Theorem grpid 13844

Description: Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011.)

Hypotheses

Ref	Expression
grpinveu.b	|- B = ( Base ` G )
grpinveu.p	|- .+ = ( +g ` G )
grpinveu.o	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
grpid	|- ( ( G e. Grp /\ X e. B ) -> ( ( X .+ X ) = X <-> .0. = X ) )

Proof of Theorem grpid

Step	Hyp	Ref	Expression
1		eqcom 2240	. 2 |- ( .0. = X <-> X = .0. )
2		grpinveu.b	. . . . . . 7 |- B = ( Base ` G )
3		grpinveu.o	. . . . . . 7 |- .0. = ( 0g ` G )
4	2, 3	grpidcl 13834	. . . . . 6 |- ( G e. Grp -> .0. e. B )
5		grpinveu.p	. . . . . . . 8 |- .+ = ( +g ` G )
6	2, 5	grprcan 13842	. . . . . . 7 |- ( ( G e. Grp /\ ( X e. B /\ .0. e. B /\ X e. B ) ) -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) )
7	6	3exp2 1256	. . . . . 6 |- ( G e. Grp -> ( X e. B -> ( .0. e. B -> ( X e. B -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) ) ) ) )
8	4, 7	mpid 42	. . . . 5 |- ( G e. Grp -> ( X e. B -> ( X e. B -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) ) ) )
9	8	pm2.43d 50	. . . 4 |- ( G e. Grp -> ( X e. B -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) ) )
10	9	imp 124	. . 3 |- ( ( G e. Grp /\ X e. B ) -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) )
11	2, 5, 3	grplid 13836	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( .0. .+ X ) = X )
12	11	eqeq2d 2250	. . 3 |- ( ( G e. Grp /\ X e. B ) -> ( ( X .+ X ) = ( .0. .+ X ) <-> ( X .+ X ) = X ) )
13	10, 12	bitr3d 190	. 2 |- ( ( G e. Grp /\ X e. B ) -> ( X = .0. <-> ( X .+ X ) = X ) )
14	1, 13	bitr2id 193	1 |- ( ( G e. Grp /\ X e. B ) -> ( ( X .+ X ) = X <-> .0. = X ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   0gc0g 13610   Grpcgrp 13805

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9305  df-2 9363  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808

This theorem is used by:  isgrpid2  13845  grpidd2  13846  subg0  13983  qus0  14038  ghmid  14052  lmod0vid  14657  cnfld0  14908  psr0  15077