Theorem prstcleval

Description: Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024) (Proof shortened by AV, 12-Nov-2024)

	Ref	Expression
Hypotheses	prstcnid.c	$$⊢\phi \to C=\mathrm{ProsetToCat}\left(K\right)$$
	prstcnid.k	$$⊢\phi \to K\in \mathrm{Proset}$$
	prstcle.l	$$⊢\phi \to \underset{\dot{}}{\le }={\le }_K$$
Assertion	prstcleval	$$⊢\phi \to \underset{\dot{}}{\le }={\le }_C$$

Proof

Step	Hyp	Ref	Expression
1		prstcnid.c	$$⊢\phi \to C=\mathrm{ProsetToCat}\left(K\right)$$
2		prstcnid.k	$$⊢\phi \to K\in \mathrm{Proset}$$
3		prstcle.l	$$⊢\phi \to \underset{\dot{}}{\le }={\le }_K$$
4		pleid	$$⊢\mathrm{le}=\mathrm{Slot}{\le }_{\mathrm{ndx}}$$
5		slotsdifplendx2	$$⊢\left({\le }_{\mathrm{ndx}}\ne \mathrm{comp}\left(\mathrm{ndx}\right)\wedge {\le }_{\mathrm{ndx}}\ne \mathrm{Hom}\left(\mathrm{ndx}\right)\right)$$
6	5	simpli	$$⊢{\le }_{\mathrm{ndx}}\ne \mathrm{comp}\left(\mathrm{ndx}\right)$$
7	5	simpri	$$⊢{\le }_{\mathrm{ndx}}\ne \mathrm{Hom}\left(\mathrm{ndx}\right)$$
8	1 2 4 6 7	prstcnid	$$⊢\phi \to {\le }_K={\le }_C$$
9	3 8	eqtrd	$$⊢\phi \to \underset{\dot{}}{\le }={\le }_C$$