Theorem grpinvssd 13659

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 2231	. . . . . . 7 |- ( invg ` S ) = ( invg ` S )
4	2, 3	grpinvcl 13630	. . . . . 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 6024	. . . . . . 7 |- ( x = ( ( invg ` S ) ` X ) -> ( x ( +g ` M ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` M ) y ) )
10		oveq1 6024	. . . . . . 7 |- ( x = ( ( invg ` S ) ` X ) -> ( x ( +g ` S ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) y ) )
11	9, 10	eqeq12d 2246	. . . . . 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 6025	. . . . . . 7 |- ( y = X -> ( ( ( invg ` S ) ` X ) ( +g ` M ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` M ) X ) )
13		oveq2 6025	. . . . . . 7 |- ( y = X -> ( ( ( invg ` S ) ` X ) ( +g ` S ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) X ) )
14	12, 13	eqeq12d 2246	. . . . . 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 2924	. . . . 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 1272	. . . 4 |- ( ( ph /\ X e. B ) -> ( ( ( invg ` S ) ` X ) ( +g ` M ) X ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) X ) )
17		eqid 2231	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
18		eqid 2231	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
19	2, 17, 18, 3	grplinv 13632	. . . . 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 3227	. . . . . 6 |- ( ( ph /\ X e. B ) -> X e. ( Base ` M ) )
24		eqid 2231	. . . . . . 7 |- ( Base ` M ) = ( Base ` M )
25		eqid 2231	. . . . . . 7 |- ( +g ` M ) = ( +g ` M )
26		eqid 2231	. . . . . . 7 |- ( 0g ` M ) = ( 0g ` M )
27		eqid 2231	. . . . . . 7 |- ( invg ` M ) = ( invg ` M )
28	24, 25, 26, 27	grplinv 13632	. . . . . 6 |- ( ( M e. Grp /\ X e. ( Base ` M ) ) -> ( ( ( invg ` M ) ` X ) ( +g ` M ) X ) = ( 0g ` M ) )
29	21, 23, 28	syl2an2r 599	. . . . 5 |- ( ( ph /\ X e. B ) -> ( ( ( invg ` M ) ` X ) ( +g ` M ) X ) = ( 0g ` M ) )
30	21, 1, 2, 22, 7	grpidssd 13658	. . . . . 6 |- ( ph -> ( 0g ` M ) = ( 0g ` S ) )
31	30	adantr 276	. . . . 5 |- ( ( ph /\ X e. B ) -> ( 0g ` M ) = ( 0g ` S ) )
32	29, 31	eqtr2d 2265	. . . 4 |- ( ( ph /\ X e. B ) -> ( 0g ` S ) = ( ( ( invg ` M ) ` X ) ( +g ` M ) X ) )
33	16, 20, 32	3eqtrd 2268	. . 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 3228	. . . 4 |- ( ( ph /\ X e. B ) -> ( ( invg ` S ) ` X ) e. ( Base ` M ) )
37	24, 27	grpinvcl 13630	. . . . 5 |- ( ( M e. Grp /\ X e. ( Base ` M ) ) -> ( ( invg ` M ) ` X ) e. ( Base ` M ) )
38	21, 23, 37	syl2an2r 599	. . . 4 |- ( ( ph /\ X e. B ) -> ( ( invg ` M ) ` X ) e. ( Base ` M ) )
39	24, 25	grprcan 13619	. . . 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 1275	. . 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 1397   e. wcel 2202   A.wral 2510   C_ wss 3200   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   0gc0g 13338   Grpcgrp 13582   invgcminusg 13583

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586

This theorem is referenced by:  grpissubg  13780