Theorem joindm3 50176

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

Hypotheses

Ref	Expression
joindm2.b	⊢ 𝐵 = (Base‘𝐾)
joindm2.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
joindm2.u	⊢ 𝑈 = (lub‘𝐾)
joindm2.j	⊢ ∨ = (join‘𝐾)
joindm3.l	⊢ ≤ = (le‘𝐾)

Assertion

Ref	Expression
joindm3	⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ∧ ∀𝑤 ∈ 𝐵 ((𝑥 ≤ 𝑤 ∧ 𝑦 ≤ 𝑤) → 𝑧 ≤ 𝑤))))

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

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

Proof of Theorem joindm3

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		joindm2.u	. . 3 ⊢ 𝑈 = (lub‘𝐾)
4		joindm2.j	. . 3 ⊢ ∨ = (join‘𝐾)
5	1, 2, 3, 4	joindm2 50175	. 2 ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈))
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 4777	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → {𝑥, 𝑦} ⊆ 𝐵)
9		joindm3.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
10		biid 264	. . . . . . 7 ⊢ ((∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑤 → 𝑧 ≤ 𝑤)) ↔ (∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑤 → 𝑧 ≤ 𝑤)))
11	1, 9, 3, 10, 2	lubeldm 18530	. . . . . 6 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃!𝑧 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑤 → 𝑧 ≤ 𝑤)))))
12	11	baibd 549	. . . . 5 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ⊆ 𝐵) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ ∃!𝑧 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑤 → 𝑧 ≤ 𝑤))))
13	8, 12	syldan 603	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ ∃!𝑧 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑤 → 𝑧 ≤ 𝑤))))
14	2	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ 𝑉)
15	1, 9, 4, 14, 6, 7	joinval2lem 18557	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑤 → 𝑧 ≤ 𝑤)) ↔ ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ∧ ∀𝑤 ∈ 𝐵 ((𝑥 ≤ 𝑤 ∧ 𝑦 ≤ 𝑤) → 𝑧 ≤ 𝑤))))
16	15	reubidv 3381	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (∃!𝑧 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑤 → 𝑧 ≤ 𝑤)) ↔ ∃!𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ∧ ∀𝑤 ∈ 𝐵 ((𝑥 ≤ 𝑤 ∧ 𝑦 ≤ 𝑤) → 𝑧 ≤ 𝑤))))
17	16	adantl 487	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃!𝑧 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 ≤ 𝑤 → 𝑧 ≤ 𝑤)) ↔ ∃!𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ∧ ∀𝑤 ∈ 𝐵 ((𝑥 ≤ 𝑤 ∧ 𝑦 ≤ 𝑤) → 𝑧 ≤ 𝑤))))
18	13, 17	bitrd 282	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ ∃!𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ∧ ∀𝑤 ∈ 𝐵 ((𝑥 ≤ 𝑤 ∧ 𝑦 ≤ 𝑤) → 𝑧 ≤ 𝑤))))
19	18	2ralbidva 3224	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ∧ ∀𝑤 ∈ 𝐵 ((𝑥 ≤ 𝑤 ∧ 𝑦 ≤ 𝑤) → 𝑧 ≤ 𝑤))))
20	5, 19	bitrd 282	1 ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ∧ ∀𝑤 ∈ 𝐵 ((𝑥 ≤ 𝑤 ∧ 𝑦 ≤ 𝑤) → 𝑧 ≤ 𝑤))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃!wreu 3363   ⊆ wss 3895  {cpr 4586   class class class wbr 5102   × cxp 5648  dom cdm 5650  ‘cfv 6538  Basecbs 17392  lecple 17440  lubclub 18488  joincjn 18490

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-oprab 7424  df-lub 18523  df-join 18525

This theorem is used by: (None)