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Theorem thlval 21115
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 (𝑊𝑉𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⟩))

Proof of Theorem thlval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3462 . 2 (𝑊𝑉𝑊 ∈ V)
2 thlval.k . . 3 𝐾 = (toHL‘𝑊)
3 fveq2 6843 . . . . . . . 8 ( = 𝑊 → (ClSubSp‘) = (ClSubSp‘𝑊))
4 thlval.c . . . . . . . 8 𝐶 = (ClSubSp‘𝑊)
53, 4eqtr4di 2791 . . . . . . 7 ( = 𝑊 → (ClSubSp‘) = 𝐶)
65fveq2d 6847 . . . . . 6 ( = 𝑊 → (toInc‘(ClSubSp‘)) = (toInc‘𝐶))
7 thlval.i . . . . . 6 𝐼 = (toInc‘𝐶)
86, 7eqtr4di 2791 . . . . 5 ( = 𝑊 → (toInc‘(ClSubSp‘)) = 𝐼)
9 fveq2 6843 . . . . . . 7 ( = 𝑊 → (ocv‘) = (ocv‘𝑊))
10 thlval.o . . . . . . 7 = (ocv‘𝑊)
119, 10eqtr4di 2791 . . . . . 6 ( = 𝑊 → (ocv‘) = )
1211opeq2d 4838 . . . . 5 ( = 𝑊 → ⟨(oc‘ndx), (ocv‘)⟩ = ⟨(oc‘ndx), ⟩)
138, 12oveq12d 7376 . . . 4 ( = 𝑊 → ((toInc‘(ClSubSp‘)) sSet ⟨(oc‘ndx), (ocv‘)⟩) = (𝐼 sSet ⟨(oc‘ndx), ⟩))
14 df-thl 21085 . . . 4 toHL = ( ∈ V ↦ ((toInc‘(ClSubSp‘)) sSet ⟨(oc‘ndx), (ocv‘)⟩))
15 ovex 7391 . . . 4 (𝐼 sSet ⟨(oc‘ndx), ⟩) ∈ V
1613, 14, 15fvmpt 6949 . . 3 (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet ⟨(oc‘ndx), ⟩))
172, 16eqtrid 2785 . 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 1542  wcel 2107  Vcvv 3444  cop 4593  cfv 6497  (class class class)co 7358   sSet csts 17040  ndxcnx 17070  occoc 17146  toInccipo 18421  ocvcocv 21080  ClSubSpccss 21081  toHLcthl 21082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-thl 21085
This theorem is referenced by:  thlbas  21116  thlbasOLD  21117  thlle  21118  thlleOLD  21119  thloc  21121
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