Theorem isgrpid2 13904

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 13903	. . . 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 13893	. . . 4 |- ( G e. Grp -> .0. e. B )
8	1, 2, 3	grplid 13895	. . . . 5 |- ( ( G e. Grp /\ .0. e. B ) -> ( .0. .+ .0. ) = .0. )
9	7, 8	mpdan 425	. . . 4 |- ( G e. Grp -> ( .0. .+ .0. ) = .0. )
10	7, 9	jca 306	. . 3 |- ( G e. Grp -> ( .0. e. B /\ ( .0. .+ .0. ) = .0. ) )
11		eleq1 2301	. . . 4 |- ( .0. = Z -> ( .0. e. B <-> Z e. B ) )
12		id 19	. . . . . 6 |- ( .0. = Z -> .0. = Z )
13	12, 12	oveq12d 6106	. . . . 5 |- ( .0. = Z -> ( .0. .+ .0. ) = ( Z .+ Z ) )
14	13, 12	eqeq12d 2253	. . . 4 |- ( .0. = Z -> ( ( .0. .+ .0. ) = .0. <-> ( Z .+ Z ) = Z ) )
15	11, 14	anbi12d 477	. . 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   ` cfv 5380  (class class class)co 6088   Basecbs 13410   +g cplusg 13490   0gc0g 13669   Grpcgrp 13864

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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

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 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fn 5383  df-fv 5388  df-riota 6041  df-ov 6091  df-inn 9314  df-2 9372  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867

This theorem is used by: (None)