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Theorem isclatd 47864
Description: The predicate "is a complete lattice" (deduction form). (Contributed by Zhi Wang, 29-Sep-2024.)
Hypotheses
Ref Expression
isclatd.b (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))
isclatd.u (πœ‘ β†’ π‘ˆ = (lubβ€˜πΎ))
isclatd.g (πœ‘ β†’ 𝐺 = (glbβ€˜πΎ))
isclatd.k (πœ‘ β†’ 𝐾 ∈ Poset)
isclatd.1 ((πœ‘ ∧ 𝑠 βŠ† 𝐡) β†’ 𝑠 ∈ dom π‘ˆ)
isclatd.2 ((πœ‘ ∧ 𝑠 βŠ† 𝐡) β†’ 𝑠 ∈ dom 𝐺)
Assertion
Ref Expression
isclatd (πœ‘ β†’ 𝐾 ∈ CLat)
Distinct variable groups:   𝐡,𝑠   𝐺,𝑠   π‘ˆ,𝑠   πœ‘,𝑠
Allowed substitution hint:   𝐾(𝑠)

Proof of Theorem isclatd
Dummy variables 𝑑 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isclatd.k . 2 (πœ‘ β†’ 𝐾 ∈ Poset)
2 eqid 2726 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3 eqid 2726 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
4 eqid 2726 . . . . 5 (lubβ€˜πΎ) = (lubβ€˜πΎ)
5 biid 261 . . . . 5 ((βˆ€π‘¦ ∈ 𝑑 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑑 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)) ↔ (βˆ€π‘¦ ∈ 𝑑 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑑 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)))
62, 3, 4, 5, 1lubdm 18313 . . . 4 (πœ‘ β†’ dom (lubβ€˜πΎ) = {𝑑 ∈ 𝒫 (Baseβ€˜πΎ) ∣ βˆƒ!π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑑 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑑 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧))})
7 ssrab2 4072 . . . 4 {𝑑 ∈ 𝒫 (Baseβ€˜πΎ) ∣ βˆƒ!π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑑 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑑 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧))} βŠ† 𝒫 (Baseβ€˜πΎ)
86, 7eqsstrdi 4031 . . 3 (πœ‘ β†’ dom (lubβ€˜πΎ) βŠ† 𝒫 (Baseβ€˜πΎ))
9 elpwi 4604 . . . . . . 7 (𝑠 ∈ 𝒫 𝐡 β†’ 𝑠 βŠ† 𝐡)
10 isclatd.1 . . . . . . 7 ((πœ‘ ∧ 𝑠 βŠ† 𝐡) β†’ 𝑠 ∈ dom π‘ˆ)
119, 10sylan2 592 . . . . . 6 ((πœ‘ ∧ 𝑠 ∈ 𝒫 𝐡) β†’ 𝑠 ∈ dom π‘ˆ)
1211ralrimiva 3140 . . . . 5 (πœ‘ β†’ βˆ€π‘  ∈ 𝒫 𝐡𝑠 ∈ dom π‘ˆ)
13 dfss3 3965 . . . . 5 (𝒫 𝐡 βŠ† dom π‘ˆ ↔ βˆ€π‘  ∈ 𝒫 𝐡𝑠 ∈ dom π‘ˆ)
1412, 13sylibr 233 . . . 4 (πœ‘ β†’ 𝒫 𝐡 βŠ† dom π‘ˆ)
15 isclatd.b . . . . 5 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))
1615pweqd 4614 . . . 4 (πœ‘ β†’ 𝒫 𝐡 = 𝒫 (Baseβ€˜πΎ))
17 isclatd.u . . . . 5 (πœ‘ β†’ π‘ˆ = (lubβ€˜πΎ))
1817dmeqd 5898 . . . 4 (πœ‘ β†’ dom π‘ˆ = dom (lubβ€˜πΎ))
