Theorem grpinvssd 13327

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 2204	. . . . . . 7 |- ( invg ` S ) = ( invg ` S )
4	2, 3	grpinvcl 13298	. . . . . 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 5941	. . . . . . 7 |- ( x = ( ( invg ` S ) ` X ) -> ( x ( +g ` M ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` M ) y ) )
10		oveq1 5941	. . . . . . 7 |- ( x = ( ( invg ` S ) ` X ) -> ( x ( +g ` S ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) y ) )
11	9, 10	eqeq12d 2219	. . . . . 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 5942	. . . . . . 7 |- ( y = X -> ( ( ( invg ` S ) ` X ) ( +g ` M ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` M ) X ) )
13		oveq2 5942	. . . . . . 7 |- ( y = X -> ( ( ( invg ` S ) ` X ) ( +g ` S ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) X ) )
14	12, 13	eqeq12d 2219	. . . . . 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 2890	. . . . 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 1248	. . . 4 |- ( ( ph /\ X e. B ) -> ( ( ( invg ` S ) ` X ) ( +g ` M ) X ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) X ) )
17		eqid 2204	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
18		eqid 2204	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
19	2, 17, 18, 3	grplinv 13300	. . . . 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 3192	. . . . . 6 |- ( ( ph /\ X e. B ) -> X e. ( Base ` M ) )
24		eqid 2204	. . . . . . 7 |- ( Base ` M ) = ( Base ` M )
25		eqid 2204	. . . . . . 7 |- ( +g ` M ) = ( +g ` M )
26		eqid 2204	. . . . . . 7 |- ( 0g ` M ) = ( 0g ` M )
27		eqid 2204	. . . . . . 7 |- ( invg ` M ) = ( invg ` M )
28	24, 25, 26, 27	grplinv 13300	. . . . . 6 |- ( ( M e. Grp /\ X e. ( Base ` M ) ) -> ( ( ( invg ` M ) ` X ) ( +g ` M ) X ) = ( 0g ` M ) )
29	21, 23, 28	syl2an2r 595	. . . . 5 |- ( ( ph /\ X e. B ) -> ( ( ( invg ` M ) ` X ) ( +g ` M ) X ) = ( 0g ` M ) )
30	21, 1, 2, 22, 7	grpidssd 13326	. . . . . 6 |- ( ph -> ( 0g ` M ) = ( 0g ` S ) )
31	30	adantr 276	. . . . 5 |- ( ( ph /\ X e. B ) -> ( 0g ` M ) = ( 0g ` S ) )
32	29, 31	eqtr2d 2238	. . . 4 |- ( ( ph /\ X e. B ) -> ( 0g ` S ) = ( ( ( invg ` M ) ` X ) ( +g ` M ) X ) )
33	16, 20, 32	3eqtrd 2241	. . 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 3193	. . . 4 |- ( ( ph /\ X e. B ) -> ( ( invg ` S ) ` X ) e. ( Base ` M ) )
37	24, 27	grpinvcl 13298	. . . . 5 |- ( ( M e. Grp /\ X e. ( Base ` M ) ) -> ( ( invg ` M ) ` X ) e. ( Base ` M ) )
38	21, 23, 37	syl2an2r 595	. . . 4 |- ( ( ph /\ X e. B ) -> ( ( invg ` M ) ` X ) e. ( Base ` M ) )
39	24, 25	grprcan 13287	. . . 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 1251	. . 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   A.wral 2483   C_ wss 3165   ` cfv 5268  (class class class)co 5934   Basecbs 12751   +g cplusg 12828   0gc0g 13006   Grpcgrp 13250   invgcminusg 13251

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-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-cnex 7998  ax-resscn 7999  ax-1re 8001  ax-addrcl 8004

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-riota 5889  df-ov 5937  df-inn 9019  df-2 9077  df-ndx 12754  df-slot 12755  df-base 12757  df-plusg 12841  df-0g 13008  df-mgm 13106  df-sgrp 13152  df-mnd 13167  df-grp 13253  df-minusg 13254

This theorem is referenced by:  grpissubg  13448