Theorem evls1dm

Description: The domain of the subring polynomial evaluation function. (Contributed by Thierry Arnoux, 9-Feb-2025)

	Ref	Expression
Hypotheses	evls1fn.o	$$⊢O=R{\mathrm{evalSub}}_{1}S$$
	evls1fn.p	$$⊢P={\mathrm{Poly}}_1\left(R{\upharpoonright }_{𝑠}S\right)$$
	evls1fn.u	$$⊢U={\mathrm{Base}}_P$$
	evls1fn.1	$$⊢\phi \to R\in \mathrm{CRing}$$
	evls1fn.2	$$⊢\phi \to S\in \mathrm{SubRing}\left(R\right)$$
Assertion	evls1dm	$$⊢\phi \to \mathrm{dom}O=U$$

Proof

Step	Hyp	Ref	Expression
1		evls1fn.o	$$⊢O=R{\mathrm{evalSub}}_{1}S$$
2		evls1fn.p	$$⊢P={\mathrm{Poly}}_1\left(R{\upharpoonright }_{𝑠}S\right)$$
3		evls1fn.u	$$⊢U={\mathrm{Base}}_P$$
4		evls1fn.1	$$⊢\phi \to R\in \mathrm{CRing}$$
5		evls1fn.2	$$⊢\phi \to S\in \mathrm{SubRing}\left(R\right)$$
6		eqid	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_R$$
7		eqid	$$⊢R{\uparrow }_{𝑠}{\mathrm{Base}}_R=R{\uparrow }_{𝑠}{\mathrm{Base}}_R$$
8		eqid	$$⊢R{\upharpoonright }_{𝑠}S=R{\upharpoonright }_{𝑠}S$$
9	1 6 7 8 2	evls1rhm	$$⊢\left(R\in \mathrm{CRing}\wedge S\in \mathrm{SubRing}\left(R\right)\right)\to O\in \left(P\mathrm{RingHom}\left(R{\uparrow }_{𝑠}{\mathrm{Base}}_R\right)\right)$$
10	4 5 9	syl2anc	$$⊢\phi \to O\in \left(P\mathrm{RingHom}\left(R{\uparrow }_{𝑠}{\mathrm{Base}}_R\right)\right)$$
11		eqid	$$⊢{\mathrm{Base}}_{\left(R{\uparrow }_{𝑠}{\mathrm{Base}}_R\right)}={\mathrm{Base}}_{\left(R{\uparrow }_{𝑠}{\mathrm{Base}}_R\right)}$$
12	3 11	rhmf	$$⊢O\in \left(P\mathrm{RingHom}\left(R{\uparrow }_{𝑠}{\mathrm{Base}}_R\right)\right)\to O:U⟶{\mathrm{Base}}_{\left(R{\uparrow }_{𝑠}{\mathrm{Base}}_R\right)}$$
13	10 12	syl	$$⊢\phi \to O:U⟶{\mathrm{Base}}_{\left(R{\uparrow }_{𝑠}{\mathrm{Base}}_R\right)}$$
14	13	fdmd	$$⊢\phi \to \mathrm{dom}O=U$$