  13215  ghmmulg  13220  ghmrn  13221  ghmpreima  13230  ghmeql  13231