Metamath Proof Explorer


Theorem mplvsca

Description: The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015) (Revised by Mario Carneiro, 2-Oct-2015)

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

Proof

Step Hyp Ref Expression
1 mplvsca.p
 |-  P = ( I mPoly R )
2 mplvsca.n
 |-  .xb = ( .s ` P )
3 mplvsca.k
 |-  K = ( Base ` R )
4 mplvsca.b
 |-  B = ( Base ` P )
5 mplvsca.m
 |-  .x. = ( .r ` R )
6 mplvsca.d
 |-  D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
7 mplvsca.x
 |-  ( ph -> X e. K )
8 mplvsca.f
 |-  ( ph -> F e. B )
9 eqid
 |-  ( I mPwSer R ) = ( I mPwSer R )
10 1 9 2 mplvsca2
 |-  .xb = ( .s ` ( I mPwSer R ) )
11 eqid
 |-  ( Base ` ( I mPwSer R ) ) = ( Base ` ( I mPwSer R ) )
12 1 9 4 11 mplbasss
 |-  B C_ ( Base ` ( I mPwSer R ) )
13 12 8 sselid
 |-  ( ph -> F e. ( Base ` ( I mPwSer R ) ) )
14 9 10 3 11 5 6 7 13 psrvsca
 |-  ( ph -> ( X .xb F ) = ( ( D X. { X } ) oF .x. F ) )