Theorem grpinvssd 13279

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 2196	. . . . . . 7 |- ( invg ` S ) = ( invg ` S )
4	2, 3	grpinvcl 13250	. . . . . 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 5932	. . . . . . 7 |- ( x = ( ( invg ` S ) ` X ) -> ( x ( +g ` M ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` M ) y ) )
10		oveq1 5932	. . . . . . 7 |- ( x = ( ( invg ` S ) ` X ) -> ( x ( +g ` S ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) y ) )
11	9, 10	eqeq12d 2211	. . . . . 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 5933	. . . . . . 7 |- ( y = X -> ( ( ( invg ` S ) ` X ) ( +g ` M ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` M ) X ) )
13		oveq2 5933	. . . . . . 7 |- ( y = X -> ( ( ( invg ` S ) ` X ) ( +g ` S ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) X ) )
14	12, 13	eqeq12d 2211	. . . . . 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 2882	. . . . 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 2196	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
18		eqid 2196	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
19	2, 17, 18, 3	grplinv 13252	. . . . 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 3184	. . . . . 6 |- ( ( ph /\ X e. B ) -> X e. ( Base ` M ) )
24		eqid 2196	. . . . . . 7 |- ( Base ` M ) = ( Base ` M )
25		eqid 2196	. . . . . . 7 |- ( +g ` M ) = ( +g ` M )
26		eqid 2196	. . . . . . 7 |- ( 0g ` M ) = ( 0g ` M )
27		eqid 2196	. . . . . . 7 |- ( invg ` M ) = ( invg ` M )
28	24, 25, 26, 27	grplinv 13252	. . . . . 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 13278	. . . . . 6 |- ( ph -> ( 0g ` M ) = ( 0g ` S ) )
31	30	adantr 276	. . . . 5 |- ( ( ph /\ X e. B ) -> ( 0g ` M ) = ( 0g ` S ) )
32	29, 31	eqtr2d 2230	. . . 4 |- ( ( ph /\ X e. B ) -> ( 0g ` S ) = ( ( ( invg ` M ) ` X ) ( +g ` M ) X ) )
33	16, 20, 32	3eqtrd 2233	. . 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 3185	. . . 4 |- ( ( ph /\ X e. B ) -> ( ( invg ` S ) ` X ) e. ( Base ` M ) )
37	24, 27	grpinvcl 13250	. . . . 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 13239	. . . 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 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   0gc0g 12958   Grpcgrp 13202   invgcminusg 13203

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206

This theorem is referenced by:  grpissubg  13400