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Metamath Proof Explorer

Theorem mat1rhmcl

Description: The value of the ring homomorphism F is a matrix with dimension 1. (Contributed by AV, 22-Dec-2019)

Ref Expression
Hypotheses mat1rhmval.k 
|- K = ( Base ` R )
mat1rhmval.a 
|- A = ( { E } Mat R )
mat1rhmval.b 
|- B = ( Base ` A )
mat1rhmval.o 
|- O = <. E , E >.
mat1rhmval.f 
|- F = ( x e. K |-> { <. O , x >. } )
Assertion mat1rhmcl 
|- ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( F ` X ) e. B )

Proof

Step Hyp Ref Expression
1 mat1rhmval.k 
 |-  K = ( Base ` R )
2 mat1rhmval.a 
 |-  A = ( { E } Mat R )
3 mat1rhmval.b 
 |-  B = ( Base ` A )
4 mat1rhmval.o 
 |-  O = <. E , E >.
5 mat1rhmval.f 
 |-  F = ( x e. K |-> { <. O , x >. } )
6 2 1 4 mat1dimbas 
 |-  ( ( R e. Ring /\ E e. V /\ X e. K ) -> { <. O , X >. } e. ( Base ` A ) )
7 1 2 3 4 5 mat1rhmval 
 |-  ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( F ` X ) = { <. O , X >. } )
8 3 a1i 
 |-  ( ( R e. Ring /\ E e. V /\ X e. K ) -> B = ( Base ` A ) )
9 6 7 8 3eltr4d 
 |-  ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( F ` X ) e. B )