Theorem grpidssd 13609

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 2229	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
4	2, 3	grpidcl 13562	. . . . . 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 6008	. . . . . . 7 |- ( x = ( 0g ` S ) -> ( x ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` M ) y ) )
8		oveq1 6008	. . . . . . 7 |- ( x = ( 0g ` S ) -> ( x ( +g ` S ) y ) = ( ( 0g ` S ) ( +g ` S ) y ) )
9	7, 8	eqeq12d 2244	. . . . . 6 |- ( x = ( 0g ` S ) -> ( ( x ( +g ` M ) y ) = ( x ( +g ` S ) y ) <-> ( ( 0g ` S ) ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` S ) y ) ) )
10		oveq2 6009	. . . . . . 7 |- ( y = ( 0g ` S ) -> ( ( 0g ` S ) ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) )
11		oveq2 6009	. . . . . . 7 |- ( y = ( 0g ` S ) -> ( ( 0g ` S ) ( +g ` S ) y ) = ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) )
12	10, 11	eqeq12d 2244	. . . . . 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 2921	. . . . 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 1270	. . . 4 |- ( ph -> ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) = ( ( 0g ` S ) ( +g ` S ) ( 0g ` S ) ) )
15		eqid 2229	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
16	2, 15, 3	grplid 13564	. . . . 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 2262	. . 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 3225	. . . 4 |- ( ph -> ( 0g ` S ) e. ( Base ` M ) )
22		eqid 2229	. . . . 5 |- ( Base ` M ) = ( Base ` M )
23		eqid 2229	. . . . 5 |- ( +g ` M ) = ( +g ` M )
24		eqid 2229	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
25	22, 23, 24	grpidlcan 13599	. . . 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 1271	. . 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 2235	1 |- ( ph -> ( 0g ` M ) = ( 0g ` S ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   ` cfv 5318  (class class class)co 6001   Basecbs 13032   +g cplusg 13110   0gc0g 13289   Grpcgrp 13533

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-riota 5954  df-ov 6004  df-inn 9111  df-2 9169  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536

This theorem is referenced by:  grpinvssd  13610