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

Theorem thlval 19958
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 3198 . 2 (𝑊𝑉𝑊 ∈ V)
2 thlval.k . . 3 𝐾 = (toHL‘𝑊)
3 fveq2 6148 . . . . . . . 8 ( = 𝑊 → (CSubSp‘) = (CSubSp‘𝑊))
4 thlval.c . . . . . . . 8 𝐶 = (CSubSp‘𝑊)
53, 4syl6eqr 2673 . . . . . . 7 ( = 𝑊 → (CSubSp‘) = 𝐶)
65fveq2d 6152 . . . . . 6 ( = 𝑊 → (toInc‘(CSubSp‘)) = (toInc‘𝐶))
7 thlval.i . . . . . 6 𝐼 = (toInc‘𝐶)
86, 7syl6eqr 2673 . . . . 5 ( = 𝑊 → (toInc‘(CSubSp‘)) = 𝐼)
9 fveq2 6148 . . . . . . 7 ( = 𝑊 → (ocv‘) = (ocv‘𝑊))
10 thlval.o . . . . . . 7 = (ocv‘𝑊)
119, 10syl6eqr 2673 . . . . . 6 ( = 𝑊 → (ocv‘) = )
1211opeq2d 4377 . . . . 5 ( = 𝑊 → ⟨(oc‘ndx), (ocv‘)⟩ = ⟨(oc‘ndx), ⟩)
138, 12oveq12d 6622 . . . 4 ( = 𝑊 → ((toInc‘(CSubSp‘)) sSet ⟨(oc‘ndx), (ocv‘)⟩) = (𝐼 sSet ⟨(oc‘ndx), ⟩))
14 df-thl 19928 . . . 4 toHL = ( ∈ V ↦ ((toInc‘(CSubSp‘)) sSet ⟨(oc‘ndx), (ocv‘)⟩))
15 ovex 6632 . . . 4 (𝐼 sSet ⟨(oc‘ndx), ⟩) ∈ V
1613, 14, 15fvmpt 6239 . . 3 (𝑊 ∈ V → (toHL‘𝑊) = (𝐼 sSet ⟨(oc‘ndx), ⟩))
172, 16syl5eq 2667 . 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 1480  wcel 1987  Vcvv 3186  cop 4154  cfv 5847  (class class class)co 6604  ndxcnx 15778   sSet csts 15779  occoc 15870  toInccipo 17072  ocvcocv 19923  CSubSpccss 19924  toHLcthl 19925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-thl 19928
This theorem is referenced by:  thlbas  19959  thlle  19960  thloc  19962
  Copyright terms: Public domain W3C validator