Theorem isgrpid2 13242

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 13241	. . . 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 13231	. . . 4 |- ( G e. Grp -> .0. e. B )
8	1, 2, 3	grplid 13233	. . . . 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 2259	. . . 4 |- ( .0. = Z -> ( .0. e. B <-> Z e. B ) )
12		id 19	. . . . . 6 |- ( .0. = Z -> .0. = Z )
13	12, 12	oveq12d 5943	. . . . 5 |- ( .0. = Z -> ( .0. .+ .0. ) = ( Z .+ Z ) )
14	13, 12	eqeq12d 2211	. . . 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 2167   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   0gc0g 12958   Grpcgrp 13202

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-riota 5880  df-ov 5928  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205

This theorem is referenced by: (None)