Description: The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015) (Revised by Mario Carneiro, 2-Oct-2015)
Ref | Expression | ||
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Hypotheses | mplvsca.p | |- P = ( I mPoly R ) |
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mplvsca.n | |- .xb = ( .s ` P ) |
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mplvsca.k | |- K = ( Base ` R ) |
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mplvsca.b | |- B = ( Base ` P ) |
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mplvsca.m | |- .x. = ( .r ` R ) |
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mplvsca.d | |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } |
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mplvsca.x | |- ( ph -> X e. K ) |
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mplvsca.f | |- ( ph -> F e. B ) |
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Assertion | mplvsca | |- ( ph -> ( X .xb F ) = ( ( D X. { X } ) oF .x. F ) ) |
Step | Hyp | Ref | Expression |
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1 | mplvsca.p | |- P = ( I mPoly R ) |
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2 | mplvsca.n | |- .xb = ( .s ` P ) |
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3 | mplvsca.k | |- K = ( Base ` R ) |
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4 | mplvsca.b | |- B = ( Base ` P ) |
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5 | mplvsca.m | |- .x. = ( .r ` R ) |
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6 | mplvsca.d | |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } |
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7 | mplvsca.x | |- ( ph -> X e. K ) |
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8 | mplvsca.f | |- ( ph -> F e. B ) |
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9 | eqid | |- ( I mPwSer R ) = ( I mPwSer R ) |
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10 | 1 9 2 | mplvsca2 | |- .xb = ( .s ` ( I mPwSer R ) ) |
11 | eqid | |- ( Base ` ( I mPwSer R ) ) = ( Base ` ( I mPwSer R ) ) |
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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 ) ) |