Theorem grpidssd 13151

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 2193	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
4	2, 3	grpidcl 13104	. . . . . 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 5926	. . . . . . 7 |- ( x = ( 0g ` S ) -> ( x ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` M ) y ) )
8		oveq1 5926	. . . . . . 7 |- ( x = ( 0g ` S ) -> ( x ( +g ` S ) y ) = ( ( 0g ` S ) ( +g ` S ) y ) )
9	7, 8	eqeq12d 2208	. . . . . 6 |- ( x = ( 0g ` S ) -> ( ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) <-> ( ( 0g ` S ) ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` S ) y ) ) )
10		oveq2 5927	. . . . . . 7 |- ( y = ( 0g ` S ) -> ( ( 0g ` S ) ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) )
11		oveq2 5927	. . . . . . 7 |- ( y = ( 0g ` S ) -> ( ( 0g ` S ) ( +g ` S ) y ) = ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) )
12	10, 11	eqeq12d 2208	. . . . . 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 2879	. . . . 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 1248	. . . 4 |- ( ph -> ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) = ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) )
15		eqid 2193	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
16	2, 15, 3	grplid 13106	. . . . 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 2226	. . 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 3181	. . . 4 |- ( ph -> ( 0g ` S ) e. ( Base ` M ) )
22		eqid 2193	. . . . 5 |- ( Base ` M ) = ( Base ` M )
23		eqid 2193	. . . . 5 |- ( +g ` M ) = ( +g ` M )
24		eqid 2193	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
25	22, 23, 24	grpidlcan 13141	. . . 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 1249	. . 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 2199	1 |- ( ph -> ( 0g ` M ) = ( 0g ` S ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   A.wral 2472   C_ wss 3154   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   0gc0g 12870   Grpcgrp 13075

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078

This theorem is referenced by:  grpinvssd  13152