Theorem joinval2lem 16831

Description: Lemma for joinval2 16832 and joineu 16833. (Contributed by NM, 12-Sep-2018.) TODO: combine this through joineu into joinlem?

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
joinval2lem	⊢ ((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))))

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

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

Proof of Theorem joinval2lem

Step	Hyp	Ref	Expression
1		breq1 4586	. . 3 ⊢ (𝑦 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑋 ≤ 𝑥))
2		breq1 4586	. . 3 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑥 ↔ 𝑌 ≤ 𝑥))
3	1, 2	ralprg 4181	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ↔ (𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥)))
4		breq1 4586	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≤ 𝑧 ↔ 𝑋 ≤ 𝑧))
5		breq1 4586	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑧 ↔ 𝑌 ≤ 𝑧))
6	4, 5	ralprg 4181	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 ↔ (𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧)))
7	6	imbi1d 330	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
8	7	ralbidv 2969	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
9	3, 8	anbi12d 743	1 ⊢ ((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))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {cpr 4127   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775  joincjn 16767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584

This theorem is referenced by:  joinval2  16832  joineu  16833