Theorem grpidd2 12919

Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 12904. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
grpidd2.b	|- ( ph -> B = ( Base ` G ) )
grpidd2.p	|- ( ph -> .+ = ( +g ` G ) )
grpidd2.z	|- ( ph -> .0. e. B )
grpidd2.i	|- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )
grpidd2.j	|- ( ph -> G e. Grp )

Assertion

Ref	Expression
grpidd2	|- ( ph -> .0. = ( 0g ` G ) )

Distinct variable groups:   x, B   x, .+   ph, x   x, .0.

Allowed substitution hint:   G( x)

Proof of Theorem grpidd2

Step	Hyp	Ref	Expression
1		grpidd2.p	. . . . 5 |- ( ph -> .+ = ( +g ` G ) )
2	1	oveqd 5894	. . . 4 |- ( ph -> ( .0. .+ .0. ) = ( .0. ( +g ` G ) .0. ) )
3		oveq2 5885	. . . . . 6 |- ( x = .0. -> ( .0. .+ x ) = ( .0. .+ .0. ) )
4		id 19	. . . . . 6 |- ( x = .0. -> x = .0. )
5	3, 4	eqeq12d 2192	. . . . 5 |- ( x = .0. -> ( ( .0. .+ x ) = x <-> ( .0. .+ .0. ) = .0. ) )
6		grpidd2.i	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )
7	6	ralrimiva 2550	. . . . 5 |- ( ph -> A. x e. B ( .0. .+ x ) = x )
8		grpidd2.z	. . . . 5 |- ( ph -> .0. e. B )
9	5, 7, 8	rspcdva 2848	. . . 4 |- ( ph -> ( .0. .+ .0. ) = .0. )
10	2, 9	eqtr3d 2212	. . 3 |- ( ph -> ( .0. ( +g ` G ) .0. ) = .0. )
11		grpidd2.j	. . . 4 |- ( ph -> G e. Grp )
12		grpidd2.b	. . . . 5 |- ( ph -> B = ( Base ` G ) )
13	8, 12	eleqtrd 2256	. . . 4 |- ( ph -> .0. e. ( Base ` G ) )
14		eqid 2177	. . . . 5 |- ( Base ` G ) = ( Base ` G )
15		eqid 2177	. . . . 5 |- ( +g ` G ) = ( +g ` G )
16		eqid 2177	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
17	14, 15, 16	grpid 12917	. . . 4 |- ( ( G e. Grp /\ .0. e. ( Base ` G ) ) -> ( ( .0. ( +g ` G ) .0. ) = .0. <-> ( 0g ` G ) = .0. ) )
18	11, 13, 17	syl2anc 411	. . 3 |- ( ph -> ( ( .0. ( +g ` G ) .0. ) = .0. <-> ( 0g ` G ) = .0. ) )
19	10, 18	mpbid 147	. 2 |- ( ph -> ( 0g ` G ) = .0. )
20	19	eqcomd 2183	1 |- ( ph -> .0. = ( 0g ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   0gc0g 12710   Grpcgrp 12882

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885

This theorem is referenced by: (None)