Theorem meetdm3 49592

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 49591	. 2 ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺))
6		simprl 780	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
7		simprr 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
8	6, 7	prssd 4780	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → {𝑥, 𝑦} ⊆ 𝐵)
9		meetdm3.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
10		biid 263	. . . . . . 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 18396	. . . . . 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 547	. . . . 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 600	. . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ 𝑉)
15	1, 9, 4, 14, 6, 7	meetval2lem 18424	. . . . . 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 3383	. . . . 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 485	. . . 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 281	. . 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 3224	. 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 281	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 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃!wreu 3365   ⊆ wss 3904  {cpr 4584   class class class wbr 5100   × cxp 5645  dom cdm 5647  ‘cfv 6521  Basecbs 17245  lecple 17293  glbcglb 18342  meetcmee 18344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-oprab 7400  df-glb 18377  df-meet 18379

This theorem is referenced by: (None)