Theorem meetval2lem 18027

Description: Lemma for meetval2 18028 and meeteu 18029. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu 18029 into meetlem 18030?

Hypotheses

Ref	Expression
meetval2.b	⊢ 𝐵 = (Base‘𝐾)
meetval2.l	⊢ ≤ = (le‘𝐾)
meetval2.m	⊢ ∧ = (meet‘𝐾)
meetval2.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
meetval2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
meetval2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
meetval2lem	⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))))

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)   ≤ (𝑥,𝑧)   ∧ (𝑦)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem meetval2lem

Step	Hyp	Ref	Expression
1		breq2 5074	. . 3 ⊢ (𝑦 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑋))
2		breq2 5074	. . 3 ⊢ (𝑦 = 𝑌 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑌))
3	1, 2	ralprg 4627	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ↔ (𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌)))
4		breq2 5074	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 𝑋))
5		breq2 5074	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 𝑌))
6	4, 5	ralprg 4627	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 ↔ (𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌)))
7	6	imbi1d 341	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)))
8	7	ralbidv 3120	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)))
9	3, 8	anbi12d 630	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {cpr 4560   class class class wbr 5070  ‘cfv 6418  Basecbs 16840  lecple 16895  meetcmee 17945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071

This theorem is referenced by:  meetval2  18028  meeteu  18029  meetdm3  46153