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Theorem isclatd 48887
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 2736 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2736 . . . . 5 (le‘𝐾) = (le‘𝐾)
4 eqid 2736 . . . . 5 (lub‘𝐾) = (lub‘𝐾)
5 biid 261 . . . . 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 18397 . . . 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 4079 . . . 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 4027 . . 3 (𝜑 → dom (lub‘𝐾) ⊆ 𝒫 (Base‘𝐾))
9 elpwi 4606 . . . . . . 7 (𝑠 ∈ 𝒫 𝐵𝑠𝐵)
10 isclatd.1 . . . . . . 7 ((𝜑𝑠𝐵) → 𝑠 ∈ dom 𝑈)
119, 10sylan2 593 . . . . . 6 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ dom 𝑈)
1211ralrimiva 3145 . . . . 5 (𝜑 → ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝑈)
13 dfss3 3971 . . . . 5 (𝒫 𝐵 ⊆ dom 𝑈 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝑈)
1412, 13sylibr 234 . . . 4 (𝜑 → 𝒫 𝐵 ⊆ dom 𝑈)
15 isclatd.b . . . . 5 (𝜑𝐵 = (Base‘𝐾))
1615pweqd 4616 . . . 4 (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17 isclatd.u . . . . 5 (𝜑𝑈 = (lub‘𝐾))
1817dmeqd 5915 . . . 4 (𝜑 → dom 𝑈 = dom (lub‘𝐾))
1914, 16, 183sstr3d 4037 . . 3 (𝜑 → 𝒫 (Base‘𝐾) ⊆ dom (lub‘𝐾))
208, 19eqssd 4000 . 2 (𝜑 → dom (lub‘𝐾) = 𝒫 (Base‘𝐾))
21 eqid 2736 . . . . 5 (glb‘𝐾) = (glb‘𝐾)
22 biid 261 . . . . 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 18410 . . . 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 4079 . . . 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 4027 . . 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 3145 . . . . 5 (𝜑 → ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝐺)
29 dfss3 3971 . . . . 5 (𝒫 𝐵 ⊆ dom 𝐺 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝐺)
3028, 29sylibr 234 . . . 4 (𝜑 → 𝒫 𝐵 ⊆ dom 𝐺)
31 isclatd.g . . . . 5 (𝜑𝐺 = (glb‘𝐾))
3231dmeqd 5915 . . . 4 (𝜑 → dom 𝐺 = dom (glb‘𝐾))
3330, 16, 323sstr3d 4037 . . 3 (𝜑 → 𝒫 (Base‘𝐾) ⊆ dom (glb‘𝐾))
3425, 33eqssd 4000 . 2 (𝜑 → dom (glb‘𝐾) = 𝒫 (Base‘𝐾))
352, 4, 21isclat 18546 . . 3 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
3635biimpri 228 . 2 ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))) → 𝐾 ∈ CLat)
371, 20, 34, 36syl12anc 836 1 (𝜑𝐾 ∈ CLat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  ∃!wreu 3377  {crab 3435  wss 3950  𝒫 cpw 4599   class class class wbr 5142  dom cdm 5684  cfv 6560  Basecbs 17248  lecple 17305  Posetcpo 18354  lubclub 18356  glbcglb 18357  CLatccla 18544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-lub 18392  df-glb 18393  df-clat 18545
This theorem is referenced by:  mreclat  48901  topclat  48902
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