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Theorem isclatd 46226
Description: The predicate "is a complete lattice" (deduction form). (Contributed by Zhi Wang, 29-Sep-2024.)
Hypotheses
Ref Expression
isclatd.b (𝜑𝐵 = (Base‘𝐾))
isclatd.u (𝜑𝑈 = (lub‘𝐾))
isclatd.g (𝜑𝐺 = (glb‘𝐾))
isclatd.k (𝜑𝐾 ∈ Poset)
isclatd.1 ((𝜑𝑠𝐵) → 𝑠 ∈ dom 𝑈)
isclatd.2 ((𝜑𝑠𝐵) → 𝑠 ∈ dom 𝐺)
Assertion
Ref Expression
isclatd (𝜑𝐾 ∈ CLat)
Distinct variable groups:   𝐵,𝑠   𝐺,𝑠   𝑈,𝑠   𝜑,𝑠
Allowed substitution hint:   𝐾(𝑠)

Proof of Theorem isclatd
Dummy variables 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isclatd.k . 2 (𝜑𝐾 ∈ Poset)
2 eqid 2738 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2738 . . . . 5 (le‘𝐾) = (le‘𝐾)
4 eqid 2738 . . . . 5 (lub‘𝐾) = (lub‘𝐾)
5 biid 260 . . . . 5 ((∀𝑦𝑡 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑡 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦𝑡 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑡 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))
62, 3, 4, 5, 1lubdm 18058 . . . 4 (𝜑 → dom (lub‘𝐾) = {𝑡 ∈ 𝒫 (Base‘𝐾) ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑡 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑡 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))})
7 ssrab2 4014 . . . 4 {𝑡 ∈ 𝒫 (Base‘𝐾) ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑡 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑡 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))} ⊆ 𝒫 (Base‘𝐾)
86, 7eqsstrdi 3976 . . 3 (𝜑 → dom (lub‘𝐾) ⊆ 𝒫 (Base‘𝐾))
9 elpwi 4544 . . . . . . 7 (𝑠 ∈ 𝒫 𝐵𝑠𝐵)
10 isclatd.1 . . . . . . 7 ((𝜑𝑠𝐵) → 𝑠 ∈ dom 𝑈)
119, 10sylan2 593 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ dom 𝑈)
1211ralrimiva 3103 . . . . 5 (𝜑 → ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝑈)
13 dfss3 3910 . . . . 5 (𝒫 𝐵 ⊆ dom 𝑈 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝑈)
1412, 13sylibr 233 . . . 4 (𝜑 → 𝒫 𝐵 ⊆ dom 𝑈)
15 isclatd.b . . . . 5 (𝜑𝐵 = (Base‘𝐾))
1615pweqd 4554 . . . 4 (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17 isclatd.u . . . . 5 (𝜑𝑈 = (lub‘𝐾))
1817dmeqd 5809 . . . 4 (𝜑 → dom 𝑈 = dom (lub‘𝐾))
1914, 16, 183sstr3d 3968 . . 3 (𝜑 → 𝒫 (Base‘𝐾) ⊆ dom (lub‘𝐾))
208, 19eqssd 3939 . 2 (𝜑 → dom (lub‘𝐾) = 𝒫 (Base‘𝐾))
21 eqid 2738 . . . . 5 (glb‘𝐾) = (glb‘𝐾)
22 biid 260 . . . . 5 ((∀𝑦𝑡 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑡 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)) ↔ (∀𝑦𝑡 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑡 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)))
232, 3, 21, 22, 1glbdm 18071 . . . 4 (𝜑 → dom (glb‘𝐾) = {𝑡 ∈ 𝒫 (Base‘𝐾) ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑡 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑡 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))})
24 ssrab2 4014 . . . 4 {𝑡 ∈ 𝒫 (Base‘𝐾) ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑡 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑡 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))} ⊆ 𝒫 (Base‘𝐾)
2523, 24eqsstrdi 3976 . . 3 (𝜑 → dom (glb‘𝐾) ⊆ 𝒫 (Base‘𝐾))
26 isclatd.2 . . . . . . 7 ((𝜑𝑠𝐵) → 𝑠 ∈ dom 𝐺)
279, 26sylan2 593 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ dom 𝐺)
2827ralrimiva 3103 . . . . 5 (𝜑 → ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝐺)
29 dfss3 3910 . . . . 5 (𝒫 𝐵 ⊆ dom 𝐺 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝐺)
3028, 29sylibr 233 . . . 4 (𝜑 → 𝒫 𝐵 ⊆ dom 𝐺)
31 isclatd.g . . . . 5 (𝜑𝐺 = (glb‘𝐾))
3231dmeqd 5809 . . . 4 (𝜑 → dom 𝐺 = dom (glb‘𝐾))
3330, 16, 323sstr3d 3968 . . 3 (𝜑 → 𝒫 (Base‘𝐾) ⊆ dom (glb‘𝐾))
3425, 33eqssd 3939 . 2 (𝜑 → dom (glb‘𝐾) = 𝒫 (Base‘𝐾))
352, 4, 21isclat 18207 . . 3 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
3635biimpri 227 . 2 ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))) → 𝐾 ∈ CLat)
371, 20, 34, 36syl12anc 834 1 (𝜑𝐾 ∈ CLat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wral 3064  ∃!wreu 3066  {crab 3068  wss 3888  𝒫 cpw 4535   class class class wbr 5075  dom cdm 5586  cfv 6428  Basecbs 16901  lecple 16958  Posetcpo 18014  lubclub 18016  glbcglb 18017  CLatccla 18205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5210  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5486  df-xp 5592  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-res 5598  df-ima 5599  df-iota 6386  df-fun 6430  df-fn 6431  df-f 6432  df-f1 6433  df-fo 6434  df-f1o 6435  df-fv 6436  df-riota 7226  df-lub 18053  df-glb 18054  df-clat 18206
This theorem is referenced by:  mreclat  46240  topclat  46241
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