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Metamath Proof Explorer

Theorem ressply1bas

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

Ref Expression
Hypotheses ressply1.s ⊢ 𝑆 = ( Poly1 ‘ 𝑅 )
ressply1.h ⊢ 𝐻 = ( 𝑅 ↾s 𝑇 )
ressply1.u ⊢ 𝑈 = ( Poly1 ‘ 𝐻 )
ressply1.b ⊢ 𝐵 = ( Base ‘ 𝑈 )
ressply1.2 ⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) )
ressply1.p ⊢ 𝑃 = ( 𝑆 ↾s 𝐵 )
Assertion ressply1bas ( 𝜑 → 𝐵 = ( Base ‘ 𝑃 ) )

Proof

Step Hyp Ref Expression
1 ressply1.s ⊢ 𝑆 = ( Poly1 ‘ 𝑅 )
2 ressply1.h ⊢ 𝐻 = ( 𝑅 ↾s 𝑇 )
3 ressply1.u ⊢ 𝑈 = ( Poly1 ‘ 𝐻 )
4 ressply1.b ⊢ 𝐵 = ( Base ‘ 𝑈 )
5 ressply1.2 ⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) )
6 ressply1.p ⊢ 𝑃 = ( 𝑆 ↾s 𝐵 )
7 eqid ⊢ ( PwSer1 ‘ 𝐻 ) = ( PwSer1 ‘ 𝐻 )
8 eqid ⊢ ( Base ‘ ( PwSer1 ‘ 𝐻 ) ) = ( Base ‘ ( PwSer1 ‘ 𝐻 ) )
9 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
10 1 2 3 4 5 7 8 9 ressply1bas2 ⊢ ( 𝜑 → 𝐵 = ( ( Base ‘ ( PwSer1 ‘ 𝐻 ) ) ∩ ( Base ‘ 𝑆 ) ) )
11 inss2 ⊢ ( ( Base ‘ ( PwSer1 ‘ 𝐻 ) ) ∩ ( Base ‘ 𝑆 ) ) ⊆ ( Base ‘ 𝑆 )
12 10 11 eqsstrdi ⊢ ( 𝜑 → 𝐵 ⊆ ( Base ‘ 𝑆 ) )
13 6 9 ressbas2 ⊢ ( 𝐵 ⊆ ( Base ‘ 𝑆 ) → 𝐵 = ( Base ‘ 𝑃 ) )
14 12 13 syl ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑃 ) )