Theorem isclatd 45885

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 2736	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2736	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2736	. . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾)
5		biid 264	. . . . 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 17811	. . . 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 3979	. . . 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 3941	. . 3 ⊢ (𝜑 → dom (lub‘𝐾) ⊆ 𝒫 (Base‘𝐾))
9		elpwi 4508	. . . . . . 7 ⊢ (𝑠 ∈ 𝒫 𝐵 → 𝑠 ⊆ 𝐵)
10		isclatd.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝑈)
11	9, 10	sylan2 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ dom 𝑈)
12	11	ralrimiva 3095	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝑈)
13		dfss3 3875	. . . . 5 ⊢ (𝒫 𝐵 ⊆ dom 𝑈 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝑈)
14	12, 13	sylibr 237	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 ⊆ dom 𝑈)
15		isclatd.b	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
16	15	pweqd 4518	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17		isclatd.u	. . . . 5 ⊢ (𝜑 → 𝑈 = (lub‘𝐾))
18	17	dmeqd 5759	. . . 4 ⊢ (𝜑 → dom 𝑈 = dom (lub‘𝐾))
19	14, 16, 18	3sstr3d 3933	. . 3 ⊢ (𝜑 → 𝒫 (Base‘𝐾) ⊆ dom (lub‘𝐾))
20	8, 19	eqssd 3904	. 2 ⊢ (𝜑 → dom (lub‘𝐾) = 𝒫 (Base‘𝐾))
21		eqid 2736	. . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾)
22		biid 264	. . . . 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 17824	. . . 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 3979	. . . 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 3941	. . 3 ⊢ (𝜑 → dom (glb‘𝐾) ⊆ 𝒫 (Base‘𝐾))
26		isclatd.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝐺)
27	9, 26	sylan2 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ dom 𝐺)
28	27	ralrimiva 3095	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝐺)
29		dfss3 3875	. . . . 5 ⊢ (𝒫 𝐵 ⊆ dom 𝐺 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝐺)
30	28, 29	sylibr 237	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 ⊆ dom 𝐺)
31		isclatd.g	. . . . 5 ⊢ (𝜑 → 𝐺 = (glb‘𝐾))
32	31	dmeqd 5759	. . . 4 ⊢ (𝜑 → dom 𝐺 = dom (glb‘𝐾))
33	30, 16, 32	3sstr3d 3933	. . 3 ⊢ (𝜑 → 𝒫 (Base‘𝐾) ⊆ dom (glb‘𝐾))
34	25, 33	eqssd 3904	. 2 ⊢ (𝜑 → dom (glb‘𝐾) = 𝒫 (Base‘𝐾))
35	2, 4, 21	isclat 17960	. . 3 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
36	35	biimpri 231	. 2 ⊢ ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))) → 𝐾 ∈ CLat)
37	1, 20, 34, 36	syl12anc 837	1 ⊢ (𝜑 → 𝐾 ∈ CLat)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∀wral 3051  ∃!wreu 3053  {crab 3055   ⊆ wss 3853  𝒫 cpw 4499   class class class wbr 5039  dom cdm 5536  ‘cfv 6358  Basecbs 16666  lecple 16756  Posetcpo 17768  lubclub 17770  glbcglb 17771  CLatccla 17958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-lub 17806  df-glb 17807  df-clat 17959

This theorem is referenced by:  mreclat  45899  topclat  45900