Theorem grpidd2 13823

Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 13805. (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 6092	. . . 4 |- ( ph -> ( .0. .+ .0. ) = ( .0. ( +g ` G ) .0. ) )
3		oveq2 6083	. . . . . 6 |- ( x = .0. -> ( .0. .+ x ) = ( .0. .+ .0. ) )
4		id 19	. . . . . 6 |- ( x = .0. -> x = .0. )
5	3, 4	eqeq12d 2253	. . . . 5 |- ( x = .0. -> ( ( .0. .+ x ) = x <-> ( .0. .+ .0. ) = .0. ) )
6		grpidd2.i	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )
7	6	ralrimiva 2623	. . . . 5 |- ( ph -> A. x e. B ( .0. .+ x ) = x )
8		grpidd2.z	. . . . 5 |- ( ph -> .0. e. B )
9	5, 7, 8	rspcdva 2934	. . . 4 |- ( ph -> ( .0. .+ .0. ) = .0. )
10	2, 9	eqtr3d 2273	. . 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 2317	. . . 4 |- ( ph -> .0. e. ( Base ` G ) )
14		eqid 2238	. . . . 5 |- ( Base ` G ) = ( Base ` G )
15		eqid 2238	. . . . 5 |- ( +g ` G ) = ( +g ` G )
16		eqid 2238	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
17	14, 15, 16	grpid 13821	. . . 4 |- ( ( G e. Grp /\ .0. e. ( Base ` G ) ) -> ( ( .0. ( +g ` G ) .0. ) = .0. <-> ( 0g ` G ) = .0. ) )
18	11, 13, 17	syl2anc 415	. . 3 |- ( ph -> ( ( .0. ( +g ` G ) .0. ) = .0. <-> ( 0g ` G ) = .0. ) )
19	10, 18	mpbid 147	. 2 |- ( ph -> ( 0g ` G ) = .0. )
20	19	eqcomd 2244	1 |- ( ph -> .0. = ( 0g ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   0gc0g 13587   Grpcgrp 13782

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-riota 6028  df-ov 6078  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785

This theorem is referenced by:  imasgrp2  13890