Theorem grpidssd 12970

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 both groups have the same identity element. (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
grpidssd	|- ( ph -> ( 0g ` M ) = ( 0g ` S ) )

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

Allowed substitution hints:   ph( x, y)

Proof of Theorem grpidssd

Step	Hyp	Ref	Expression
1		grpidssd.s	. . . . . 6 |- ( ph -> S e. Grp )
2		grpidssd.b	. . . . . . 7 |- B = ( Base ` S )
3		eqid 2187	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
4	2, 3	grpidcl 12923	. . . . . 6 |- ( S e. Grp -> ( 0g ` S ) e. B )
5	1, 4	syl 14	. . . . 5 |- ( ph -> ( 0g ` S ) e. B )
6		grpidssd.o	. . . . 5 |- ( ph -> A. x e. B A. y e. B ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) )
7		oveq1 5895	. . . . . . 7 |- ( x = ( 0g ` S ) -> ( x ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` M ) y ) )
8		oveq1 5895	. . . . . . 7 |- ( x = ( 0g ` S ) -> ( x ( +g ` S ) y ) = ( ( 0g ` S ) ( +g ` S ) y ) )
9	7, 8	eqeq12d 2202	. . . . . 6 |- ( x = ( 0g ` S ) -> ( ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) <-> ( ( 0g ` S ) ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` S ) y ) ) )
10		oveq2 5896	. . . . . . 7 |- ( y = ( 0g ` S ) -> ( ( 0g ` S ) ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) )
11		oveq2 5896	. . . . . . 7 |- ( y = ( 0g ` S ) -> ( ( 0g ` S ) ( +g ` S ) y ) = ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) )
12	10, 11	eqeq12d 2202	. . . . . 6 |- ( y = ( 0g ` S ) -> ( ( ( 0g ` S ) ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` S ) y ) <-> ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) = ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) ) )
13	9, 12	rspc2va 2867	. . . . 5 |- ( ( ( ( 0g ` S ) e. B /\ ( 0g ` S ) e. B ) /\ A. x e. B A. y e. B ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) ) -> ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) = ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) )
14	5, 5, 6, 13	syl21anc 1247	. . . 4 |- ( ph -> ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) = ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) )
15		eqid 2187	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
16	2, 15, 3	grplid 12925	. . . . 5 |- ( ( S e. Grp /\ ( 0g ` S ) e. B ) -> ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) = ( 0g ` S ) )
17	1, 4, 16	syl2anc2 412	. . . 4 |- ( ph -> ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) = ( 0g ` S ) )
18	14, 17	eqtrd 2220	. . 3 |- ( ph -> ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) = ( 0g ` S ) )
19		grpidssd.m	. . . 4 |- ( ph -> M e. Grp )
20		grpidssd.c	. . . . 5 |- ( ph -> B C_ ( Base ` M ) )
21	20, 5	sseldd 3168	. . . 4 |- ( ph -> ( 0g ` S ) e. ( Base ` M ) )
22		eqid 2187	. . . . 5 |- ( Base ` M ) = ( Base ` M )
23		eqid 2187	. . . . 5 |- ( +g ` M ) = ( +g ` M )
24		eqid 2187	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
25	22, 23, 24	grpidlcan 12960	. . . 4 |- ( ( M e. Grp /\ ( 0g ` S ) e. ( Base ` M ) /\ ( 0g ` S ) e. ( Base ` M ) ) -> ( ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) = ( 0g ` S ) <-> ( 0g ` S ) = ( 0g ` M ) ) )
26	19, 21, 21, 25	syl3anc 1248	. . 3 |- ( ph -> ( ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) = ( 0g ` S ) <-> ( 0g ` S ) = ( 0g ` M ) ) )
27	18, 26	mpbid 147	. 2 |- ( ph -> ( 0g ` S ) = ( 0g ` M ) )
28	27	eqcomd 2193	1 |- ( ph -> ( 0g ` M ) = ( 0g ` S ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1363   e. wcel 2158   A.wral 2465   C_ wss 3141   ` cfv 5228  (class class class)co 5888   Basecbs 12475   +g cplusg 12550   0gc0g 12722   Grpcgrp 12896

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-riota 5844  df-ov 5891  df-inn 8933  df-2 8991  df-ndx 12478  df-slot 12479  df-base 12481  df-plusg 12563  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12837  df-grp 12899

This theorem is referenced by:  grpinvssd  12971