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Metamath Proof Explorer

Theorem gsumvallem2

Description: Lemma for properties of the set of identities of G . The set of identities of a monoid is exactly the unique identity element. (Contributed by Mario Carneiro, 7-Dec-2014)

Ref Expression
Hypotheses gsumvallem2.b 
|- B = ( Base ` G )
gsumvallem2.z 
|- .0. = ( 0g ` G )
gsumvallem2.p 
|- .+ = ( +g ` G )
gsumvallem2.o 
|- O = { x e. B | A. y e. B ( ( x .+ y ) = y /\ ( y .+ x ) = y ) }
Assertion gsumvallem2 
|- ( G e. Mnd -> O = { .0. } )

Proof

Step Hyp Ref Expression
1 gsumvallem2.b 
 |-  B = ( Base ` G )
2 gsumvallem2.z 
 |-  .0. = ( 0g ` G )
3 gsumvallem2.p 
 |-  .+ = ( +g ` G )
4 gsumvallem2.o 
 |-  O = { x e. B | A. y e. B ( ( x .+ y ) = y /\ ( y .+ x ) = y ) }
5 1 2 3 4 mgmidsssn0 
 |-  ( G e. Mnd -> O C_ { .0. } )
6 1 2 mndidcl 
 |-  ( G e. Mnd -> .0. e. B )
7 1 3 2 mndlrid 
 |-  ( ( G e. Mnd /\ y e. B ) -> ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) )
8 7 ralrimiva 
 |-  ( G e. Mnd -> A. y e. B ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) )
9 oveq1 
 |-  ( x = .0. -> ( x .+ y ) = ( .0. .+ y ) )
10 9 eqeq1d 
 |-  ( x = .0. -> ( ( x .+ y ) = y <-> ( .0. .+ y ) = y ) )
11 10 ovanraleqv 
 |-  ( x = .0. -> ( A. y e. B ( ( x .+ y ) = y /\ ( y .+ x ) = y ) <-> A. y e. B ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) ) )
12 11 4 elrab2 
 |-  ( .0. e. O <-> ( .0. e. B /\ A. y e. B ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) ) )
13 6 8 12 sylanbrc 
 |-  ( G e. Mnd -> .0. e. O )
14 13 snssd 
 |-  ( G e. Mnd -> { .0. } C_ O )
15 5 14 eqssd 
 |-  ( G e. Mnd -> O = { .0. } )