Theorem joindm3 49093

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 49092	. 2 ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈))
6		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
7		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
8	6, 7	prssd 4773	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → {𝑥, 𝑦} ⊆ 𝐵)
9		joindm3.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
10		biid 261	. . . . . . 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 18259	. . . . . 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 539	. . . . 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 591	. . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ 𝑉)
15	1, 9, 4, 14, 6, 7	joinval2lem 18286	. . . . . 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 3363	. . . . 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 481	. . . 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 279	. . 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 3195	. 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 279	1 ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ∧ ∀𝑤 ∈ 𝐵 ((𝑥 ≤ 𝑤 ∧ 𝑦 ≤ 𝑤) → 𝑧 ≤ 𝑤))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃!wreu 3345   ⊆ wss 3898  {cpr 4577   class class class wbr 5093   × cxp 5617  dom cdm 5619  ‘cfv 6486  Basecbs 17122  lecple 17170  lubclub 18217  joincjn 18219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-oprab 7356  df-lub 18252  df-join 18254

This theorem is referenced by: (None)