Theorem grpinvssd 13832

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 2234	. . . . . . 7 |- ( invg ` S ) = ( invg ` S )
4	2, 3	grpinvcl 13803	. . . . . 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 6065	. . . . . . 7 |- ( x = ( ( invg ` S ) ` X ) -> ( x ( +g ` M ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` M ) y ) )
10		oveq1 6065	. . . . . . 7 |- ( x = ( ( invg ` S ) ` X ) -> ( x ( +g ` S ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) y ) )
11	9, 10	eqeq12d 2249	. . . . . 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 6066	. . . . . . 7 |- ( y = X -> ( ( ( invg ` S ) ` X ) ( +g ` M ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` M ) X ) )
13		oveq2 6066	. . . . . . 7 |- ( y = X -> ( ( ( invg ` S ) ` X ) ( +g ` S ) y ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) X ) )
14	12, 13	eqeq12d 2249	. . . . . 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 2938	. . . . 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 1273	. . . 4 |- ( ( ph /\ X e. B ) -> ( ( ( invg ` S ) ` X ) ( +g ` M ) X ) = ( ( ( invg ` S ) ` X ) ( +g ` S ) X ) )
17		eqid 2234	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
18		eqid 2234	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
19	2, 17, 18, 3	grplinv 13805	. . . . 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 3242	. . . . . 6 |- ( ( ph /\ X e. B ) -> X e. ( Base ` M ) )
24		eqid 2234	. . . . . . 7 |- ( Base ` M ) = ( Base ` M )
25		eqid 2234	. . . . . . 7 |- ( +g ` M ) = ( +g ` M )
26		eqid 2234	. . . . . . 7 |- ( 0g ` M ) = ( 0g ` M )
27		eqid 2234	. . . . . . 7 |- ( invg ` M ) = ( invg ` M )
28	24, 25, 26, 27	grplinv 13805	. . . . . 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 13831	. . . . . 6 |- ( ph -> ( 0g ` M ) = ( 0g ` S ) )
31	30	adantr 276	. . . . 5 |- ( ( ph /\ X e. B ) -> ( 0g ` M ) = ( 0g ` S ) )
32	29, 31	eqtr2d 2268	. . . 4 |- ( ( ph /\ X e. B ) -> ( 0g ` S ) = ( ( ( invg ` M ) ` X ) ( +g ` M ) X ) )
33	16, 20, 32	3eqtrd 2271	. . 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 3243	. . . 4 |- ( ( ph /\ X e. B ) -> ( ( invg ` S ) ` X ) e. ( Base ` M ) )
37	24, 27	grpinvcl 13803	. . . . 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 13792	. . . 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 1276	. . 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 1398   e. wcel 2205   A.wral 2522   C_ wss 3214   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   0gc0g 13553   Grpcgrp 13755   invgcminusg 13756

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759

This theorem is referenced by:  grpissubg  13947