Theorem isclatd 48971

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 2729	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2729	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2729	. . . . 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 18310	. . . 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 4043	. . . 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 3991	. . 3 ⊢ (𝜑 → dom (lub‘𝐾) ⊆ 𝒫 (Base‘𝐾))
9		elpwi 4570	. . . . . . 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 3935	. . . . 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 4580	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17		isclatd.u	. . . . 5 ⊢ (𝜑 → 𝑈 = (lub‘𝐾))
18	17	dmeqd 5869	. . . 4 ⊢ (𝜑 → dom 𝑈 = dom (lub‘𝐾))
19	14, 16, 18	3sstr3d 4001	. . 3 ⊢ (𝜑 → 𝒫 (Base‘𝐾) ⊆ dom (lub‘𝐾))
20	8, 19	eqssd 3964	. 2 ⊢ (𝜑 → dom (lub‘𝐾) = 𝒫 (Base‘𝐾))
21		eqid 2729	. . . . 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 18323	. . . 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 4043	. . . 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 3991	. . 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 3935	. . . . 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 5869	. . . 4 ⊢ (𝜑 → dom 𝐺 = dom (glb‘𝐾))
33	30, 16, 32	3sstr3d 4001	. . 3 ⊢ (𝜑 → 𝒫 (Base‘𝐾) ⊆ dom (glb‘𝐾))
34	25, 33	eqssd 3964	. 2 ⊢ (𝜑 → dom (glb‘𝐾) = 𝒫 (Base‘𝐾))
35	2, 4, 21	isclat 18459	. . 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 1540   ∈ wcel 2109  ∀wral 3044  ∃!wreu 3352  {crab 3405   ⊆ wss 3914  𝒫 cpw 4563   class class class wbr 5107  dom cdm 5638  ‘cfv 6511  Basecbs 17179  lecple 17227  Posetcpo 18268  lubclub 18270  glbcglb 18271  CLatccla 18457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-lub 18305  df-glb 18306  df-clat 18458

This theorem is referenced by:  mreclat  48985  topclat  48986