Theorem grpinvssd 13882

Description: If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the elements of the first group have the same inverses in both groups. (Contributed by AV, 15-Mar-2019.)

Hypotheses

Ref	Expression
grpidssd.m	|- ( ph -> M e. Grp )
grpidssd.s	|- ( ph -> S e. Grp )
grpidssd.b	|- B = ( Base ` S )
grpidssd.c	|- ( ph -> B C_ ( Base ` M ) )
grpidssd.o	|- ( ph -> A. x e. B A. y e. B ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) )

Assertion

Ref	Expression
grpinvssd	|- ( ph -> ( X e. B -> ( ( invg ` S ) ` X ) = ( ( invg ` M ) ` X ) ) )

Distinct variable groups:   x, B, y   x, M, y   x, S, y   x, X, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem grpinvssd

Step	Hyp	Ref	Expression
1		grpidssd.s	. . . . . 6 |- ( ph -> S e. Grp )
2		grpidssd.b	. . . . . . 7 |- B = ( Base ` S )
3		eqid 2238	. . . . . . 7 |- ( invg ` S ) = ( invg ` S )
4	2, 3	grpinvcl 13853	. . . . . 6 |- ( ( S e. Grp /\ X e. B ) -> ( ( invg ` S ) ` X ) e. B )
5	1, 4	sylan 283	. . . . 5 |- ( ( ph /\ X e. B ) -> ( ( invg ` S ) ` X ) e. B )
6		simpr 110	. . . . 5 |- ( ( ph /\ X e. B ) -> X e. B )
7		grpidssd.o	. . . . . 6 |- ( ph -> A. x e. B A. y e. B ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) )
8	7	adantr 276	. . . . 5 |- ( ( ph /\ X e. B ) -> A. x e. B A. y e. B ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) )
9		oveq1 6092	. . . . . . 7 |- ( x = ( ( invg ` S ) ` X ) -> ( x ( +g ` M ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` M ) y ) )
10		oveq1 6092	. . . . . . 7 |- ( x = ( ( invg ` S ) ` X ) -> ( x ( +g ` S ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) y ) )
11	9, 10	eqeq12d 2253	. . . . . 6 |- ( x = ( ( invg ` S ) ` X ) -> ( ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) <-> ( ( ( invg ` S ) ` X ) ( +g ` M ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) y ) ) )
12		oveq2 6093	. . . . . . 7 |- ( y = X -> ( ( ( invg ` S ) ` X ) ( +g ` M ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` M ) X ) )
13		oveq2 6093	. . . . . . 7 |- ( y = X -> ( ( ( invg ` S ) ` X ) ( +g ` S ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) X ) )
14	12, 13	eqeq12d 2253	. . . . . 6 |- ( y = X -> ( ( ( ( invg ` S ) ` X ) ( +g ` M ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) y ) <-> ( ( ( invg ` S ) ` X ) ( +g ` M ) X ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) X ) ) )
15	11, 14	rspc2va 2944	. . . . 5 |- ( ( ( ( ( invg ` S ) ` X ) e. B /\ X e. B ) /\ A. x e. B A. y e. B ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) ) -> ( ( ( invg ` S ) ` X ) ( +g ` M ) X ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) X ) )
16	5, 6, 8, 15	syl21anc 1277	. . . 4 |- ( ( ph /\ X e. B ) -> ( ( ( invg ` S ) ` X ) ( +g ` M ) X ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) X ) )
17		eqid 2238	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
18		eqid 2238	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
