Theorem meetdm3 50416

Description: The meet of any two elements always exists iff all unordered pairs have GLB (expanded version). (Contributed by Zhi Wang, 25-Sep-2024.)

Hypotheses

Ref	Expression
joindm2.b	⊢ 𝐵 = (Base‘𝐾)
joindm2.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
meetdm2.g	⊢ 𝐺 = (glb‘𝐾)
meetdm2.m	⊢ ∧ = (meet‘𝐾)
meetdm3.l	⊢ ≤ = (le‘𝐾)

Assertion

Ref	Expression
meetdm3	⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑥 ∧ 𝑧 ≤ 𝑦) ∧ ∀𝑤 ∈ 𝐵 ((𝑤 ≤ 𝑥 ∧ 𝑤 ≤ 𝑦) → 𝑤 ≤ 𝑧))))

Distinct variable groups:   𝑤, ∧ ,𝑥,𝑦,𝑧   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝐾,𝑧   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑧, 𝑤)   𝐺(𝑥, 𝑦, 𝑧, 𝑤)   𝐾(𝑥, 𝑦)   ≤ (𝑥, 𝑦, 𝑧, 𝑤)   𝑉(𝑥, 𝑦, 𝑧, 𝑤)

Proof of Theorem meetdm3

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		joindm2.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		joindm2.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
3		meetdm2.g	. . 3 ⊢ 𝐺 = (glb‘𝐾)
4		meetdm2.m	. . 3 ⊢ ∧ = (meet‘𝐾)
5	1, 2, 3, 4	meetdm2 50415	. 2 ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺))
6		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
7		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
8	6, 7	prssd 4783	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → {𝑥, 𝑦} ⊆ 𝐵)
9		meetdm3.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
10		biid 264	. . . . . . 7 ⊢ ((∀𝑣 ∈ {𝑥, 𝑦}𝑧 ≤ 𝑣 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑤 ≤ 𝑣 → 𝑤 ≤ 𝑧)) ↔ (∀𝑣 ∈ {𝑥, 𝑦}𝑧 ≤ 𝑣 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑤 ≤ 𝑣 → 𝑤 ≤ 𝑧)))
11	1, 9, 3, 10, 2	glbeldm 18588	. . . . . 6 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃!𝑧 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑧 ≤ 𝑣 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑤 ≤ 𝑣 → 𝑤 ≤ 𝑧)))))
12	11	baibd 549	. . . . 5 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ⊆ 𝐵) → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ ∃!𝑧 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑧 ≤ 𝑣 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑤 ≤ 𝑣 → 𝑤 ≤ 𝑧))))
13	8, 12	syldan 603	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ ∃!𝑧 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑧 ≤ 𝑣 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑤 ≤ 𝑣 → 𝑤 ≤ 𝑧))))
14	2	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ 𝑉)
15	1, 9, 4, 14, 6, 7	meetval2lem 18616	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((∀𝑣 ∈ {𝑥, 𝑦}𝑧 ≤ 𝑣 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑤 ≤ 𝑣 → 𝑤 ≤ 𝑧)) ↔ ((𝑧 ≤ 𝑥 ∧ 𝑧 ≤ 𝑦) ∧ ∀𝑤 ∈ 𝐵 ((𝑤 ≤ 𝑥 ∧ 𝑤 ≤ 𝑦) → 𝑤 ≤ 𝑧))))
16	15	reubidv 3385	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (∃!𝑧 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑧 ≤ 𝑣 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑤 ≤ 𝑣 → 𝑤 ≤ 𝑧)) ↔ ∃!𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑥 ∧ 𝑧 ≤ 𝑦) ∧ ∀𝑤 ∈ 𝐵 ((𝑤 ≤ 𝑥 ∧ 𝑤 ≤ 𝑦) → 𝑤 ≤ 𝑧))))
17	16	adantl 487	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃!𝑧 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑧 ≤ 𝑣 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑤 ≤ 𝑣 → 𝑤 ≤ 𝑧)) ↔ ∃!𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑥 ∧ 𝑧 ≤ 𝑦) ∧ ∀𝑤 ∈ 𝐵 ((𝑤 ≤ 𝑥 ∧ 𝑤 ≤ 𝑦) → 𝑤 ≤ 𝑧))))
18	13, 17	bitrd 282	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ ∃!𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑥 ∧ 𝑧 ≤ 𝑦) ∧ ∀𝑤 ∈ 𝐵 ((𝑤 ≤ 𝑥 ∧ 𝑤 ≤ 𝑦) → 𝑤 ≤ 𝑧))))
19	18	2ralbidva 3228	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑥 ∧ 𝑧 ≤ 𝑦) ∧ ∀𝑤 ∈ 𝐵 ((𝑤 ≤ 𝑥 ∧ 𝑤 ≤ 𝑦) → 𝑤 ≤ 𝑧))))
20	5, 19	bitrd 282	1 ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑥 ∧ 𝑧 ≤ 𝑦) ∧ ∀𝑤 ∈ 𝐵 ((𝑤 ≤ 𝑥 ∧ 𝑤 ≤ 𝑦) → 𝑤 ≤ 𝑧))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃!wreu 3367   ⊆ wss 3900  {cpr 4590   class class class wbr 5108   × cxp 5656  dom cdm 5658  ‘cfv 6566  Basecbs 17431  lecple 17479  glbcglb 18534  meetcmee 18536

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  ax-un 7783

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-oprab 7455  df-glb 18569  df-meet 18571

This theorem is used by: (None)