Metamath Proof Explorer

Theorem cpm2mvalel

Description: A (matrix) element of the result of an inverse matrix transformation. (Contributed by AV, 14-Dec-2019)

Ref Expression
Hypotheses cpm2mfval.i 
|- I = ( N cPolyMatToMat R )
cpm2mfval.s 
|- S = ( N ConstPolyMat R )
Assertion cpm2mvalel 
|- ( ( ( N e. Fin /\ R e. V /\ M e. S ) /\ ( X e. N /\ Y e. N ) ) -> ( X ( I ` M ) Y ) = ( ( coe1 ` ( X M Y ) ) ` 0 ) )

Proof

Step Hyp Ref Expression
1 cpm2mfval.i 
 |-  I = ( N cPolyMatToMat R )
2 cpm2mfval.s 
 |-  S = ( N ConstPolyMat R )
3 1 2 cpm2mval 
 |-  ( ( N e. Fin /\ R e. V /\ M e. S ) -> ( I ` M ) = ( x e. N , y e. N |-> ( ( coe1 ` ( x M y ) ) ` 0 ) ) )
4 3 adantr 
 |-  ( ( ( N e. Fin /\ R e. V /\ M e. S ) /\ ( X e. N /\ Y e. N ) ) -> ( I ` M ) = ( x e. N , y e. N |-> ( ( coe1 ` ( x M y ) ) ` 0 ) ) )
5 oveq12 
 |-  ( ( x = X /\ y = Y ) -> ( x M y ) = ( X M Y ) )
6 5 fveq2d 
 |-  ( ( x = X /\ y = Y ) -> ( coe1 ` ( x M y ) ) = ( coe1 ` ( X M Y ) ) )
7 6 fveq1d 
 |-  ( ( x = X /\ y = Y ) -> ( ( coe1 ` ( x M y ) ) ` 0 ) = ( ( coe1 ` ( X M Y ) ) ` 0 ) )
8 7 adantl 
 |-  ( ( ( ( N e. Fin /\ R e. V /\ M e. S ) /\ ( X e. N /\ Y e. N ) ) /\ ( x = X /\ y = Y ) ) -> ( ( coe1 ` ( x M y ) ) ` 0 ) = ( ( coe1 ` ( X M Y ) ) ` 0 ) )
9 simprl 
 |-  ( ( ( N e. Fin /\ R e. V /\ M e. S ) /\ ( X e. N /\ Y e. N ) ) -> X e. N )
10 simprr 
 |-  ( ( ( N e. Fin /\ R e. V /\ M e. S ) /\ ( X e. N /\ Y e. N ) ) -> Y e. N )
11 fvexd 
 |-  ( ( ( N e. Fin /\ R e. V /\ M e. S ) /\ ( X e. N /\ Y e. N ) ) -> ( ( coe1 ` ( X M Y ) ) ` 0 ) e. _V )
12 4 8 9 10 11 ovmpod 
 |-  ( ( ( N e. Fin /\ R e. V /\ M e. S ) /\ ( X e. N /\ Y e. N ) ) -> ( X ( I ` M ) Y ) = ( ( coe1 ` ( X M Y ) ) ` 0 ) )