19	2, 17, 18, 3	grplinv 13855	. . . . 5 |- ( ( S e. Grp /\ X e. B ) -> ( ( ( invg ` S ) ` X ) ( +g ` S ) X ) = ( 0g ` S ) )
20	1, 19	sylan 283	. . . 4 |- ( ( ph /\ X e. B ) -> ( ( ( invg ` S ) ` X ) ( +g ` S ) X ) = ( 0g ` S ) )
21		grpidssd.m	. . . . . 6 |- ( ph -> M e. Grp )
22		grpidssd.c	. . . . . . 7 |- ( ph -> B C_ ( Base ` M ) )
23	22	sselda 3248	. . . . . 6 |- ( ( ph /\ X e. B ) -> X e. ( Base ` M ) )
24		eqid 2238	. . . . . . 7 |- ( Base ` M ) = ( Base ` M )
25		eqid 2238	. . . . . . 7 |- ( +g ` M ) = ( +g ` M )
26		eqid 2238	. . . . . . 7 |- ( 0g ` M ) = ( 0g ` M )
27		eqid 2238	. . . . . . 7 |- ( invg ` M ) = ( invg ` M )
28	24, 25, 26, 27	grplinv 13855	. . . . . 6 |- ( ( M e. Grp /\ X e. ( Base ` M ) ) -> ( ( ( invg ` M ) ` X ) ( +g ` M ) X ) = ( 0g ` M ) )
29	21, 23, 28	syl2an2r 603	. . . . 5 |- ( ( ph /\ X e. B ) -> ( ( ( invg ` M ) ` X ) ( +g ` M ) X ) = ( 0g ` M ) )
30	21, 1, 2, 22, 7	grpidssd 13881	. . . . . 6 |- ( ph -> ( 0g ` M ) = ( 0g ` S ) )
31	30	adantr 276	. . . . 5 |- ( ( ph /\ X e. B ) -> ( 0g ` M ) = ( 0g ` S ) )
32	29, 31	eqtr2d 2272	. . . 4 |- ( ( ph /\ X e. B ) -> ( 0g ` S ) = ( ( ( invg ` M ) ` X ) ( +g ` M ) X ) )
33	16, 20, 32	3eqtrd 2275	. . 3 |- ( ( ph /\ X e. B ) -> ( ( ( invg ` S ) ` X ) ( +g ` M ) X ) = ( ( ( invg ` M ) ` X ) ( +g ` M ) X ) )
34	21	adantr 276	. . . 4 |- ( ( ph /\ X e. B ) -> M e. Grp )
35	22	adantr 276	. . . . 5 |- ( ( ph /\ X e. B ) -> B C_ ( Base ` M ) )
36	35, 5	sseldd 3249	. . . 4 |- ( ( ph /\ X e. B ) -> ( ( invg ` S ) ` X ) e. ( Base ` M ) )
37	24, 27	grpinvcl 13853	. . . . 5 |- ( ( M e. Grp /\ X e. ( Base ` M ) ) -> ( ( invg ` M ) ` X ) e. ( Base ` M ) )
38	21, 23, 37	syl2an2r 603	. . . 4 |- ( ( ph /\ X e. B ) -> ( ( invg ` M ) ` X ) e. ( Base ` M ) )
39	24, 25	grprcan 13842	. . . 4 |- ( ( M e. Grp /\ ( ( ( invg ` S ) ` X ) e. ( Base ` M ) /\ ( ( invg ` M ) ` X ) e. ( Base ` M ) /\ X e. ( Base ` M ) ) ) -> ( ( ( ( invg ` S ) ` X ) ( +g ` M ) X ) = ( ( ( invg ` M ) ` X ) ( +g ` M ) X ) <-> ( ( invg ` S ) ` X ) = ( ( invg ` M ) ` X ) ) )
40	34, 36, 38, 23, 39	syl13anc 1280	. . 3 |- ( ( ph /\ X e. B ) -> ( ( ( ( invg ` S ) ` X ) ( +g ` M ) X ) = ( ( ( invg ` M ) ` X ) ( +g ` M ) X ) <-> ( ( invg ` S ) ` X ) = ( ( invg ` M ) ` X ) ) )
41	33, 40	mpbid 147	. 2 |- ( ( ph /\ X e. B ) -> ( ( invg ` S ) ` X ) = ( ( invg ` M ) ` X ) )
42	41	ex 115	1 |- ( ph -> ( X e. B -> ( ( invg ` S ) ` X ) = ( ( invg ` M ) ` X ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   0gc0g 13610   Grpcgrp 13805   invgcminusg 13806

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  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 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  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 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9305  df-2 9363  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809

This theorem is used by:  grpissubg  13997