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Theorem clatl 17715
Description: A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.) TODO: use eqrelrdv2 5649 to shorten proof and eliminate joindmss 17606 and meetdmss 17620?
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 2824 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2824 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
3 simpl 486 . . . . . . 7 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → 𝐾 ∈ Poset)
41, 2, 3joindmss 17606 . . . . . 6 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → dom (join‘𝐾) ⊆ ((Base‘𝐾) × (Base‘𝐾)))
5 relxp 5554 . . . . . . . 8 Rel ((Base‘𝐾) × (Base‘𝐾))
65a1i 11 . . . . . . 7 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → Rel ((Base‘𝐾) × (Base‘𝐾)))
7 opelxp 5572 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)))
8 vex 3482 . . . . . . . . . . . . 13 𝑥 ∈ V
9 vex 3482 . . . . . . . . . . . . 13 𝑦 ∈ V
108, 9prss 4734 . . . . . . . . . . . 12 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
117, 10sylbb 222 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ⊆ (Base‘𝐾))
12 prex 5314 . . . . . . . . . . . 12 {𝑥, 𝑦} ∈ V
1312elpw 4524 . . . . . . . . . . 11 ({𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
1411, 13sylibr 237 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾))
15 eleq2 2904 . . . . . . . . . 10 (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) ↔ {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾)))
1614, 15syl5ibr 249 . . . . . . . . 9 (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
1716adantl 485 . . . . . . . 8 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
18 eqid 2824 . . . . . . . . 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 17603 . . . . . . . 8 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ dom (join‘𝐾) ↔ {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
2217, 21sylibrd 262 . . . . . . 7 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → ⟨𝑥, 𝑦⟩ ∈ dom (join‘𝐾)))
236, 22relssdv 5642 . . . . . 6 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → ((Base‘𝐾) × (Base‘𝐾)) ⊆ dom (join‘𝐾))
244, 23eqssd 3968 . . . . 5 ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))
2524ex 416 . . . 4 (𝐾 ∈ Poset → (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) → dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾))))
26 eqid 2824 . . . . . . 7 (meet‘𝐾) = (meet‘𝐾)
27 simpl 486 . . . . . . 7 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → 𝐾 ∈ Poset)
281, 26, 27meetdmss 17620 . . . . . 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 2904 . . . . . . . . . 10 (dom (glb‘𝐾) = 𝒫 (Base‘𝐾) → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) ↔ {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾)))
3114, 30syl5ibr 249 . . . . . . . . 9 (dom (glb‘𝐾) = 𝒫 (Base‘𝐾) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
3231adantl 485 . . . . . . . 8 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
33 eqid 2824 . . . . . . . . 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 17617 . . . . . . . 8 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ dom (meet‘𝐾) ↔ {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
3732, 36sylibrd 262 . . . . . . 7 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → ⟨𝑥, 𝑦⟩ ∈ dom (meet‘𝐾)))
3829, 37relssdv 5642 . . . . . 6 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → ((Base‘𝐾) × (Base‘𝐾)) ⊆ dom (meet‘𝐾))
3928, 38eqssd 3968 . . . . 5 ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))
4039ex 416 . . . 4 (𝐾 ∈ Poset → (dom (glb‘𝐾) = 𝒫 (Base‘𝐾) → dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾))))
4125, 40anim12d 611 . . 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 572 . 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 17708 . 2 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
441, 2, 26islat 17646 . 2 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)) ∧ dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))))
4542, 43, 443imtr4i 295 1 (𝐾 ∈ CLat → 𝐾 ∈ Lat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3479  wss 3918  𝒫 cpw 4520  {cpr 4550  cop 4554   × cxp 5534  dom cdm 5536  Rel wrel 5541  cfv 6336  Basecbs 16472  Posetcpo 17539  lubclub 17541  glbcglb 17542  joincjn 17543  meetcmee 17544  Latclat 17644  CLatccla 17706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-oprab 7142  df-lub 17573  df-glb 17574  df-join 17575  df-meet 17576  df-lat 17645  df-clat 17707
This theorem is referenced by:  lubel  17721  lubun  17722  clatleglb  17725
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