Theorem grpidssd 13208

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 2196	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
4	2, 3	grpidcl 13161	. . . . . 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 5929	. . . . . . 7 |- ( x = ( 0g ` S ) -> ( x ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` M ) y ) )
8		oveq1 5929	. . . . . . 7 |- ( x = ( 0g ` S ) -> ( x ( +g ` S ) y ) = ( ( 0g ` S ) ( +g ` S ) y ) )
9	7, 8	eqeq12d 2211	. . . . . 6 |- ( x = ( 0g ` S ) -> ( ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) <-> ( ( 0g ` S ) ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` S ) y ) ) )
10		oveq2 5930	. . . . . . 7 |- ( y = ( 0g ` S ) -> ( ( 0g ` S ) ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) )
11		oveq2 5930	. . . . . . 7 |- ( y = ( 0g ` S ) -> ( ( 0g ` S ) ( +g ` S ) y ) = ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) )
12	10, 11	eqeq12d 2211	. . . . . 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 2882	. . . . 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 2196	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
16	2, 15, 3	grplid 13163	. . . . 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 2229	. . 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 3184	. . . 4 |- ( ph -> ( 0g ` S ) e. ( Base ` M ) )
22		eqid 2196	. . . . 5 |- ( Base ` M ) = ( Base ` M )
23		eqid 2196	. . . . 5 |- ( +g ` M ) = ( +g ` M )
24		eqid 2196	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
25	22, 23, 24	grpidlcan 13198	. . . 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 2202	1 |- ( ph -> ( 0g ` M ) = ( 0g ` S ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927   Grpcgrp 13132

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-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

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 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135

This theorem is referenced by:  grpinvssd  13209