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Metamath Proof Explorer

Theorem ply1bas

Description: The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015) Remove hypothesis. (Revised by SN, 20-May-2025)

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

Proof

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