Theorem isgrpid2 13112

Description: Properties showing that an element Z is the identity element of a group. (Contributed by NM, 7-Aug-2013.)

Hypotheses

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

Assertion

Ref	Expression
isgrpid2	|- ( G e. Grp -> ( ( Z e. B /\ ( Z .+ Z ) = Z ) <-> .0. = Z ) )

Proof of Theorem isgrpid2

Step	Hyp	Ref	Expression
1		grpinveu.b	. . . . 5 |- B = ( Base ` G )
2		grpinveu.p	. . . . 5 |- .+ = ( +g ` G )
3		grpinveu.o	. . . . 5 |- .0. = ( 0g ` G )
4	1, 2, 3	grpid 13111	. . . 4 |- ( ( G e. Grp /\ Z e. B ) -> ( ( Z .+ Z ) = Z <-> .0. = Z ) )
5	4	biimpd 144	. . 3 |- ( ( G e. Grp /\ Z e. B ) -> ( ( Z .+ Z ) = Z -> .0. = Z ) )
6	5	expimpd 363	. 2 |- ( G e. Grp -> ( ( Z e. B /\ ( Z .+ Z ) = Z ) -> .0. = Z ) )
7	1, 3	grpidcl 13101	. . . 4 |- ( G e. Grp -> .0. e. B )
8	1, 2, 3	grplid 13103	. . . . 5 |- ( ( G e. Grp /\ .0. e. B ) -> ( .0. .+ .0. ) = .0. )
9	7, 8	mpdan 421	. . . 4 |- ( G e. Grp -> ( .0. .+ .0. ) = .0. )
10	7, 9	jca 306	. . 3 |- ( G e. Grp -> ( .0. e. B /\ ( .0. .+ .0. ) = .0. ) )
11		eleq1 2256	. . . 4 |- ( .0. = Z -> ( .0. e. B <-> Z e. B ) )
12		id 19	. . . . . 6 |- ( .0. = Z -> .0. = Z )
13	12, 12	oveq12d 5936	. . . . 5 |- ( .0. = Z -> ( .0. .+ .0. ) = ( Z .+ Z ) )
14	13, 12	eqeq12d 2208	. . . 4 |- ( .0. = Z -> ( ( .0. .+ .0. ) = .0. <-> ( Z .+ Z ) = Z ) )
15	11, 14	anbi12d 473	. . 3 |- ( .0. = Z -> ( ( .0. e. B /\ ( .0. .+ .0. ) = .0. ) <-> ( Z e. B /\ ( Z .+ Z ) = Z ) ) )
16	10, 15	syl5ibcom 155	. 2 |- ( G e. Grp -> ( .0. = Z -> ( Z e. B /\ ( Z .+ Z ) = Z ) ) )
17	6, 16	impbid 129	1 |- ( G e. Grp -> ( ( Z e. B /\ ( Z .+ Z ) = Z ) <-> .0. = Z ) )

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: (None)