Theorem hlhilslem

Description: Lemma for hlhilsbase etc. (Contributed by Mario Carneiro, 28-Jun-2015) (Revised by AV, 6-Nov-2024)

	Ref	Expression
Hypotheses	hlhilslem.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	hlhilslem.e	$$⊢E=\mathrm{EDRing}\left(K\right)\left(W\right)$$
	hlhilslem.u	$$⊢U=\mathrm{HLHil}\left(K\right)\left(W\right)$$
	hlhilslem.r	$$⊢R=\mathrm{Scalar}\left(U\right)$$
	hlhilslem.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	hlhilslem.f	$$⊢F=\mathrm{Slot}F\left(\mathrm{ndx}\right)$$
	hlhilslem.n	$$⊢F\left(\mathrm{ndx}\right)\ne {\mathrm{*}}_{\mathrm{ndx}}$$
	hlhilslem.c	$$⊢C=F\left(E\right)$$
Assertion	hlhilslem	$$⊢\phi \to C=F\left(R\right)$$

Proof

Step	Hyp	Ref	Expression
1		hlhilslem.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		hlhilslem.e	$$⊢E=\mathrm{EDRing}\left(K\right)\left(W\right)$$
3		hlhilslem.u	$$⊢U=\mathrm{HLHil}\left(K\right)\left(W\right)$$
4		hlhilslem.r	$$⊢R=\mathrm{Scalar}\left(U\right)$$
5		hlhilslem.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
6		hlhilslem.f	$$⊢F=\mathrm{Slot}F\left(\mathrm{ndx}\right)$$
7		hlhilslem.n	$$⊢F\left(\mathrm{ndx}\right)\ne {\mathrm{*}}_{\mathrm{ndx}}$$
8		hlhilslem.c	$$⊢C=F\left(E\right)$$
9	6 7	setsnid	$$⊢F\left(E\right)=F\left(E\mathrm{sSet}⟨{\mathrm{*}}_{\mathrm{ndx}},\mathrm{HGMap}\left(K\right)\left(W\right)⟩\right)$$
10	8 9	eqtri	$$⊢C=F\left(E\mathrm{sSet}⟨{\mathrm{*}}_{\mathrm{ndx}},\mathrm{HGMap}\left(K\right)\left(W\right)⟩\right)$$
11		eqid	$$⊢\mathrm{HGMap}\left(K\right)\left(W\right)=\mathrm{HGMap}\left(K\right)\left(W\right)$$
12		eqid	$$⊢E\mathrm{sSet}⟨{\mathrm{*}}_{\mathrm{ndx}},\mathrm{HGMap}\left(K\right)\left(W\right)⟩=E\mathrm{sSet}⟨{\mathrm{*}}_{\mathrm{ndx}},\mathrm{HGMap}\left(K\right)\left(W\right)⟩$$
13	1 3 5 2 11 12	hlhilsca	$$⊢\phi \to E\mathrm{sSet}⟨{\mathrm{*}}_{\mathrm{ndx}},\mathrm{HGMap}\left(K\right)\left(W\right)⟩=\mathrm{Scalar}\left(U\right)$$
14	13 4	eqtr4di	$$⊢\phi \to E\mathrm{sSet}⟨{\mathrm{*}}_{\mathrm{ndx}},\mathrm{HGMap}\left(K\right)\left(W\right)⟩=R$$
15	14	fveq2d	$$⊢\phi \to F\left(E\mathrm{sSet}⟨{\mathrm{*}}_{\mathrm{ndx}},\mathrm{HGMap}\left(K\right)\left(W\right)⟩\right)=F\left(R\right)$$
16	10 15	syl5eq	$$⊢\phi \to C=F\left(R\right)$$