Theorem grpidd2 17459

Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 17444. (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 6667	. . . 4 |- ( ph -> ( .0. .+ .0. ) = ( .0. ( +g ` G ) .0. ) )
3		grpidd2.z	. . . . 5 |- ( ph -> .0. e. B )
4		grpidd2.i	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )
5	4	ralrimiva 2966	. . . . 5 |- ( ph -> A. x e. B ( .0. .+ x ) = x )
6		oveq2 6658	. . . . . . 7 |- ( x = .0. -> ( .0. .+ x ) = ( .0. .+ .0. ) )
7		id 22	. . . . . . 7 |- ( x = .0. -> x = .0. )
8	6, 7	eqeq12d 2637	. . . . . 6 |- ( x = .0. -> ( ( .0. .+ x ) = x <-> ( .0. .+ .0. ) = .0. ) )
9	8	rspcv 3305	. . . . 5 |- ( .0. e. B -> ( A. x e. B ( .0. .+ x ) = x -> ( .0. .+ .0. ) = .0. ) )
10	3, 5, 9	sylc 65	. . . 4 |- ( ph -> ( .0. .+ .0. ) = .0. )
11	2, 10	eqtr3d 2658	. . 3 |- ( ph -> ( .0. ( +g ` G ) .0. ) = .0. )
12		grpidd2.j	. . . 4 |- ( ph -> G e. Grp )
13		grpidd2.b	. . . . 5 |- ( ph -> B = ( Base ` G ) )
14	3, 13	eleqtrd 2703	. . . 4 |- ( ph -> .0. e. ( Base ` G ) )
15		eqid 2622	. . . . 5 |- ( Base ` G ) = ( Base ` G )
16		eqid 2622	. . . . 5 |- ( +g ` G ) = ( +g ` G )
17		eqid 2622	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
18	15, 16, 17	grpid 17457	. . . 4 |- ( ( G e. Grp /\ .0. e. ( Base ` G ) ) -> ( ( .0. ( +g ` G ) .0. ) = .0. <-> ( 0g ` G ) = .0. ) )
19	12, 14, 18	syl2anc 693	. . 3 |- ( ph -> ( ( .0. ( +g ` G ) .0. ) = .0. <-> ( 0g ` G ) = .0. ) )
20	11, 19	mpbid 222	. 2 |- ( ph -> ( 0g ` G ) = .0. )
21	20	eqcomd 2628	1 |- ( ph -> .0. = ( 0g ` G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425

This theorem is referenced by:  imasgrp2  17530