Theorem grpidd2 13315

Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 13297. (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 5960	. . . 4 |- ( ph -> ( .0. .+ .0. ) = ( .0. ( +g ` G ) .0. ) )
3		oveq2 5951	. . . . . 6 |- ( x = .0. -> ( .0. .+ x ) = ( .0. .+ .0. ) )
4		id 19	. . . . . 6 |- ( x = .0. -> x = .0. )
5	3, 4	eqeq12d 2219	. . . . 5 |- ( x = .0. -> ( ( .0. .+ x ) = x <-> ( .0. .+ .0. ) = .0. ) )
6		grpidd2.i	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )
7	6	ralrimiva 2578	. . . . 5 |- ( ph -> A. x e. B ( .0. .+ x ) = x )
8		grpidd2.z	. . . . 5 |- ( ph -> .0. e. B )
9	5, 7, 8	rspcdva 2881	. . . 4 |- ( ph -> ( .0. .+ .0. ) = .0. )
10	2, 9	eqtr3d 2239	. . 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 2283	. . . 4 |- ( ph -> .0. e. ( Base ` G ) )
14		eqid 2204	. . . . 5 |- ( Base ` G ) = ( Base ` G )
15		eqid 2204	. . . . 5 |- ( +g ` G ) = ( +g ` G )
16		eqid 2204	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
17	14, 15, 16	grpid 13313	. . . 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 2210	1 |- ( ph -> .0. = ( 0g ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   ` cfv 5270  (class class class)co 5943   Basecbs 12774   +g cplusg 12851   0gc0g 13030   Grpcgrp 13274

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-cnex 8015  ax-resscn 8016  ax-1re 8018  ax-addrcl 8021

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-fv 5278  df-riota 5898  df-ov 5946  df-inn 9036  df-2 9094  df-ndx 12777  df-slot 12778  df-base 12780  df-plusg 12864  df-0g 13032  df-mgm 13130  df-sgrp 13176  df-mnd 13191  df-grp 13277

This theorem is referenced by:  imasgrp2  13388