Theorem grpidd2 13113

Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 13095. (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 5935	. . . 4 |- ( ph -> ( .0. .+ .0. ) = ( .0. ( +g ` G ) .0. ) )
3		oveq2 5926	. . . . . 6 |- ( x = .0. -> ( .0. .+ x ) = ( .0. .+ .0. ) )
4		id 19	. . . . . 6 |- ( x = .0. -> x = .0. )
5	3, 4	eqeq12d 2208	. . . . 5 |- ( x = .0. -> ( ( .0. .+ x ) = x <-> ( .0. .+ .0. ) = .0. ) )
6		grpidd2.i	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )
7	6	ralrimiva 2567	. . . . 5 |- ( ph -> A. x e. B ( .0. .+ x ) = x )
8		grpidd2.z	. . . . 5 |- ( ph -> .0. e. B )
9	5, 7, 8	rspcdva 2869	. . . 4 |- ( ph -> ( .0. .+ .0. ) = .0. )
10	2, 9	eqtr3d 2228	. . 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 2272	. . . 4 |- ( ph -> .0. e. ( Base ` G ) )
14		eqid 2193	. . . . 5 |- ( Base ` G ) = ( Base ` G )
15		eqid 2193	. . . . 5 |- ( +g ` G ) = ( +g ` G )
16		eqid 2193	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
17	14, 15, 16	grpid 13111	. . . 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 2199	1 |- ( ph -> .0. = ( 0g ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   0gc0g 12867   Grpcgrp 13072

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075

This theorem is referenced by:  imasgrp2  13180