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Theorem isclat 17310
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 6352 . . . . . 6 (𝑙 = 𝐾 → (lub‘𝑙) = (lub‘𝐾))
2 isclat.u . . . . . 6 𝑈 = (lub‘𝐾)
31, 2syl6eqr 2812 . . . . 5 (𝑙 = 𝐾 → (lub‘𝑙) = 𝑈)
43dmeqd 5481 . . . 4 (𝑙 = 𝐾 → dom (lub‘𝑙) = dom 𝑈)
5 fveq2 6352 . . . . . 6 (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6 isclat.b . . . . . 6 𝐵 = (Base‘𝐾)
75, 6syl6eqr 2812 . . . . 5 (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
87pweqd 4307 . . . 4 (𝑙 = 𝐾 → 𝒫 (Base‘𝑙) = 𝒫 𝐵)
94, 8eqeq12d 2775 . . 3 (𝑙 = 𝐾 → (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝑈 = 𝒫 𝐵))
10 fveq2 6352 . . . . . 6 (𝑙 = 𝐾 → (glb‘𝑙) = (glb‘𝐾))
11 isclat.g . . . . . 6 𝐺 = (glb‘𝐾)
1210, 11syl6eqr 2812 . . . . 5 (𝑙 = 𝐾 → (glb‘𝑙) = 𝐺)
1312dmeqd 5481 . . . 4 (𝑙 = 𝐾 → dom (glb‘𝑙) = dom 𝐺)
1413, 8eqeq12d 2775 . . 3 (𝑙 = 𝐾 → (dom (glb‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝐺 = 𝒫 𝐵))
159, 14anbi12d 749 . 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 17309 . 2 CLat = {𝑙 ∈ Poset ∣ (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙))}
1715, 16elrab2 3507 1 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wcel 2139  𝒫 cpw 4302  dom cdm 5266  cfv 6049  Basecbs 16059  Posetcpo 17141  lubclub 17143  glbcglb 17144  CLatccla 17308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-dm 5276  df-iota 6012  df-fv 6057  df-clat 17309
This theorem is referenced by:  clatpos  17311  clatlem  17312  clatlubcl2  17314  clatglbcl2  17316  clatl  17317  oduclatb  17345  xrsclat  29989
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