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Metamath Proof Explorer

Theorem mplbasss

Description: The set of polynomials is a subset of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015) (Revised by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses mplval2.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
mplval2.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
mplval2.u ⊢ 𝑈 = ( Base ‘ 𝑃 )
mplbasss.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
Assertion mplbasss 𝑈 ⊆ 𝐵

Proof

Step Hyp Ref Expression
1 mplval2.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
2 mplval2.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
3 mplval2.u ⊢ 𝑈 = ( Base ‘ 𝑃 )
4 mplbasss.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
5 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
6 1 2 4 5 3 mplbas ⊢ 𝑈 = { 𝑓 ∈ 𝐵 ∣ 𝑓 finSupp ( 0g ‘ 𝑅 ) }
7 6 ssrab3 ⊢ 𝑈 ⊆ 𝐵