Theorem ply1bas

Description: The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015)

	Ref	Expression
Hypotheses	ply1val.1	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	ply1val.2	⊢ 𝑆 = ( PwSer1 ‘ 𝑅 )
	ply1bas.u	⊢ 𝑈 = ( Base ‘ 𝑃 )
Assertion	ply1bas	⊢ 𝑈 = ( Base ‘ ( 1o mPoly 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		ply1val.1	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2		ply1val.2	⊢ 𝑆 = ( PwSer1 ‘ 𝑅 )
3		ply1bas.u	⊢ 𝑈 = ( Base ‘ 𝑃 )
4		eqid	⊢ ( 1o mPoly 𝑅 ) = ( 1o mPoly 𝑅 )
5		eqid	⊢ ( 1o mPwSer 𝑅 ) = ( 1o mPwSer 𝑅 )
6		eqid	⊢ ( Base ‘ ( 1o mPoly 𝑅 ) ) = ( Base ‘ ( 1o mPoly 𝑅 ) )
7		eqid	⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
8	2 7 5	psr1bas2	⊢ ( Base ‘ 𝑆 ) = ( Base ‘ ( 1o mPwSer 𝑅 ) )
9	4 5 6 8	mplbasss	⊢ ( Base ‘ ( 1o mPoly 𝑅 ) ) ⊆ ( Base ‘ 𝑆 )
10	1 2	ply1val	⊢ 𝑃 = ( 𝑆 ↾s ( Base ‘ ( 1o mPoly 𝑅 ) ) )
11	10 7	ressbas2	⊢ ( ( Base ‘ ( 1o mPoly 𝑅 ) ) ⊆ ( Base ‘ 𝑆 ) → ( Base ‘ ( 1o mPoly 𝑅 ) ) = ( Base ‘ 𝑃 ) )
12	9 11	ax-mp	⊢ ( Base ‘ ( 1o mPoly 𝑅 ) ) = ( Base ‘ 𝑃 )
13	3 12	eqtr4i	⊢ 𝑈 = ( Base ‘ ( 1o mPoly 𝑅 ) )