Theorem thlval 20842

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 3515	. 2 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2		thlval.k	. . 3 ⊢ 𝐾 = (toHL‘𝑊)
3		fveq2 6673	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = (ClSubSp‘𝑊))
4		thlval.c	. . . . . . . 8 ⊢ 𝐶 = (ClSubSp‘𝑊)
5	3, 4	syl6eqr 2877	. . . . . . 7 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = 𝐶)
6	5	fveq2d 6677	. . . . . 6 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = (toInc‘𝐶))
7		thlval.i	. . . . . 6 ⊢ 𝐼 = (toInc‘𝐶)
8	6, 7	syl6eqr 2877	. . . . 5 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = 𝐼)
9		fveq2 6673	. . . . . . 7 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = (ocv‘𝑊))
10		thlval.o	. . . . . . 7 ⊢ ⊥ = (ocv‘𝑊)
11	9, 10	syl6eqr 2877	. . . . . 6 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = ⊥ )
12	11	opeq2d 4813	. . . . 5 ⊢ (ℎ = 𝑊 → ⟨(oc‘ndx), (ocv‘ℎ)⟩ = ⟨(oc‘ndx), ⊥ ⟩)
13	8, 12	oveq12d 7177	. . . 4 ⊢ (ℎ = 𝑊 → ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩) = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
14		df-thl 20812	. . . 4 ⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩))
15		ovex 7192	. . . 4 ⊢ (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩) ∈ V
16	13, 14, 15	fvmpt 6771	. . 3 ⊢ (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
17	2, 16	syl5eq 2871	. 2 ⊢ (𝑊 ∈ V → 𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
18	1, 17	syl 17	1 ⊢ (𝑊 ∈ 𝑉 → 𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113  Vcvv 3497  ⟨cop 4576  ‘cfv 6358  (class class class)co 7159  ndxcnx 16483   sSet csts 16484  occoc 16576  toInccipo 17764  ocvcocv 20807  ClSubSpccss 20808  toHLcthl 20809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fv 6366  df-ov 7162  df-thl 20812

This theorem is referenced by:  thlbas  20843  thlle  20844  thloc  20846