Theorem clatl 18732

Description: A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.) TODO: use eqrelrdv2 5780 to shorten proof and eliminate joindmss 18601 and meetdmss 18615?

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 2763	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2763	. . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾)
3		simpl 488	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → 𝐾 ∈ Poset)
4	1, 2, 3	joindmss 18601	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → dom (join‘𝐾) ⊆ ((Base‘𝐾) × (Base‘𝐾)))
5		relxp 5676	. . . . . . . 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 5694	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)))
8		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
9		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
10	8, 9	prss 4781	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
11	7, 10	sylbb 222	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ⊆ (Base‘𝐾))
12		prex 5403	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑦} ∈ V
13	12	elpw 4564	. . . . . . . . . . 11 ⊢ ({𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
14	11, 13	sylibr 237	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾))
15		eleq2 2853	. . . . . . . . . 10 ⊢ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) ↔ {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾)))
16	14, 15	imbitrrid 249	. . . . . . . . 9 ⊢ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
17	16	adantl 487	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
18		eqid 2763	. . . . . . . . 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 18598	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ dom (join‘𝐾) ↔ {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
22	17, 21	sylibrd 262	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → ⟨𝑥, 𝑦⟩ ∈ dom (join‘𝐾)))
23	6, 22	relssdv 5773	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → ((Base‘𝐾) × (Base‘𝐾)) ⊆ dom (join‘𝐾))
24	4, 23	eqssd 3930	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))
25	24	ex 418	. . . 4 ⊢ (𝐾 ∈ Poset → (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) → dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾))))
26		eqid 2763	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
27		simpl 488	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → 𝐾 ∈ Poset)
28	1, 26, 27	meetdmss 18615	. . . . . 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 2853	. . . . . . . . . 10 ⊢ (dom (glb‘𝐾) = 𝒫 (Base‘𝐾) → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) ↔ {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾)))
31	14, 30	imbitrrid 249	. . . . . . . . 9 ⊢ (dom (glb‘𝐾) = 𝒫 (Base‘𝐾) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
32	31	adantl 487	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
33		eqid 2763	. . . . . . . . 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 18612	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ dom (meet‘𝐾) ↔ {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
37	32, 36	sylibrd 262	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → ⟨𝑥, 𝑦⟩ ∈ dom (meet‘𝐾)))
38	29, 37	relssdv 5773	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → ((Base‘𝐾) × (Base‘𝐾)) ⊆ dom (meet‘𝐾))
39	28, 38	eqssd 3930	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))
40	39	ex 418	. . . 4 ⊢ (𝐾 ∈ Poset → (dom (glb‘𝐾) = 𝒫 (Base‘𝐾) → dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾))))
41	25, 40	anim12d 621	. . 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 579	. 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 18724	. 2 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
44	1, 2, 26	islat 18657	. 2 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)) ∧ dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))))
45	42, 43, 44	3imtr4i 295	1 ⊢ (𝐾 ∈ CLat → 𝐾 ∈ Lat)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560  {cpr 4590  ⟨cop 4840   × cxp 5656  dom cdm 5658  Rel wrel 5663  ‘cfv 6566  Basecbs 17431  Posetcpo 18531  lubclub 18533  glbcglb 18534  joincjn 18535  meetcmee 18536  Latclat 18655  CLatccla 18722

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-oprab 7455  df-lub 18568  df-glb 18569  df-join 18570  df-meet 18571  df-lat 18656  df-clat 18723

This theorem is used by:  lubel  18738  lubun  18739  clatleglb  18742  topdlat  50449