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Metamath Proof Explorer

Theorem mgmlrid

Description: The identity element of a magma, if it exists, is a left and right identity. (Contributed by Mario Carneiro, 27-Dec-2014)

Ref Expression
Hypotheses ismgmid.b 
|- B = ( Base ` G )
ismgmid.o 
|- .0. = ( 0g ` G )
ismgmid.p 
|- .+ = ( +g ` G )
mgmidcl.e 
|- ( ph -> E. e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) )
Assertion mgmlrid 
|- ( ( ph /\ X e. B ) -> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) )

Proof

Step Hyp Ref Expression
1 ismgmid.b 
 |-  B = ( Base ` G )
2 ismgmid.o 
 |-  .0. = ( 0g ` G )
3 ismgmid.p 
 |-  .+ = ( +g ` G )
4 mgmidcl.e 
 |-  ( ph -> E. e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) )
5 eqid 
 |-  .0. = .0.
6 1 2 3 4 ismgmid 
 |-  ( ph -> ( ( .0. e. B /\ A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) <-> .0. = .0. ) )
7 5 6 mpbiri 
 |-  ( ph -> ( .0. e. B /\ A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) )
8 7 simprd 
 |-  ( ph -> A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
9 oveq2 
 |-  ( x = X -> ( .0. .+ x ) = ( .0. .+ X ) )
10 id 
 |-  ( x = X -> x = X )
11 9 10 eqeq12d 
 |-  ( x = X -> ( ( .0. .+ x ) = x <-> ( .0. .+ X ) = X ) )
12 oveq1 
 |-  ( x = X -> ( x .+ .0. ) = ( X .+ .0. ) )
13 12 10 eqeq12d 
 |-  ( x = X -> ( ( x .+ .0. ) = x <-> ( X .+ .0. ) = X ) )
14 11 13 anbi12d 
 |-  ( x = X -> ( ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) <-> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) ) )
15 14 rspccva 
 |-  ( ( A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) /\ X e. B ) -> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) )
16 8 15 sylan 
 |-  ( ( ph /\ X e. B ) -> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) )