MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  thlval Structured version   Visualization version   GIF version

Theorem thlval 21713
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 3501 . 2 (𝑊𝑉𝑊 ∈ V)
2 thlval.k . . 3 𝐾 = (toHL‘𝑊)
3 fveq2 6906 . . . . . . . 8 ( = 𝑊 → (ClSubSp‘) = (ClSubSp‘𝑊))
4 thlval.c . . . . . . . 8 𝐶 = (ClSubSp‘𝑊)
53, 4eqtr4di 2795 . . . . . . 7 ( = 𝑊 → (ClSubSp‘) = 𝐶)
65fveq2d 6910 . . . . . 6 ( = 𝑊 → (toInc‘(ClSubSp‘)) = (toInc‘𝐶))
7 thlval.i . . . . . 6 𝐼 = (toInc‘𝐶)
86, 7eqtr4di 2795 . . . . 5 ( = 𝑊 → (toInc‘(ClSubSp‘)) = 𝐼)
9 fveq2 6906 . . . . . . 7 ( = 𝑊 → (ocv‘) = (ocv‘𝑊))
10 thlval.o . . . . . . 7 = (ocv‘𝑊)
119, 10eqtr4di 2795 . . . . . 6 ( = 𝑊 → (ocv‘) = )
1211opeq2d 4880 . . . . 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 21683 . . . 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 2789 . 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 1540  wcel 2108  Vcvv 3480  cop 4632  cfv 6561  (class class class)co 7431   sSet csts 17200  ndxcnx 17230  occoc 17305  toInccipo 18572  ocvcocv 21678  ClSubSpccss 21679  toHLcthl 21680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-thl 21683
This theorem is referenced by:  thlbas  21714  thlbasOLD  21715  thlle  21716  thlleOLD  21717  thloc  21719
  Copyright terms: Public domain W3C validator