Theorem meetval2lem 18298

Description: Lemma for meetval2 18299 and meeteu 18300. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu 18300 into meetlem 18301?

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 5095	. . 3 ⊢ (𝑦 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑋))
2		breq2 5095	. . 3 ⊢ (𝑦 = 𝑌 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑌))
3	1, 2	ralprg 4649	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ↔ (𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌)))
4		breq2 5095	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 𝑋))
5		breq2 5095	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 𝑌))
6	4, 5	ralprg 4649	. . . 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 3155	. 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 632	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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {cpr 4578   class class class wbr 5091  ‘cfv 6481  Basecbs 17120  lecple 17168  meetcmee 18218

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 2113  ax-9 2121  ax-ext 2703

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-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092

This theorem is referenced by:  meetval2  18299  meeteu  18300  meetdm3  49008