Theorem evlcl

Description: A polynomial over the ring R evaluates to an element in R . (Contributed by SN, 12-Mar-2025)

	Ref	Expression
Hypotheses	evlcl.q	$$⊢Q=I\mathrm{eval}R$$
	evlcl.p	$$⊢P=I\mathrm{mPoly}R$$
	evlcl.b	$$⊢B={\mathrm{Base}}_P$$
	evlcl.k	$$⊢K={\mathrm{Base}}_R$$
	evlcl.i	$$⊢\phi \to I\in V$$
	evlcl.r	$$⊢\phi \to R\in \mathrm{CRing}$$
	evlcl.f	$$⊢\phi \to F\in B$$
	evlcl.a	$$⊢\phi \to A\in \left(K^I\right)$$
Assertion	evlcl	$$⊢\phi \to Q\left(F\right)\left(A\right)\in K$$

Proof

Step	Hyp	Ref	Expression
1		evlcl.q	$$⊢Q=I\mathrm{eval}R$$
2		evlcl.p	$$⊢P=I\mathrm{mPoly}R$$
3		evlcl.b	$$⊢B={\mathrm{Base}}_P$$
4		evlcl.k	$$⊢K={\mathrm{Base}}_R$$
5		evlcl.i	$$⊢\phi \to I\in V$$
6		evlcl.r	$$⊢\phi \to R\in \mathrm{CRing}$$
7		evlcl.f	$$⊢\phi \to F\in B$$
8		evlcl.a	$$⊢\phi \to A\in \left(K^I\right)$$
9		eqid	$$⊢R{\uparrow }_{𝑠}\left(K^I\right)=R{\uparrow }_{𝑠}\left(K^I\right)$$
10		eqid	$$⊢{\mathrm{Base}}_{\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)}={\mathrm{Base}}_{\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)}$$
11		ovexd	$$⊢\phi \to K^I\in \mathrm{V}$$
12	1 4 2 9	evlrhm	$$⊢\left(I\in V\wedge R\in \mathrm{CRing}\right)\to Q\in \left(P\mathrm{RingHom}\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)\right)$$
13	5 6 12	syl2anc	$$⊢\phi \to Q\in \left(P\mathrm{RingHom}\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)\right)$$
14	3 10	rhmf	$$⊢Q\in \left(P\mathrm{RingHom}\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)\right)\to Q:B⟶{\mathrm{Base}}_{\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)}$$
15	13 14	syl	$$⊢\phi \to Q:B⟶{\mathrm{Base}}_{\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)}$$
16	15 7	ffvelcdmd	$$⊢\phi \to Q\left(F\right)\in {\mathrm{Base}}_{\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)}$$
17	9 4 10 6 11 16	pwselbas	$$⊢\phi \to Q\left(F\right):K^I⟶K$$
18	17 8	ffvelcdmd	$$⊢\phi \to Q\left(F\right)\left(A\right)\in K$$