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Metamath Proof Explorer

Theorem psrvsca

Description: The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014)

Ref Expression
Hypotheses psrvsca.s 
|- S = ( I mPwSer R )
psrvsca.n 
|- .xb = ( .s ` S )
psrvsca.k 
|- K = ( Base ` R )
psrvsca.b 
|- B = ( Base ` S )
psrvsca.m 
|- .x. = ( .r ` R )
psrvsca.d 
|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
psrvsca.x 
|- ( ph -> X e. K )
psrvsca.y 
|- ( ph -> F e. B )
Assertion psrvsca 
|- ( ph -> ( X .xb F ) = ( ( D X. { X } ) oF .x. F ) )

Proof

Step Hyp Ref Expression
1 psrvsca.s 
 |-  S = ( I mPwSer R )
2 psrvsca.n 
 |-  .xb = ( .s ` S )
3 psrvsca.k 
 |-  K = ( Base ` R )
4 psrvsca.b 
 |-  B = ( Base ` S )
5 psrvsca.m 
 |-  .x. = ( .r ` R )
6 psrvsca.d 
 |-  D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
7 psrvsca.x 
 |-  ( ph -> X e. K )
8 psrvsca.y 
 |-  ( ph -> F e. B )
9 sneq 
 |-  ( x = X -> { x } = { X } )
10 9 xpeq2d 
 |-  ( x = X -> ( D X. { x } ) = ( D X. { X } ) )
11 10 oveq1d 
 |-  ( x = X -> ( ( D X. { x } ) oF .x. f ) = ( ( D X. { X } ) oF .x. f ) )
12 oveq2 
 |-  ( f = F -> ( ( D X. { X } ) oF .x. f ) = ( ( D X. { X } ) oF .x. F ) )
13 1 2 3 4 5 6 psrvscafval 
 |-  .xb = ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) )
14 ovex 
 |-  ( ( D X. { X } ) oF .x. F ) e. _V
15 11 12 13 14 ovmpo 
 |-  ( ( X e. K /\ F e. B ) -> ( X .xb F ) = ( ( D X. { X } ) oF .x. F ) )
16 7 8 15 syl2anc 
 |-  ( ph -> ( X .xb F ) = ( ( D X. { X } ) oF .x. F ) )