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Metamath Proof Explorer

Theorem isgrpid2

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

Ref Expression
Hypotheses grpinveu.b 
|- B = ( Base ` G )
grpinveu.p 
|- .+ = ( +g ` G )
grpinveu.o 
|- .0. = ( 0g ` G )
Assertion isgrpid2 
|- ( G e. Grp -> ( ( Z e. B /\ ( Z .+ Z ) = Z ) <-> .0. = Z ) )

Proof

Step Hyp Ref Expression
1 grpinveu.b 
 |-  B = ( Base ` G )
2 grpinveu.p 
 |-  .+ = ( +g ` G )
3 grpinveu.o 
 |-  .0. = ( 0g ` G )
4 1 2 3 grpid 
 |-  ( ( G e. Grp /\ Z e. B ) -> ( ( Z .+ Z ) = Z <-> .0. = Z ) )
5 4 biimpd 
 |-  ( ( G e. Grp /\ Z e. B ) -> ( ( Z .+ Z ) = Z -> .0. = Z ) )
6 5 expimpd 
 |-  ( G e. Grp -> ( ( Z e. B /\ ( Z .+ Z ) = Z ) -> .0. = Z ) )
7 1 3 grpidcl 
 |-  ( G e. Grp -> .0. e. B )
8 1 2 3 grplid 
 |-  ( ( G e. Grp /\ .0. e. B ) -> ( .0. .+ .0. ) = .0. )
9 7 8 mpdan 
 |-  ( G e. Grp -> ( .0. .+ .0. ) = .0. )
10 7 9 jca 
 |-  ( G e. Grp -> ( .0. e. B /\ ( .0. .+ .0. ) = .0. ) )
11 eleq1 
 |-  ( .0. = Z -> ( .0. e. B <-> Z e. B ) )
12 id 
 |-  ( .0. = Z -> .0. = Z )
13 12 12 oveq12d 
 |-  ( .0. = Z -> ( .0. .+ .0. ) = ( Z .+ Z ) )
14 13 12 eqeq12d 
 |-  ( .0. = Z -> ( ( .0. .+ .0. ) = .0. <-> ( Z .+ Z ) = Z ) )
15 11 14 anbi12d 
 |-  ( .0. = Z -> ( ( .0. e. B /\ ( .0. .+ .0. ) = .0. ) <-> ( Z e. B /\ ( Z .+ Z ) = Z ) ) )
16 10 15 syl5ibcom 
 |-  ( G e. Grp -> ( .0. = Z -> ( Z e. B /\ ( Z .+ Z ) = Z ) ) )
17 6 16 impbid 
 |-  ( G e. Grp -> ( ( Z e. B /\ ( Z .+ Z ) = Z ) <-> .0. = Z ) )