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Theorem clatl 17044
Description: A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.) TODO: use eqrelrdv2 5185 to shorten proof and eliminate joindmss 16935 and meetdmss 16949?
Assertion
Ref Expression
clatl (𝐾 ∈ CLat → 𝐾 ∈ Lat)

Proof of Theorem clatl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2621 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
3 simpl 473 . . . . . . 7 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → 𝐾 ∈ Poset)
41, 2, 3joindmss 16935 . . . . . 6 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → dom (join‘𝐾) ⊆ ((Base‘𝐾) × (Base‘𝐾)))
5 relxp 5193 . . . . . . . 8 Rel ((Base‘𝐾) × (Base‘𝐾))
65a1i 11 . . . . . . 7 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → Rel ((Base‘𝐾) × (Base‘𝐾)))
7 opelxp 5111 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)))
8 vex 3192 . . . . . . . . . . . . 13 𝑥 ∈ V
9 vex 3192 . . . . . . . . . . . . 13 𝑦 ∈ V
108, 9prss 4324 . . . . . . . . . . . 12 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
117, 10sylbb 209 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ⊆ (Base‘𝐾))
12 prex 4875 . . . . . . . . . . . 12 {𝑥, 𝑦} ∈ V
1312elpw 4141 . . . . . . . . . . 11 ({𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
1411, 13sylibr 224 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾))
15 eleq2 2687 . . . . . . . . . 10 (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) ↔ {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾)))
1614, 15syl5ibr 236 . . . . . . . . 9 (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
1716adantl 482 . . . . . . . 8 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
18 eqid 2621 . . . . . . . . 9 (lub‘𝐾) = (lub‘𝐾)
198a1i 11 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → 𝑥 ∈ V)
209a1i 11 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → 𝑦 ∈ V)
2118, 2, 3, 19, 20joindef 16932 . . . . . . . 8 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ dom (join‘𝐾) ↔ {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
2217, 21sylibrd 249 . . . . . . 7 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → ⟨𝑥, 𝑦⟩ ∈ dom (join‘𝐾)))
236, 22relssdv 5178 . . . . . 6 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → ((Base‘𝐾) × (Base‘𝐾)) ⊆ dom (join‘𝐾))
244, 23eqssd 3604 . . . . 5 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))
2524ex 450 . . . 4 (𝐾 ∈ Poset → (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) → dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾))))
26 eqid 2621 . . . . . . 7 (meet‘𝐾) = (meet‘𝐾)
27 simpl 473 . . . . . . 7 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → 𝐾 ∈ Poset)
281, 26, 27meetdmss 16949 . . . . . 6 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → dom (meet‘𝐾) ⊆ ((Base‘𝐾) × (Base‘𝐾)))
295a1i 11 . . . . . . 7 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → Rel ((Base‘𝐾) × (Base‘𝐾)))
30 eleq2 2687 . . . . . . . . . 10 (dom (glb‘𝐾) = 𝒫 (Base‘𝐾) → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) ↔ {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾)))
3114, 30syl5ibr 236 . . . . . . . . 9 (dom (glb‘𝐾) = 𝒫 (Base‘𝐾) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
3231adantl 482 . . . . . . . 8 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
33 eqid 2621 . . . . . . . . 9 (glb‘𝐾) = (glb‘𝐾)
348a1i 11 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → 𝑥 ∈ V)
359a1i 11 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → 𝑦 ∈ V)
3633, 26, 27, 34, 35meetdef 16946 . . . . . . . 8 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ dom (meet‘𝐾) ↔ {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
3732, 36sylibrd 249 . . . . . . 7 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → ⟨𝑥, 𝑦⟩ ∈ dom (meet‘𝐾)))
3829, 37relssdv 5178 . . . . . 6 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → ((Base‘𝐾) × (Base‘𝐾)) ⊆ dom (meet‘𝐾))
3928, 38eqssd 3604 . . . . 5 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))
4039ex 450 . . . 4 (𝐾 ∈ Poset → (dom (glb‘𝐾) = 𝒫 (Base‘𝐾) → dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾))))
4125, 40anim12d 585 . . 3 (𝐾 ∈ Poset → ((dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)) ∧ dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))))
4241imdistani 725 . 2 ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))) → (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)) ∧ dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))))
431, 18, 33isclat 17037 . 2 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
441, 2, 26islat 16975 . 2 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)) ∧ dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))))
4542, 43, 443imtr4i 281 1 (𝐾 ∈ CLat → 𝐾 ∈ Lat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3189  wss 3559  𝒫 cpw 4135  {cpr 4155  cop 4159   × cxp 5077  dom cdm 5079  Rel wrel 5084  cfv 5852  Basecbs 15788  Posetcpo 16868  lubclub 16870  glbcglb 16871  joincjn 16872  meetcmee 16873  Latclat 16973  CLatccla 17035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-oprab 6614  df-lub 16902  df-glb 16903  df-join 16904  df-meet 16905  df-lat 16974  df-clat 17036
This theorem is referenced by:  lubel  17050  lubun  17051  clatleglb  17054
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