Theorem thlval 21641

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 3458	. 2 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2		thlval.k	. . 3 ⊢ 𝐾 = (toHL‘𝑊)
3		fveq2 6831	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = (ClSubSp‘𝑊))
4		thlval.c	. . . . . . . 8 ⊢ 𝐶 = (ClSubSp‘𝑊)
5	3, 4	eqtr4di 2786	. . . . . . 7 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = 𝐶)
6	5	fveq2d 6835	. . . . . 6 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = (toInc‘𝐶))
7		thlval.i	. . . . . 6 ⊢ 𝐼 = (toInc‘𝐶)
8	6, 7	eqtr4di 2786	. . . . 5 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = 𝐼)
9		fveq2 6831	. . . . . . 7 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = (ocv‘𝑊))
10		thlval.o	. . . . . . 7 ⊢ ⊥ = (ocv‘𝑊)
11	9, 10	eqtr4di 2786	. . . . . 6 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = ⊥ )
12	11	opeq2d 4833	. . . . 5 ⊢ (ℎ = 𝑊 → ⟨(oc‘ndx), (ocv‘ℎ)⟩ = ⟨(oc‘ndx), ⊥ ⟩)
13	8, 12	oveq12d 7373	. . . 4 ⊢ (ℎ = 𝑊 → ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩) = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
14		df-thl 21611	. . . 4 ⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩))
15		ovex 7388	. . . 4 ⊢ (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩) ∈ V
16	13, 14, 15	fvmpt 6938	. . 3 ⊢ (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
17	2, 16	eqtrid 2780	. 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 1541   ∈ wcel 2113  Vcvv 3437  ⟨cop 4583  ‘cfv 6489  (class class class)co 7355   sSet csts 17081  ndxcnx 17111  occoc 17176  toInccipo 18441  ocvcocv 21606  ClSubSpccss 21607  toHLcthl 21608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358  df-thl 21611

This theorem is referenced by:  thlbas  21642  thlle  21643  thloc  21645