Theorem ply1sclcl

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

	Ref	Expression
Hypotheses	ply1scl.p	$$⊢P={\mathrm{Poly}}_1\left(R\right)$$
	ply1scl.a	$$⊢A=\mathrm{algSc}\left(P\right)$$
	coe1scl.k	$$⊢K={\mathrm{Base}}_R$$
	ply1sclf.b	$$⊢B={\mathrm{Base}}_P$$
Assertion	ply1sclcl	$$⊢\left(R\in \mathrm{Ring}\wedge S\in K\right)\to A\left(S\right)\in B$$

Proof

Step	Hyp	Ref	Expression
1		ply1scl.p	$$⊢P={\mathrm{Poly}}_1\left(R\right)$$
2		ply1scl.a	$$⊢A=\mathrm{algSc}\left(P\right)$$
3		coe1scl.k	$$⊢K={\mathrm{Base}}_R$$
4		ply1sclf.b	$$⊢B={\mathrm{Base}}_P$$
5	1 2 3 4	ply1sclf	$$⊢R\in \mathrm{Ring}\to A:K⟶B$$
6	5	ffvelcdmda	$$⊢\left(R\in \mathrm{Ring}\wedge S\in K\right)\to A\left(S\right)\in B$$