Theorem isclatd 47561

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 2732	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2732	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2732	. . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾)
5		biid 260	. . . . 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 18300	. . . 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 4076	. . . 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 4035	. . 3 ⊢ (𝜑 → dom (lub‘𝐾) ⊆ 𝒫 (Base‘𝐾))
9		elpwi 4608	. . . . . . 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 3146	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝑈)
13		dfss3 3969	. . . . 5 ⊢ (𝒫 𝐵 ⊆ dom 𝑈 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝑈)
14	12, 13	sylibr 233	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 ⊆ dom 𝑈)
15		isclatd.b	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
16	15	pweqd 4618	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17		isclatd.u	. . . . 5 ⊢ (𝜑 → 𝑈 = (lub‘𝐾))
18	17	dmeqd 5903	. . . 4 ⊢ (𝜑 → dom 𝑈 = dom (lub‘𝐾))
19	14, 16, 18	3sstr3d 4027	. . 3 ⊢ (𝜑 → 𝒫 (Base‘𝐾) ⊆ dom (lub‘𝐾))
20	8, 19	eqssd 3998	. 2 ⊢ (𝜑 → dom (lub‘𝐾) = 𝒫 (Base‘𝐾))
21		eqid 2732	. . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾)
22		biid 260	. . . . 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 18313	. . . 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 4076	. . . 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 4035	. . 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 3146	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝐺)
29		dfss3 3969	. . . . 5 ⊢ (𝒫 𝐵 ⊆ dom 𝐺 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝐺)
30	28, 29	sylibr 233	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 ⊆ dom 𝐺)
31		isclatd.g	. . . . 5 ⊢ (𝜑 → 𝐺 = (glb‘𝐾))
32	31	dmeqd 5903	. . . 4 ⊢ (𝜑 → dom 𝐺 = dom (glb‘𝐾))
33	30, 16, 32	3sstr3d 4027	. . 3 ⊢ (𝜑 → 𝒫 (Base‘𝐾) ⊆ dom (glb‘𝐾))
34	25, 33	eqssd 3998	. 2 ⊢ (𝜑 → dom (glb‘𝐾) = 𝒫 (Base‘𝐾))
35	2, 4, 21	isclat 18449	. . 3 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
36	35	biimpri 227	. 2 ⊢ ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))) → 𝐾 ∈ CLat)
37	1, 20, 34, 36	syl12anc 835	1 ⊢ (𝜑 → 𝐾 ∈ CLat)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃!wreu 3374  {crab 3432   ⊆ wss 3947  𝒫 cpw 4601   class class class wbr 5147  dom cdm 5675  ‘cfv 6540  Basecbs 17140  lecple 17200  Posetcpo 18256  lubclub 18258  glbcglb 18259  CLatccla 18447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-lub 18295  df-glb 18296  df-clat 18448

This theorem is referenced by:  mreclat  47575  topclat  47576