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Theorem thlval 19858
 Description: Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.)
Hypotheses
Ref Expression
thlval.k 𝐾 = (toHL‘𝑊)
thlval.c 𝐶 = (CSubSp‘𝑊)
thlval.i 𝐼 = (toInc‘𝐶)
thlval.o = (ocv‘𝑊)
Assertion
Ref Expression
thlval (𝑊𝑉𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⟩))

Proof of Theorem thlval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝑊𝑉𝑊 ∈ V)
2 thlval.k . . 3 𝐾 = (toHL‘𝑊)
3 fveq2 6103 . . . . . . . 8 ( = 𝑊 → (CSubSp‘) = (CSubSp‘𝑊))
4 thlval.c . . . . . . . 8 𝐶 = (CSubSp‘𝑊)
53, 4syl6eqr 2662 . . . . . . 7 ( = 𝑊 → (CSubSp‘) = 𝐶)
65fveq2d 6107 . . . . . 6 ( = 𝑊 → (toInc‘(CSubSp‘)) = (toInc‘𝐶))
7 thlval.i . . . . . 6 𝐼 = (toInc‘𝐶)
86, 7syl6eqr 2662 . . . . 5 ( = 𝑊 → (toInc‘(CSubSp‘)) = 𝐼)
9 fveq2 6103 . . . . . . 7 ( = 𝑊 → (ocv‘) = (ocv‘𝑊))
10 thlval.o . . . . . . 7 = (ocv‘𝑊)
119, 10syl6eqr 2662 . . . . . 6 ( = 𝑊 → (ocv‘) = )
1211opeq2d 4347 . . . . 5 ( = 𝑊 → ⟨(oc‘ndx), (ocv‘)⟩ = ⟨(oc‘ndx), ⟩)
138, 12oveq12d 6567 . . . 4 ( = 𝑊 → ((toInc‘(CSubSp‘)) sSet ⟨(oc‘ndx), (ocv‘)⟩) = (𝐼 sSet ⟨(oc‘ndx), ⟩))
14 df-thl 19828 . . . 4 toHL = ( ∈ V ↦ ((toInc‘(CSubSp‘)) sSet ⟨(oc‘ndx), (ocv‘)⟩))
15 ovex 6577 . . . 4 (𝐼 sSet ⟨(oc‘ndx), ⟩) ∈ V
1613, 14, 15fvmpt 6191 . . 3 (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet ⟨(oc‘ndx), ⟩))
172, 16syl5eq 2656 . 2 (𝑊 ∈ V → 𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⟩))
181, 17syl 17 1 (𝑊𝑉𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549  ndxcnx 15692   sSet csts 15693  occoc 15776  toInccipo 16974  ocvcocv 19823  CSubSpccss 19824  toHLcthl 19825 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-thl 19828 This theorem is referenced by:  thlbas  19859  thlle  19860  thloc  19862
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