Theorem isgrpid2 13115

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 13114	. . . 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 13104	. . . 4 |- ( G e. Grp -> .0. e. B )
8	1, 2, 3	grplid 13106	. . . . 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 5937	. . . . 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 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   0gc0g 12870   Grpcgrp 13075

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 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

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 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078

This theorem is referenced by: (None)