1914, 16, 183sstr3d 4023 . . 3 (πœ‘ β†’ 𝒫 (Baseβ€˜πΎ) βŠ† dom (lubβ€˜πΎ))
208, 19eqssd 3994 . 2 (πœ‘ β†’ dom (lubβ€˜πΎ) = 𝒫 (Baseβ€˜πΎ))
21 eqid 2726 . . . . 5 (glbβ€˜πΎ) = (glbβ€˜πΎ)
22 biid 261 . . . . 5 ((βˆ€π‘¦ ∈ 𝑑 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑑 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)) ↔ (βˆ€π‘¦ ∈ 𝑑 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑑 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)))
232, 3, 21, 22, 1glbdm 18326 . . . 4 (πœ‘ β†’ dom (glbβ€˜πΎ) = {𝑑 ∈ 𝒫 (Baseβ€˜πΎ) ∣ βˆƒ!π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑑 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑑 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯))})
24 ssrab2 4072 . . . 4 {𝑑 ∈ 𝒫 (Baseβ€˜πΎ) ∣ βˆƒ!π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑑 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑑 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯))} βŠ† 𝒫 (Baseβ€˜πΎ)
2523, 24eqsstrdi 4031 . . 3 (πœ‘ β†’ dom (glbβ€˜πΎ) βŠ† 𝒫 (Baseβ€˜πΎ))
26 isclatd.2 . . . . . . 7 ((πœ‘ ∧ 𝑠 βŠ† 𝐡) β†’ 𝑠 ∈ dom 𝐺)
279, 26sylan2 592 . . . . . 6 ((πœ‘ ∧ 𝑠 ∈ 𝒫 𝐡) β†’ 𝑠 ∈ dom 𝐺)
2827ralrimiva 3140 . . . . 5 (πœ‘ β†’ βˆ€π‘  ∈ 𝒫 𝐡𝑠 ∈ dom 𝐺)
29 dfss3 3965 . . . . 5 (𝒫 𝐡 βŠ† dom 𝐺 ↔ βˆ€π‘  ∈ 𝒫 𝐡𝑠 ∈ dom 𝐺)
3028, 29sylibr 233 . . . 4 (πœ‘ β†’ 𝒫 𝐡 βŠ† dom 𝐺)
31 isclatd.g . . . . 5 (πœ‘ β†’ 𝐺 = (glbβ€˜πΎ))
3231dmeqd 5898 . . . 4 (πœ‘ β†’ dom 𝐺 = dom (glbβ€˜πΎ))
3330, 16, 323sstr3d 4023 . . 3 (πœ‘ β†’ 𝒫 (Baseβ€˜πΎ) βŠ† dom (glbβ€˜πΎ))
3425, 33eqssd 3994 . 2 (πœ‘ β†’ dom (glbβ€˜πΎ) = 𝒫 (Baseβ€˜πΎ))
352, 4, 21isclat 18462 . . 3 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lubβ€˜πΎ) = 𝒫 (Baseβ€˜πΎ) ∧ dom (glbβ€˜πΎ) = 𝒫 (Baseβ€˜πΎ))))
3635biimpri 227 . 2 ((𝐾 ∈ Poset ∧ (dom (lubβ€˜πΎ) = 𝒫 (Baseβ€˜πΎ) ∧ dom (glbβ€˜πΎ) = 𝒫 (Baseβ€˜πΎ))) β†’ 𝐾 ∈ CLat)
371, 20, 34, 36syl12anc 834 1 (πœ‘ β†’ 𝐾 ∈ CLat)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒ!wreu 3368  {crab 3426   βŠ† wss 3943  π’« cpw 4597   class class class wbr 5141  dom cdm 5669  β€˜cfv 6536  Basecbs 17150  lecple 17210  Posetcpo 18269  lubclub 18271  glbcglb 18272  CLatccla 18460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-lub 18308  df-glb 18309  df-clat 18461
This theorem is referenced by:  mreclat  47878  topclat  47879
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