This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem mplbas

Description: Base set of the set of multivariate polynomials. (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 mplbas 𝑈 = { 𝑓 ∈ 𝐵 ∣ 𝑓 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 ssrab2 ⊢ { 𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } ⊆ 𝐵
7 eqid ⊢ { 𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } = { 𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }
8 1 2 3 4 7 mplval ⊢ 𝑃 = ( 𝑆 ↾s { 𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } )
9 8 3 ressbas2 ⊢ ( { 𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } ⊆ 𝐵 → { 𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } = ( Base ‘ 𝑃 ) )
10 6 9 ax-mp ⊢ { 𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } = ( Base ‘ 𝑃 )
11 5 10 eqtr4i ⊢ 𝑈 = { 𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }