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Metamath Proof Explorer

Theorem ressmplbas

Description: A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015)

Ref Expression
Hypotheses ressmpl.s ⊢ 𝑆 = ( 𝐼 mPoly 𝑅 )
ressmpl.h ⊢ 𝐻 = ( 𝑅 ↾s 𝑇 )
ressmpl.u ⊢ 𝑈 = ( 𝐼 mPoly 𝐻 )
ressmpl.b ⊢ 𝐵 = ( Base ‘ 𝑈 )
ressmpl.1 ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
ressmpl.2 ⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) )
ressmpl.p ⊢ 𝑃 = ( 𝑆 ↾s 𝐵 )
Assertion ressmplbas ( 𝜑 → 𝐵 = ( Base ‘ 𝑃 ) )

Proof

Step Hyp Ref Expression
1 ressmpl.s ⊢ 𝑆 = ( 𝐼 mPoly 𝑅 )
2 ressmpl.h ⊢ 𝐻 = ( 𝑅 ↾s 𝑇 )
3 ressmpl.u ⊢ 𝑈 = ( 𝐼 mPoly 𝐻 )
4 ressmpl.b ⊢ 𝐵 = ( Base ‘ 𝑈 )
5 ressmpl.1 ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
6 ressmpl.2 ⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) )
7 ressmpl.p ⊢ 𝑃 = ( 𝑆 ↾s 𝐵 )
8 eqid ⊢ ( 𝐼 mPwSer 𝐻 ) = ( 𝐼 mPwSer 𝐻 )
9 eqid ⊢ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝐻 ) )
10 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
11 1 2 3 4 5 6 8 9 10 ressmplbas2 ⊢ ( 𝜑 → 𝐵 = ( ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∩ ( Base ‘ 𝑆 ) ) )
12 inss2 ⊢ ( ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∩ ( Base ‘ 𝑆 ) ) ⊆ ( Base ‘ 𝑆 )
13 11 12 eqsstrdi ⊢ ( 𝜑 → 𝐵 ⊆ ( Base ‘ 𝑆 ) )
14 7 10 ressbas2 ⊢ ( 𝐵 ⊆ ( Base ‘ 𝑆 ) → 𝐵 = ( Base ‘ 𝑃 ) )
15 13 14 syl ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑃 ) )