Theorem grpidssd 13604

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 13557	. . . . . 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 6007	. . . . . . 7 |- ( x = ( 0g ` S ) -> ( x ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` M ) y ) )
8		oveq1 6007	. . . . . . 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 6008	. . . . . . 7 |- ( y = ( 0g ` S ) -> ( ( 0g ` S ) ( +g ` M ) y ) = ( ( 0g ` S ) ( +g ` M ) ( 0g ` S ) ) )
11		oveq2 6008	. . . . . . 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 13559	. . . . 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 13594	. . . 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 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   0gc0g 13284   Grpcgrp 13528

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

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 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-riota 5953  df-ov 6003  df-inn 9107  df-2 9165  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531

This theorem is referenced by:  grpinvssd  13605