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Theorem isclat 17030
Description: The predicate "is a complete lattice." (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
isclat.b 𝐵 = (Base‘𝐾)
isclat.u 𝑈 = (lub‘𝐾)
isclat.g 𝐺 = (glb‘𝐾)
Assertion
Ref Expression
isclat (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))

Proof of Theorem isclat
Dummy variable 𝑙 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6148 . . . . . 6 (𝑙 = 𝐾 → (lub‘𝑙) = (lub‘𝐾))
2 isclat.u . . . . . 6 𝑈 = (lub‘𝐾)
31, 2syl6eqr 2673 . . . . 5 (𝑙 = 𝐾 → (lub‘𝑙) = 𝑈)
43dmeqd 5286 . . . 4 (𝑙 = 𝐾 → dom (lub‘𝑙) = dom 𝑈)
5 fveq2 6148 . . . . . 6 (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6 isclat.b . . . . . 6 𝐵 = (Base‘𝐾)
75, 6syl6eqr 2673 . . . . 5 (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
87pweqd 4135 . . . 4 (𝑙 = 𝐾 → 𝒫 (Base‘𝑙) = 𝒫 𝐵)
94, 8eqeq12d 2636 . . 3 (𝑙 = 𝐾 → (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝑈 = 𝒫 𝐵))
10 fveq2 6148 . . . . . 6 (𝑙 = 𝐾 → (glb‘𝑙) = (glb‘𝐾))
11 isclat.g . . . . . 6 𝐺 = (glb‘𝐾)
1210, 11syl6eqr 2673 . . . . 5 (𝑙 = 𝐾 → (glb‘𝑙) = 𝐺)
1312dmeqd 5286 . . . 4 (𝑙 = 𝐾 → dom (glb‘𝑙) = dom 𝐺)
1413, 8eqeq12d 2636 . . 3 (𝑙 = 𝐾 → (dom (glb‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝐺 = 𝒫 𝐵))
159, 14anbi12d 746 . 2 (𝑙 = 𝐾 → ((dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙)) ↔ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
16 df-clat 17029 . 2 CLat = {𝑙 ∈ Poset ∣ (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙))}
1715, 16elrab2 3348 1 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  𝒫 cpw 4130  dom cdm 5074  cfv 5847  Basecbs 15781  Posetcpo 16861  lubclub 16863  glbcglb 16864  CLatccla 17028
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
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-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-dm 5084  df-iota 5810  df-fv 5855  df-clat 17029
This theorem is referenced by:  clatpos  17031  clatlem  17032  clatlubcl2  17034  clatglbcl2  17036  clatl  17037  oduclatb  17065  xrsclat  29462
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