Theorem clatl 18554

Description: A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.) TODO: use eqrelrdv2 5804 to shorten proof and eliminate joindmss 18425 and meetdmss 18439?

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 2736	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2736	. . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾)
3		simpl 482	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → 𝐾 ∈ Poset)
4	1, 2, 3	joindmss 18425	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → dom (join‘𝐾) ⊆ ((Base‘𝐾) × (Base‘𝐾)))
5		relxp 5702	. . . . . . . 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 5720	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)))
8		vex 3483	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
9		vex 3483	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
10	8, 9	prss 4819	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
11	7, 10	sylbb 219	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ⊆ (Base‘𝐾))
12		prex 5436	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑦} ∈ V
13	12	elpw 4603	. . . . . . . . . . 11 ⊢ ({𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
14	11, 13	sylibr 234	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾))
15		eleq2 2829	. . . . . . . . . 10 ⊢ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) ↔ {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾)))
16	14, 15	imbitrrid 246	. . . . . . . . 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 2736	. . . . . . . . 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 18422	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ dom (join‘𝐾) ↔ {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
22	17, 21	sylibrd 259	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → ⟨𝑥, 𝑦⟩ ∈ dom (join‘𝐾)))
23	6, 22	relssdv 5797	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ dom (lub‘𝐾) = 𝒫 (Base‘𝐾)) → ((Base‘𝐾) × (Base‘𝐾)) ⊆ dom (join‘𝐾))
24	4, 23	eqssd 4000	. . . . 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 2736	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
27		simpl 482	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → 𝐾 ∈ Poset)
28	1, 26, 27	meetdmss 18439	. . . . . 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 2829	. . . . . . . . . 10 ⊢ (dom (glb‘𝐾) = 𝒫 (Base‘𝐾) → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) ↔ {𝑥, 𝑦} ∈ 𝒫 (Base‘𝐾)))
31	14, 30	imbitrrid 246	. . . . . . . . 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 2736	. . . . . . . . 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 18436	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ dom (meet‘𝐾) ↔ {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
37	32, 36	sylibrd 259	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐾) × (Base‘𝐾)) → ⟨𝑥, 𝑦⟩ ∈ dom (meet‘𝐾)))
38	29, 37	relssdv 5797	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)) → ((Base‘𝐾) × (Base‘𝐾)) ⊆ dom (meet‘𝐾))
39	28, 38	eqssd 4000	. . . . 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 609	. . 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 18546	. 2 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
44	1, 2, 26	islat 18479	. 2 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)) ∧ dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))))
45	42, 43, 44	3imtr4i 292	1 ⊢ (𝐾 ∈ CLat → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3479   ⊆ wss 3950  𝒫 cpw 4599  {cpr 4627  ⟨cop 4631   × cxp 5682  dom cdm 5684  Rel wrel 5689  ‘cfv 6560  Basecbs 17248  Posetcpo 18354  lubclub 18356  glbcglb 18357  joincjn 18358  meetcmee 18359  Latclat 18477  CLatccla 18544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-oprab 7436  df-lub 18392  df-glb 18393  df-join 18394  df-meet 18395  df-lat 18478  df-clat 18545

This theorem is referenced by:  lubel  18560  lubun  18561  clatleglb  18564  topdlat  48908