Theorem ply1sclcl

Description: The value of the algebra scalar lifting function for (univariate) polynomials applied to a scalar results in a constant polynomial. (Contributed by AV, 27-Nov-2019)

	Ref	Expression
Hypotheses	ply1scl.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	ply1scl.a	⊢ 𝐴 = ( algSc ‘ 𝑃 )
	coe1scl.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
	ply1sclf.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
Assertion	ply1sclcl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾 ) → ( 𝐴 ‘ 𝑆 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ply1scl.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2		ply1scl.a	⊢ 𝐴 = ( algSc ‘ 𝑃 )
3		coe1scl.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
4		ply1sclf.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
5	1 2 3 4	ply1sclf	⊢ ( 𝑅 ∈ Ring → 𝐴 : 𝐾 ⟶ 𝐵 )
6	5	ffvelcdmda	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾 ) → ( 𝐴 ‘ 𝑆 ) ∈ 𝐵 )