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 ) )