Theorem thlval 21810

Description: Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.)

Hypotheses

Ref	Expression
thlval.k	⊢ 𝐾 = (toHL‘𝑊)
thlval.c	⊢ 𝐶 = (ClSubSp‘𝑊)
thlval.i	⊢ 𝐼 = (toInc‘𝐶)
thlval.o	⊢ ⊥ = (ocv‘𝑊)

Assertion

Ref	Expression
thlval	⊢ (𝑊 ∈ 𝑉 → 𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))

Proof of Theorem thlval

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3484	. 2 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2		thlval.k	. . 3 ⊢ 𝐾 = (toHL‘𝑊)
3		fveq2 6879	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = (ClSubSp‘𝑊))
4		thlval.c	. . . . . . . 8 ⊢ 𝐶 = (ClSubSp‘𝑊)
5	3, 4	eqtr4di 2822	. . . . . . 7 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = 𝐶)
6	5	fveq2d 6883	. . . . . 6 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = (toInc‘𝐶))
7		thlval.i	. . . . . 6 ⊢ 𝐼 = (toInc‘𝐶)
8	6, 7	eqtr4di 2822	. . . . 5 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = 𝐼)
9		fveq2 6879	. . . . . . 7 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = (ocv‘𝑊))
10		thlval.o	. . . . . . 7 ⊢ ⊥ = (ocv‘𝑊)
11	9, 10	eqtr4di 2822	. . . . . 6 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = ⊥ )
12	11	opeq2d 4846	. . . . 5 ⊢ (ℎ = 𝑊 → ⟨(oc‘ndx), (ocv‘ℎ)⟩ = ⟨(oc‘ndx), ⊥ ⟩)
13	8, 12	oveq12d 7426	. . . 4 ⊢ (ℎ = 𝑊 → ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩) = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
14		df-thl 21780	. . . 4 ⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩))
15		ovex 7441	. . . 4 ⊢ (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩) ∈ V
16	13, 14, 15	fvmpt 6987	. . 3 ⊢ (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
17	2, 16	eqtrid 2816	. 2 ⊢ (𝑊 ∈ V → 𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
18	1, 17	syl 18	1 ⊢ (𝑊 ∈ 𝑉 → 𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4597  ‘cfv 6533  (class class class)co 7408   sSet csts 17219  ndxcnx 17249  occoc 17314  toInccipo 18579  ocvcocv 21775  ClSubSpccss 21776  toHLcthl 21777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7411  df-thl 21780

This theorem is referenced by:  thlbas  21811  thlle  21812  thloc  21814