Theorem meetdm3 46153

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 46152	. 2 ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺))
6		simprl 767	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
7		simprr 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
8	6, 7	prssd 4752	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → {𝑥, 𝑦} ⊆ 𝐵)
9		meetdm3.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
10		biid 260	. . . . . . 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 17999	. . . . . 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 539	. . . . 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 590	. . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ 𝑉)
15	1, 9, 4, 14, 6, 7	meetval2lem 18027	. . . . . 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 3315	. . . . 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 481	. . . 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 278	. . 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 3121	. 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 278	1 ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑥 ∧ 𝑧 ≤ 𝑦) ∧ ∀𝑤 ∈ 𝐵 ((𝑤 ≤ 𝑥 ∧ 𝑤 ≤ 𝑦) → 𝑤 ≤ 𝑧))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃!wreu 3065   ⊆ wss 3883  {cpr 4560   class class class wbr 5070   × cxp 5578  dom cdm 5580  ‘cfv 6418  Basecbs 16840  lecple 16895  glbcglb 17943  meetcmee 17945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-oprab 7259  df-glb 17980  df-meet 17982

This theorem is referenced by: (None)