Theorem isclatd 50428

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 2763	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2763	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
4		eqid 2763	. . . . 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 18573	. . . 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 4008	. . . 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 3957	. . 3 ⊢ (𝜑 → dom (lub‘𝐾) ⊆ 𝒫 (Base‘𝐾))
9		elpwi 4567	. . . . . . 7 ⊢ (𝑠 ∈ 𝒫 𝐵 → 𝑠 ⊆ 𝐵)
10		isclatd.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝑈)
11	9, 10	sylan2 605	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ dom 𝑈)
12	11	ralrimiva 3158	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝑈)
13		dfss3 3902	. . . . 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 4577	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17		isclatd.u	. . . . 5 ⊢ (𝜑 → 𝑈 = (lub‘𝐾))
18	17	dmeqd 5899	. . . 4 ⊢ (𝜑 → dom 𝑈 = dom (lub‘𝐾))
19	14, 16, 18	3sstr3d 3967	. . 3 ⊢ (𝜑 → 𝒫 (Base‘𝐾) ⊆ dom (lub‘𝐾))
20	8, 19	eqssd 3930	. 2 ⊢ (𝜑 → dom (lub‘𝐾) = 𝒫 (Base‘𝐾))
21		eqid 2763	. . . . 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 18586	. . . 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 4008	. . . 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 3957	. . 3 ⊢ (𝜑 → dom (glb‘𝐾) ⊆ 𝒫 (Base‘𝐾))
26		isclatd.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝐺)
27	9, 26	sylan2 605	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ dom 𝐺)
28	27	ralrimiva 3158	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝐵𝑠 ∈ dom 𝐺)
29		dfss3 3902	. . . . 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 5899	. . . 4 ⊢ (𝜑 → dom 𝐺 = dom (glb‘𝐾))
33	30, 16, 32	3sstr3d 3967	. . 3 ⊢ (𝜑 → 𝒫 (Base‘𝐾) ⊆ dom (glb‘𝐾))
34	25, 33	eqssd 3930	. 2 ⊢ (𝜑 → dom (glb‘𝐾) = 𝒫 (Base‘𝐾))
35	2, 4, 21	isclat 18724	. . 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 850	1 ⊢ (𝜑 → 𝐾 ∈ CLat)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃!wreu 3367  {crab 3416   ⊆ wss 3900  𝒫 cpw 4560   class class class wbr 5108  dom cdm 5658  ‘cfv 6566  Basecbs 17431  lecple 17479  Posetcpo 18531  lubclub 18533  glbcglb 18534  CLatccla 18722

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-lub 18568  df-glb 18569  df-clat 18723

This theorem is used by:  mreclat  50442  topclat  50443