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Theorem isclat 17722
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 6662 . . . . . 6 (𝑙 = 𝐾 → (lub‘𝑙) = (lub‘𝐾))
2 isclat.u . . . . . 6 𝑈 = (lub‘𝐾)
31, 2syl6eqr 2877 . . . . 5 (𝑙 = 𝐾 → (lub‘𝑙) = 𝑈)
43dmeqd 5762 . . . 4 (𝑙 = 𝐾 → dom (lub‘𝑙) = dom 𝑈)
5 fveq2 6662 . . . . . 6 (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6 isclat.b . . . . . 6 𝐵 = (Base‘𝐾)
75, 6syl6eqr 2877 . . . . 5 (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
87pweqd 4542 . . . 4 (𝑙 = 𝐾 → 𝒫 (Base‘𝑙) = 𝒫 𝐵)
94, 8eqeq12d 2840 . . 3 (𝑙 = 𝐾 → (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝑈 = 𝒫 𝐵))
10 fveq2 6662 . . . . . 6 (𝑙 = 𝐾 → (glb‘𝑙) = (glb‘𝐾))
11 isclat.g . . . . . 6 𝐺 = (glb‘𝐾)
1210, 11syl6eqr 2877 . . . . 5 (𝑙 = 𝐾 → (glb‘𝑙) = 𝐺)
1312dmeqd 5762 . . . 4 (𝑙 = 𝐾 → dom (glb‘𝑙) = dom 𝐺)
1413, 8eqeq12d 2840 . . 3 (𝑙 = 𝐾 → (dom (glb‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝐺 = 𝒫 𝐵))
159, 14anbi12d 633 . 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 17721 . 2 CLat = {𝑙 ∈ Poset ∣ (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙))}
1715, 16elrab2 3670 1 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2115  𝒫 cpw 4523  dom cdm 5543  cfv 6344  Basecbs 16486  Posetcpo 17553  lubclub 17555  glbcglb 17556  CLatccla 17720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3483  df-un 3925  df-in 3927  df-ss 3937  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-dm 5553  df-iota 6303  df-fv 6352  df-clat 17721
This theorem is referenced by:  clatpos  17723  clatlem  17724  clatlubcl2  17726  clatglbcl2  17728  clatl  17729  oduclatb  17757  xrsclat  30702
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