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 = ( LHyp ‘ 𝐾 )
	hlhilslem.e	⊢ 𝐸 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
	hlhilslem.u	⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
	hlhilslem.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
	hlhilslem.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hlhilslem.f	⊢ 𝐹 = Slot ( 𝐹 ‘ ndx )
	hlhilslem.n	⊢ ( 𝐹 ‘ ndx ) ≠ ( *𝑟 ‘ ndx )
	hlhilslem.c	⊢ 𝐶 = ( 𝐹 ‘ 𝐸 )
Assertion	hlhilslem	⊢ ( 𝜑 → 𝐶 = ( 𝐹 ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		hlhilslem.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hlhilslem.e	⊢ 𝐸 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
3		hlhilslem.u	⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
4		hlhilslem.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
5		hlhilslem.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6		hlhilslem.f	⊢ 𝐹 = Slot ( 𝐹 ‘ ndx )
7		hlhilslem.n	⊢ ( 𝐹 ‘ ndx ) ≠ ( *𝑟 ‘ ndx )
8		hlhilslem.c	⊢ 𝐶 = ( 𝐹 ‘ 𝐸 )
9	6 7	setsnid	⊢ ( 𝐹 ‘ 𝐸 ) = ( 𝐹 ‘ ( 𝐸 sSet ⟨ ( *𝑟 ‘ ndx ) , ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) ⟩ ) )
10	8 9	eqtri	⊢ 𝐶 = ( 𝐹 ‘ ( 𝐸 sSet ⟨ ( *𝑟 ‘ ndx ) , ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) ⟩ ) )
11		eqid	⊢ ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
12		eqid	⊢ ( 𝐸 sSet ⟨ ( *𝑟 ‘ ndx ) , ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) ⟩ ) = ( 𝐸 sSet ⟨ ( *𝑟 ‘ ndx ) , ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) ⟩ )
13	1 3 5 2 11 12	hlhilsca	⊢ ( 𝜑 → ( 𝐸 sSet ⟨ ( *𝑟 ‘ ndx ) , ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) ⟩ ) = ( Scalar ‘ 𝑈 ) )
14	13 4	eqtr4di	⊢ ( 𝜑 → ( 𝐸 sSet ⟨ ( *𝑟 ‘ ndx ) , ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) ⟩ ) = 𝑅 )
15	14	fveq2d	⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐸 sSet ⟨ ( *𝑟 ‘ ndx ) , ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) ⟩ ) ) = ( 𝐹 ‘ 𝑅 ) )
16	10 15	syl5eq	⊢ ( 𝜑 → 𝐶 = ( 𝐹 ‘ 𝑅 ) )