Theorem thlval 21874

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 3478	. 2 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2		thlval.k	. . 3 ⊢ 𝐾 = (toHL‘𝑊)
3		fveq2 6885	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = (ClSubSp‘𝑊))
4		thlval.c	. . . . . . . 8 ⊢ 𝐶 = (ClSubSp‘𝑊)
5	3, 4	eqtr4di 2818	. . . . . . 7 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = 𝐶)
6	5	fveq2d 6889	. . . . . 6 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = (toInc‘𝐶))
7		thlval.i	. . . . . 6 ⊢ 𝐼 = (toInc‘𝐶)
8	6, 7	eqtr4di 2818	. . . . 5 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = 𝐼)
9		fveq2 6885	. . . . . . 7 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = (ocv‘𝑊))
10		thlval.o	. . . . . . 7 ⊢ ⊥ = (ocv‘𝑊)
11	9, 10	eqtr4di 2818	. . . . . 6 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = ⊥ )
12	11	opeq2d 4847	. . . . 5 ⊢ (ℎ = 𝑊 → ⟨(oc‘ndx), (ocv‘ℎ)⟩ = ⟨(oc‘ndx), ⊥ ⟩)
13	8, 12	oveq12d 7434	. . . 4 ⊢ (ℎ = 𝑊 → ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩) = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
14		df-thl 21844	. . . 4 ⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩))
15		ovex 7449	. . . 4 ⊢ (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩) ∈ V
16	13, 14, 15	fvmpt 6993	. . 3 ⊢ (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
17	2, 16	eqtrid 2812	. 2 ⊢ (𝑊 ∈ V → 𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
18	1, 17	syl 18	1 ⊢ (𝑊 ∈ 𝑉 → 𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ⟨cop 4597  ‘cfv 6540  (class class class)co 7416   sSet csts 17240  ndxcnx 17270  occoc 17335  toInccipo 18600  ocvcocv 21839  ClSubSpccss 21840  toHLcthl 21841

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7419  df-thl 21844

This theorem is used by:  thlbas  21875  thlle  21876  thloc  21878