Theorem clatl 18207

Description: A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.) TODO: use eqrelrdv2 5702 to shorten proof and eliminate joindmss 18078 and meetdmss 18092?

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.

Step	Hyp	Ref	Expression
1		eqid 2739	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2739	. . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾)
3		simpl 482	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → 𝐾 ∈ Poset)
4	1, 2, 3	joindmss 18078	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → dom (join‘𝐾) ⊆ ((Base‘𝐾) × (Base‘𝐾)))
5		relxp 5606	. . . . . . . 8 ⊢ Rel ((Base‘𝐾) × (Base‘𝐾))
6	5	a1i 11	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → Rel ((Base‘𝐾) × (Base‘𝐾)))
7		opelxp 5624	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)))
8		vex 3434	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
9		vex 3434	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
10	8, 9	prss 4758	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
11	7, 10	sylbb 218	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ⊆ (Base‘𝐾))
12		prex 5358	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑦} ∈ V
13	12	elpw 4542	. . . . . . . . . . 11 ⊢ ({𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
14	11, 13	sylibr 233	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾))
15		eleq2 2828	. . . . . . . . . 10 ⊢ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) ↔ {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾)))
16	14, 15	syl5ibr 245	. . . . . . . . 9 ⊢ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
17	16	adantl 481	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
18		eqid 2739	. . . . . . . . 9 ⊢ (lub‘𝐾) = (lub‘𝐾)
19	8	a1i 11	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → 𝑥 ∈ V)
20	9	a1i 11	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → 𝑦 ∈ V)
21	18, 2, 3, 19, 20	joindef 18075	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ dom (join‘𝐾) ↔ {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
22	17, 21	sylibrd 258	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → ⟨𝑥, 𝑦⟩ ∈ dom (join‘𝐾)))
23	6, 22	relssdv 5695	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → ((Base‘𝐾) × (Base‘𝐾)) ⊆ dom (join‘𝐾))
24	4, 23	eqssd 3942	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))
25	24	ex 412	. . . 4 ⊢ (𝐾 ∈ Poset → (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) → dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾))))
26		eqid 2739	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
27		simpl 482	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → 𝐾 ∈ Poset)
28	1, 26, 27	meetdmss 18092	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → dom (meet‘𝐾) ⊆ ((Base‘𝐾) × (Base‘𝐾)))
29	5	a1i 11	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → Rel ((Base‘𝐾) × (Base‘𝐾)))
30		eleq2 2828	. . . . . . . . . 10 ⊢ (dom (glb‘𝐾) = 𝒫 (Base‘𝐾) → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) ↔ {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾)))
31	14, 30	syl5ibr 245	. . . . . . . . 9 ⊢ (dom (glb‘𝐾) = 𝒫 (Base‘𝐾) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
32	31	adantl 481	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
33		eqid 2739	. . . . . . . . 9 ⊢ (glb‘𝐾) = (glb‘𝐾)
34	8	a1i 11	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → 𝑥 ∈ V)
35	9	a1i 11	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → 𝑦 ∈ V)
36	33, 26, 27, 34, 35	meetdef 18089	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ dom (meet‘𝐾) ↔ {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
37	32, 36	sylibrd 258	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → ⟨𝑥, 𝑦⟩ ∈ dom (meet‘𝐾)))
38	29, 37	relssdv 5695	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → ((Base‘𝐾) × (Base‘𝐾)) ⊆ dom (meet‘𝐾))
39	28, 38	eqssd 3942	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))
40	39	ex 412	. . . 4 ⊢ (𝐾 ∈ Poset → (dom (glb‘𝐾) = 𝒫 (Base‘𝐾) → dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾))))
41	25, 40	anim12d 608	. . 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)))))
42	41	imdistani 568	. 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)))))
43	1, 18, 33	isclat 18199	. 2 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
44	1, 2, 26	islat 18132	. 2 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)) ∧ dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))))
45	42, 43, 44	3imtr4i 291	1 ⊢ (𝐾 ∈ CLat → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  Vcvv 3430   ⊆ wss 3891  𝒫 cpw 4538  {cpr 4568  ⟨cop 4572   × cxp 5586  dom cdm 5588  Rel wrel 5593  ‘cfv 6430  Basecbs 16893  Posetcpo 18006  lubclub 18008  glbcglb 18009  joincjn 18010  meetcmee 18011  Latclat 18130  CLatccla 18197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-oprab 7272  df-lub 18045  df-glb 18046  df-join 18047  df-meet 18048  df-lat 18131  df-clat 18198

This theorem is referenced by:  lubel  18213  lubun  18214  clatleglb  18217  topdlat  46242