Theorem selvcllem3

Description: The third argument passed to evalSub is in the domain. (Contributed by SN, 15-Dec-2023)

	Ref	Expression
Hypotheses	selvcllem2.u	$$⊢U=I\mathrm{mPoly}R$$
	selvcllem2.t	$$⊢T=J\mathrm{mPoly}U$$
	selvcllem2.c	$$⊢C=\mathrm{algSc}\left(T\right)$$
	selvcllem2.d	$$⊢D=C\circ \mathrm{algSc}\left(U\right)$$
	selvcllem2.i	$$⊢\phi \to I\in V$$
	selvcllem2.j	$$⊢\phi \to J\in W$$
	selvcllem2.r	$$⊢\phi \to R\in \mathrm{CRing}$$
Assertion	selvcllem3	$$⊢\phi \to \mathrm{ran}D\in \mathrm{SubRing}\left(T\right)$$

Proof

Step	Hyp	Ref	Expression
1		selvcllem2.u	$$⊢U=I\mathrm{mPoly}R$$
2		selvcllem2.t	$$⊢T=J\mathrm{mPoly}U$$
3		selvcllem2.c	$$⊢C=\mathrm{algSc}\left(T\right)$$
4		selvcllem2.d	$$⊢D=C\circ \mathrm{algSc}\left(U\right)$$
5		selvcllem2.i	$$⊢\phi \to I\in V$$
6		selvcllem2.j	$$⊢\phi \to J\in W$$
7		selvcllem2.r	$$⊢\phi \to R\in \mathrm{CRing}$$
8	1 2 3 4 5 6 7	selvcllem2	$$⊢\phi \to D\in \left(R\mathrm{RingHom}T\right)$$
9		rnrhmsubrg	$$⊢D\in \left(R\mathrm{RingHom}T\right)\to \mathrm{ran}D\in \mathrm{SubRing}\left(T\right)$$
10	8 9	syl	$$⊢\phi \to \mathrm{ran}D\in \mathrm{SubRing}\left(T\right)$$