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Metamath Proof Explorer

Theorem grpid

Description: Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011)

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

Proof

Step Hyp Ref Expression
1 grpinveu.b 
 |-  B = ( Base ` G )
2 grpinveu.p 
 |-  .+ = ( +g ` G )
3 grpinveu.o 
 |-  .0. = ( 0g ` G )
4 eqcom 
 |-  ( .0. = X <-> X = .0. )
5 1 3 grpidcl 
 |-  ( G e. Grp -> .0. e. B )
6 1 2 grprcan 
 |-  ( ( G e. Grp /\ ( X e. B /\ .0. e. B /\ X e. B ) ) -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) )
7 6 3exp2 
 |-  ( G e. Grp -> ( X e. B -> ( .0. e. B -> ( X e. B -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) ) ) ) )
8 5 7 mpid 
 |-  ( G e. Grp -> ( X e. B -> ( X e. B -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) ) ) )
9 8 pm2.43d 
 |-  ( G e. Grp -> ( X e. B -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) ) )
10 9 imp 
 |-  ( ( G e. Grp /\ X e. B ) -> ( ( X .+ X ) = ( .0. .+ X ) <-> X = .0. ) )
11 1 2 3 grplid 
 |-  ( ( G e. Grp /\ X e. B ) -> ( .0. .+ X ) = X )
12 11 eqeq2d 
 |-  ( ( G e. Grp /\ X e. B ) -> ( ( X .+ X ) = ( .0. .+ X ) <-> ( X .+ X ) = X ) )
13 10 12 bitr3d 
 |-  ( ( G e. Grp /\ X e. B ) -> ( X = .0. <-> ( X .+ X ) = X ) )
14 4 13 bitr2id 
 |-  ( ( G e. Grp /\ X e. B ) -> ( ( X .+ X ) = X <-> .0. = X ) )