Theorem thlval 22007

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 3471	. 2 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2		thlval.k	. . 3 ⊢ 𝐾 = (toHL‘𝑊)
3		fveq2 6886	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = (ClSubSp‘𝑊))
4		thlval.c	. . . . . . . 8 ⊢ 𝐶 = (ClSubSp‘𝑊)
5	3, 4	eqtr4di 2813	. . . . . . 7 ⊢ (ℎ = 𝑊 → (ClSubSp‘ℎ) = 𝐶)
6	5	fveq2d 6890	. . . . . 6 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = (toInc‘𝐶))
7		thlval.i	. . . . . 6 ⊢ 𝐼 = (toInc‘𝐶)
8	6, 7	eqtr4di 2813	. . . . 5 ⊢ (ℎ = 𝑊 → (toInc‘(ClSubSp‘ℎ)) = 𝐼)
9		fveq2 6886	. . . . . . 7 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = (ocv‘𝑊))
10		thlval.o	. . . . . . 7 ⊢ ⊥ = (ocv‘𝑊)
11	9, 10	eqtr4di 2813	. . . . . 6 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = ⊥ )
12	11	opeq2d 4846	. . . . 5 ⊢ (ℎ = 𝑊 → ⟨(oc‘ndx), (ocv‘ℎ)⟩ = ⟨(oc‘ndx), ⊥ ⟩)
13	8, 12	oveq12d 7439	. . . 4 ⊢ (ℎ = 𝑊 → ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩) = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
14		df-thl 21977	. . . 4 ⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩))
15		ovex 7454	. . . 4 ⊢ (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩) ∈ V
16	13, 14, 15	fvmpt 6994	. . 3 ⊢ (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
17	2, 16	eqtrid 2807	. 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 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4834  ‘cfv 6539  (class class class)co 7421   sSet csts 17347  ndxcnx 17377  occoc 17442  toInccipo 18707  ocvcocv 21972  ClSubSpccss 21973  toHLcthl 21974

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424  df-thl 21977

This theorem is used by:  thlbas  22008  thlle  22009  thloc  22011