Theorem ghminv 14112

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 14107	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2		ghminv.b	. . . . . . 7 |- B = ( Base ` S )
3		eqid 2238	. . . . . . 7 |- ( +g ` S ) = ( +g ` S )
4		eqid 2238	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
5		ghminv.y	. . . . . . 7 |- M = ( invg ` S )
6	2, 3, 4, 5	grprinv 13915	. . . . . 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 5702	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( F ` ( X ( +g ` S ) ( M ` X ) ) ) = ( F ` ( 0g ` S ) ) )
9	2, 5	grpinvcl 13912	. . . . . 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 2238	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
12	2, 3, 11	ghmlin 14110	. . . . 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 1379	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( F ` ( X ( +g ` S ) ( M ` X ) ) ) = ( ( F ` X ) ( +g ` T ) ( F ` ( M ` X ) ) ) )
14		eqid 2238	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
15	4, 14	ghmid 14111	. . . . 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 2279	. . 3 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( ( F ` X ) ( +g ` T ) ( F ` ( M ` X ) ) ) = ( 0g ` T ) )
18		ghmgrp2 14108	. . . . 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 2238	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
21	2, 20	ghmf 14109	. . . . 5 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
22	21	ffvelcdmda 5846	. . . 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 5847	. . . 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 13916	. . . 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 1278	. . 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 2244	1 |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( F ` ( M ` X ) ) = ( N ` ( F ` X ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   -->wf 5376   ` cfv 5380  (class class class)co 6088   Basecbs 13410   +g cplusg 13490   0gc0g 13669   Grpcgrp 13864   invgcminusg 13865   GrpHom cghm 14102

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-inn 9314  df-2 9372  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-minusg 13868  df-ghm 14103

This theorem is used by:  ghmsub  14113  ghmmulg  14118  ghmrn  14119  ghmpreima  14128  ghmeql  14129  asclinvg  15141