Theorem isgrpid2 17458

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 17457	. . . 4 |- ( ( G e. Grp /\ Z e. B ) -> ( ( Z .+ Z ) = Z <-> .0. = Z ) )
5	4	biimpd 219	. . 3 |- ( ( G e. Grp /\ Z e. B ) -> ( ( Z .+ Z ) = Z -> .0. = Z ) )
6	5	expimpd 629	. 2 |- ( G e. Grp -> ( ( Z e. B /\ ( Z .+ Z ) = Z ) -> .0. = Z ) )
7	1, 3	grpidcl 17450	. . . 4 |- ( G e. Grp -> .0. e. B )
8	1, 2, 3	grplid 17452	. . . . 5 |- ( ( G e. Grp /\ .0. e. B ) -> ( .0. .+ .0. ) = .0. )
9	7, 8	mpdan 702	. . . 4 |- ( G e. Grp -> ( .0. .+ .0. ) = .0. )
10	7, 9	jca 554	. . 3 |- ( G e. Grp -> ( .0. e. B /\ ( .0. .+ .0. ) = .0. ) )
11		eleq1 2689	. . . 4 |- ( .0. = Z -> ( .0. e. B <-> Z e. B ) )
12		id 22	. . . . . 6 |- ( .0. = Z -> .0. = Z )
13	12, 12	oveq12d 6668	. . . . 5 |- ( .0. = Z -> ( .0. .+ .0. ) = ( Z .+ Z ) )
14	13, 12	eqeq12d 2637	. . . 4 |- ( .0. = Z -> ( ( .0. .+ .0. ) = .0. <-> ( Z .+ Z ) = Z ) )
15	11, 14	anbi12d 747	. . 3 |- ( .0. = Z -> ( ( .0. e. B /\ ( .0. .+ .0. ) = .0. ) <-> ( Z e. B /\ ( Z .+ Z ) = Z ) ) )
16	10, 15	syl5ibcom 235	. 2 |- ( G e. Grp -> ( .0. = Z -> ( Z e. B /\ ( Z .+ Z ) = Z ) ) )
17	6, 16	impbid 202	1 |- ( G e. Grp -> ( ( Z e. B /\ ( Z .+ Z ) = Z ) <-> .0. = Z ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425

This theorem is referenced by:  drngid2  18763  dchr1  24982  erngdvlem4  36279  erngdvlem4-rN  36287