Theorem mplelbas

Description: Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015) (Revised by Mario Carneiro, 2-Oct-2015) (Revised by AV, 25-Jun-2019)

	Ref	Expression
Hypotheses	mplval.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
	mplval.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
	mplval.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
	mplval.z	⊢ 0 = ( 0g ‘ 𝑅 )
	mplbas.u	⊢ 𝑈 = ( Base ‘ 𝑃 )
Assertion	mplelbas	⊢ ( 𝑋 ∈ 𝑈 ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑋 finSupp 0 ) )

Proof

Step	Hyp	Ref	Expression
1		mplval.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
2		mplval.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
3		mplval.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
4		mplval.z	⊢ 0 = ( 0g ‘ 𝑅 )
5		mplbas.u	⊢ 𝑈 = ( Base ‘ 𝑃 )
6		breq1	⊢ ( 𝑓 = 𝑋 → ( 𝑓 finSupp 0 ↔ 𝑋 finSupp 0 ) )
7	1 2 3 4 5	mplbas	⊢ 𝑈 = { 𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }
8	6 7	elrab2	⊢ ( 𝑋 ∈ 𝑈 ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑋 finSupp 0 ) )