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Theorem thlval 21731
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 3499 . 2 (𝑊𝑉𝑊 ∈ V)
2 thlval.k . . 3 𝐾 = (toHL‘𝑊)
3 fveq2 6907 . . . . . . . 8 ( = 𝑊 → (ClSubSp‘) = (ClSubSp‘𝑊))
4 thlval.c . . . . . . . 8 𝐶 = (ClSubSp‘𝑊)
53, 4eqtr4di 2793 . . . . . . 7 ( = 𝑊 → (ClSubSp‘) = 𝐶)
65fveq2d 6911 . . . . . 6 ( = 𝑊 → (toInc‘(ClSubSp‘)) = (toInc‘𝐶))
7 thlval.i . . . . . 6 𝐼 = (toInc‘𝐶)
86, 7eqtr4di 2793 . . . . 5 ( = 𝑊 → (toInc‘(ClSubSp‘)) = 𝐼)
9 fveq2 6907 . . . . . . 7 ( = 𝑊 → (ocv‘) = (ocv‘𝑊))
10 thlval.o . . . . . . 7 = (ocv‘𝑊)
119, 10eqtr4di 2793 . . . . . 6 ( = 𝑊 → (ocv‘) = )
1211opeq2d 4885 . . . . 5 ( = 𝑊 → ⟨(oc‘ndx), (ocv‘)⟩ = ⟨(oc‘ndx), ⟩)
138, 12oveq12d 7449 . . . 4 ( = 𝑊 → ((toInc‘(ClSubSp‘)) sSet ⟨(oc‘ndx), (ocv‘)⟩) = (𝐼 sSet ⟨(oc‘ndx), ⟩))
14 df-thl 21701 . . . 4 toHL = ( ∈ V ↦ ((toInc‘(ClSubSp‘)) sSet ⟨(oc‘ndx), (ocv‘)⟩))
15 ovex 7464 . . . 4 (𝐼 sSet ⟨(oc‘ndx), ⟩) ∈ V
1613, 14, 15fvmpt 7016 . . 3 (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet ⟨(oc‘ndx), ⟩))
172, 16eqtrid 2787 . 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 1537  wcel 2106  Vcvv 3478  cop 4637  cfv 6563  (class class class)co 7431   sSet csts 17197  ndxcnx 17227  occoc 17306  toInccipo 18585  ocvcocv 21696  ClSubSpccss 21697  toHLcthl 21698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-thl 21701
This theorem is referenced by:  thlbas  21732  thlbasOLD  21733  thlle  21734  thlleOLD  21735  thloc  21737
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