Theorem isclatd 49107

Description: The predicate "is a complete lattice" (deduction form). (Contributed by Zhi Wang, 29-Sep-2024.)

Hypotheses

Ref	Expression
isclatd.b	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
isclatd.u	⊢ (𝜑 → 𝑈 = (lub‘𝐾))
isclatd.g	⊢ (𝜑 → 𝐺 = (glb‘𝐾))
isclatd.k	⊢ (𝜑 → 𝐾 ∈ Poset)
isclatd.1	⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝑈)
isclatd.2	⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝐺)

Assertion

Ref	Expression
isclatd	⊢ (𝜑 → 𝐾 ∈ CLat)

Distinct variable groups:   𝐵,𝑠   𝐺,𝑠   𝑈,𝑠   𝜑,𝑠

Allowed substitution hint:   𝐾(𝑠)

Proof of Theorem isclatd

Dummy variables 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isclatd.k	. 2 ⊢ (𝜑 → 𝐾 ∈ Poset)
2		eqid 2733	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2733	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2733	. . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾)
5		biid 261	. . . . 5 ⊢ ((∀𝑦 ∈ 𝑡 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑡 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦 ∈ 𝑡 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑡 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
6	2, 3, 4, 5, 1	lubdm 18257	. . . 4 ⊢ (𝜑 → dom (lub‘𝐾) = {𝑡 ∈ 𝒫 (Base‘𝐾) ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑡 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑡 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))})
7		ssrab2 4029	. . . 4 ⊢ {𝑡 ∈ 𝒫 (Base‘𝐾) ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑡 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑡 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))} ⊆ 𝒫 (Base‘𝐾)
8	6, 7	eqsstrdi 3975	. . 3 ⊢ (𝜑 → dom (lub‘𝐾) ⊆ 𝒫 (Base‘𝐾))
9		elpwi 4556	. . . . . . 7 ⊢ (𝑠 ∈ 𝒫 𝐵 → 𝑠 ⊆ 𝐵)
10		isclatd.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝑈)
11	9, 10	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ dom 𝑈)
12	11	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝑈)
13		dfss3 3919	. . . . 5 ⊢ (𝒫 𝐵 ⊆ dom 𝑈 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝑈)
14	12, 13	sylibr 234	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 ⊆ dom 𝑈)
15		isclatd.b	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
16	15	pweqd 4566	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17		isclatd.u	. . . . 5 ⊢ (𝜑 → 𝑈 = (lub‘𝐾))
18	17	dmeqd 5849	. . . 4 ⊢ (𝜑 → dom 𝑈 = dom (lub‘𝐾))
19	14, 16, 18	3sstr3d 3985	. . 3 ⊢ (𝜑 → 𝒫 (Base‘𝐾) ⊆ dom (lub‘𝐾))
20	8, 19	eqssd 3948	. 2 ⊢ (𝜑 → dom (lub‘𝐾) = 𝒫 (Base‘𝐾))
21		eqid 2733	. . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾)
22		biid 261	. . . . 5 ⊢ ((∀𝑦 ∈ 𝑡 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑡 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)) ↔ (∀𝑦 ∈ 𝑡 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑡 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))
23	2, 3, 21, 22, 1	glbdm 18270	. . . 4 ⊢ (𝜑 → dom (glb‘𝐾) = {𝑡 ∈ 𝒫 (Base‘𝐾) ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑡 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑡 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))})
24		ssrab2 4029	. . . 4 ⊢ {𝑡 ∈ 𝒫 (Base‘𝐾) ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑡 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑡 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))} ⊆ 𝒫 (Base‘𝐾)
25	23, 24	eqsstrdi 3975	. . 3 ⊢ (𝜑 → dom (glb‘𝐾) ⊆ 𝒫 (Base‘𝐾))
26		isclatd.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝐺)
27	9, 26	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ dom 𝐺)
28	27	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝐺)
29		dfss3 3919	. . . . 5 ⊢ (𝒫 𝐵 ⊆ dom 𝐺 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝐺)
30	28, 29	sylibr 234	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 ⊆ dom 𝐺)
31		isclatd.g	. . . . 5 ⊢ (𝜑 → 𝐺 = (glb‘𝐾))
32	31	dmeqd 5849	. . . 4 ⊢ (𝜑 → dom 𝐺 = dom (glb‘𝐾))
33	30, 16, 32	3sstr3d 3985	. . 3 ⊢ (𝜑 → 𝒫 (Base‘𝐾) ⊆ dom (glb‘𝐾))
34	25, 33	eqssd 3948	. 2 ⊢ (𝜑 → dom (glb‘𝐾) = 𝒫 (Base‘𝐾))
35	2, 4, 21	isclat 18408	. . 3 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
36	35	biimpri 228	. 2 ⊢ ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))) → 𝐾 ∈ CLat)
37	1, 20, 34, 36	syl12anc 836	1 ⊢ (𝜑 → 𝐾 ∈ CLat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃!wreu 3345  {crab 3396   ⊆ wss 3898  𝒫 cpw 4549   class class class wbr 5093  dom cdm 5619  ‘cfv 6486  Basecbs 17122  lecple 17170  Posetcpo 18215  lubclub 18217  glbcglb 18218  CLatccla 18406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-lub 18252  df-glb 18253  df-clat 18407

This theorem is referenced by:  mreclat  49121  topclat  49122