Theorem joinval2 18568

Description: Value of join for a poset with LUB expanded. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 11-Sep-2018.)

Hypotheses

Ref	Expression
joinval2.b	⊢ 𝐵 = (Base‘𝐾)
joinval2.l	⊢ ≤ = (le‘𝐾)
joinval2.j	⊢ ∨ = (join‘𝐾)
joinval2.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
joinval2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
joinval2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
joinval2	⊢ (𝜑 → (𝑋 ∨ 𝑌) = (℩𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))))

Distinct variable groups:   𝑥,𝑧,𝐵   𝑥, ∨ ,𝑧   𝑥,𝐾,𝑧   𝑥,𝑋,𝑧   𝑥,𝑌,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑧)   ≤ (𝑥, 𝑧)   𝑉(𝑥, 𝑧)

Proof of Theorem joinval2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
2		joinval2.j	. . 3 ⊢ ∨ = (join‘𝐾)
3		joinval2.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
4		joinval2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		joinval2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	1, 2, 3, 4, 5	joinval 18564	. 2 ⊢ (𝜑 → (𝑋 ∨ 𝑌) = ((lub‘𝐾)‘{𝑋, 𝑌}))
7		joinval2.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
8		joinval2.l	. . 3 ⊢ ≤ = (le‘𝐾)
9		biid 264	. . 3 ⊢ ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
10	4, 5	prssd 4781	. . 3 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝐵)
11	7, 8, 1, 9, 3, 10	lubval 18543	. 2 ⊢ (𝜑 → ((lub‘𝐾)‘{𝑋, 𝑌}) = (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
12	7, 8, 2, 3, 4, 5	joinval2lem 18567	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
13	12	riotabidv 7389	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) = (℩𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
14	4, 5, 13	syl2anc 596	. 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) = (℩𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
15	6, 11, 14	3eqtrd 2802	1 ⊢ (𝜑 → (𝑋 ∨ 𝑌) = (℩𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∀wral 3080  {cpr 4590   class class class wbr 5106  ‘cfv 6547  ℩crio 7386  (class class class)co 7430  Basecbs 17402  lecple 17450  lubclub 18498  joincjn 18500

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

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 2101  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 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-lub 18533  df-join 18535

This theorem is used by:  joinlem  18570