Theorem evlscl

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

	Ref	Expression
Hypotheses	evlscl.q	$$⊢Q=\left(I\mathrm{evalSub}R\right)\left(S\right)$$
	evlscl.p	$$⊢P=I\mathrm{mPoly}U$$
	evlscl.u	$$⊢U=R{\upharpoonright }_{𝑠}S$$
	evlscl.b	$$⊢B={\mathrm{Base}}_P$$
	evlscl.k	$$⊢K={\mathrm{Base}}_R$$
	evlscl.i	$$⊢\phi \to I\in V$$
	evlscl.r	$$⊢\phi \to R\in \mathrm{CRing}$$
	evlscl.s	$$⊢\phi \to S\in \mathrm{SubRing}\left(R\right)$$
	evlscl.f	$$⊢\phi \to F\in B$$
	evlscl.a	$$⊢\phi \to A\in \left(K^I\right)$$
Assertion	evlscl	$$⊢\phi \to Q\left(F\right)\left(A\right)\in K$$

Proof

Step	Hyp	Ref	Expression
1		evlscl.q	$$⊢Q=\left(I\mathrm{evalSub}R\right)\left(S\right)$$
2		evlscl.p	$$⊢P=I\mathrm{mPoly}U$$
3		evlscl.u	$$⊢U=R{\upharpoonright }_{𝑠}S$$
4		evlscl.b	$$⊢B={\mathrm{Base}}_P$$
5		evlscl.k	$$⊢K={\mathrm{Base}}_R$$
6		evlscl.i	$$⊢\phi \to I\in V$$
7		evlscl.r	$$⊢\phi \to R\in \mathrm{CRing}$$
8		evlscl.s	$$⊢\phi \to S\in \mathrm{SubRing}\left(R\right)$$
9		evlscl.f	$$⊢\phi \to F\in B$$
10		evlscl.a	$$⊢\phi \to A\in \left(K^I\right)$$
11		eqid	$$⊢R{\uparrow }_{𝑠}\left(K^I\right)=R{\uparrow }_{𝑠}\left(K^I\right)$$
12		eqid	$$⊢{\mathrm{Base}}_{\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)}={\mathrm{Base}}_{\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)}$$
13		ovexd	$$⊢\phi \to K^I\in \mathrm{V}$$
14	1 2 3 11 5	evlsrhm	$$⊢\left(I\in V\wedge R\in \mathrm{CRing}\wedge S\in \mathrm{SubRing}\left(R\right)\right)\to Q\in \left(P\mathrm{RingHom}\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)\right)$$
15	6 7 8 14	syl3anc	$$⊢\phi \to Q\in \left(P\mathrm{RingHom}\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)\right)$$
16	4 12	rhmf	$$⊢Q\in \left(P\mathrm{RingHom}\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)\right)\to Q:B⟶{\mathrm{Base}}_{\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)}$$
17	15 16	syl	$$⊢\phi \to Q:B⟶{\mathrm{Base}}_{\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)}$$
18	17 9	ffvelcdmd	$$⊢\phi \to Q\left(F\right)\in {\mathrm{Base}}_{\left(R{\uparrow }_{𝑠}\left(K^I\right)\right)}$$
19	11 5 12 7 13 18	pwselbas	$$⊢\phi \to Q\left(F\right):K^I⟶K$$
20	19 10	ffvelcdmd	$$⊢\phi \to Q\left(F\right)\left(A\right)\in K$$