Theorem cdleme26eALTN 41155

Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), f_z(s), f_z(t) respectively. When t ∨ v = p ∨ q, f_z(s) ≤ f_z(t) ∨ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme26.b	⊢ 𝐵 = (Base‘𝐾)
cdleme26.l	⊢ ≤ = (le‘𝐾)
cdleme26.j	⊢ ∨ = (join‘𝐾)
cdleme26.m	⊢ ∧ = (meet‘𝐾)
cdleme26.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme26.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme26eALT.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme26eALT.f	⊢ 𝐹 = ((𝑦 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑦) ∧ 𝑊)))
cdleme26eALT.g	⊢ 𝐺 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
cdleme26eALT.n	⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑦) ∧ 𝑊)))
cdleme26eALT.o	⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
cdleme26eALT.i	⊢ 𝐼 = (℩𝑢 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))
cdleme26eALT.e	⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))

Assertion

Ref	Expression
cdleme26eALTN	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝐼 ≤ (𝐸 ∨ 𝑉))

Distinct variable groups:   𝑦,𝑧,𝑢,𝐴   𝑦,𝐵,𝑧,𝑢   𝑦,𝐻,𝑧   𝑦, ∨ ,𝑧,𝑢   𝑦,𝐾,𝑧   𝑦, ≤ ,𝑧,𝑢   𝑦, ∧ ,𝑧,𝑢   𝑢,𝑁   𝑢,𝑂   𝑦,𝑃,𝑧,𝑢   𝑦,𝑄,𝑧,𝑢   𝑦,𝑆,𝑢   𝑧,𝑇,𝑢   𝑦,𝑈,𝑧,𝑢   𝑦,𝑊,𝑧,𝑢

Allowed substitution hints:   𝑆(𝑧)   𝑇(𝑦)   𝐸(𝑦,𝑧,𝑢)   𝐹(𝑦,𝑧,𝑢)   𝐺(𝑦,𝑧,𝑢)   𝐻(𝑢)   𝐼(𝑦,𝑧,𝑢)   𝐾(𝑢)   𝑁(𝑦,𝑧)   𝑂(𝑦,𝑧)   𝑉(𝑦,𝑧,𝑢)

Proof of Theorem cdleme26eALTN

Step	Hyp	Ref	Expression
1		simp11l 1303	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝐾 ∈ HL)
2		simp11r 1304	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝑊 ∈ 𝐻)
3		simp231 1336	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝑇 ∈ 𝐴)
4		simp12 1223	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
5		simp13 1224	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
6		simp21 1225	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝑃 ≠ 𝑄)
7		simp221 1333	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝑆 ∈ 𝐴)
8		simp31 1228	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)))
9		simp21 1225	. . . . 5 ⊢ (((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → 𝑦 ∈ 𝐴)
10	9	3ad2ant3 1153	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝑦 ∈ 𝐴)
11		simp322 1343	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑦 ≤ 𝑊)
12		simp31 1228	. . . . . 6 ⊢ (((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → 𝑧 ∈ 𝐴)
13	12	3ad2ant3 1153	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝑧 ∈ 𝐴)
14		simp332 1346	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑧 ≤ 𝑊)
15	13, 14	jca 520	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))
16	10, 11, 15	jca31 523	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))
17		cdleme26.l	. . . 4 ⊢ ≤ = (le‘𝐾)
18		cdleme26.j	. . . 4 ⊢ ∨ = (join‘𝐾)
19		cdleme26.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
20		cdleme26.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
21		cdleme26.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
22		cdleme26eALT.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
23		cdleme26eALT.f	. . . 4 ⊢ 𝐹 = ((𝑦 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑦) ∧ 𝑊)))
24		cdleme26eALT.g	. . . 4 ⊢ 𝐺 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
25		cdleme26eALT.n	. . . 4 ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑦) ∧ 𝑊)))
26		cdleme26eALT.o	. . . 4 ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
27	17, 18, 19, 20, 21, 22, 23, 24, 25, 26	cdleme22eALTN 41139	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑁 ≤ (𝑂 ∨ 𝑉))
28	1, 2, 3, 4, 5, 6, 7, 8, 16, 27	syl333anc 1429	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝑁 ≤ (𝑂 ∨ 𝑉))
29		simp11 1222	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
30		simp222 1334	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑆 ≤ 𝑊)
31		simp223 1335	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝑆 ≤ (𝑃 ∨ 𝑄))
32		cdleme26.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
33		cdleme26eALT.i	. . . . 5 ⊢ 𝐼 = (℩𝑢 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))
34	32, 17, 18, 19, 20, 21, 22, 23, 25, 33	cdleme25cl 41151	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐼 ∈ 𝐵)
35	29, 4, 5, 7, 30, 6, 31, 34	syl322anc 1425	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝐼 ∈ 𝐵)
36		simp323 1344	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑦 ≤ (𝑃 ∨ 𝑄))
37	32	fvexi 6895	. . . 4 ⊢ 𝐵 ∈ V
38	37, 33	riotasv 39753	. . 3 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴 ∧ (¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄))) → 𝐼 = 𝑁)
39	35, 10, 11, 36, 38	syl112anc 1401	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝐼 = 𝑁)
40		simp232 1337	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑇 ≤ 𝑊)
41		simp233 1338	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝑇 ≤ (𝑃 ∨ 𝑄))
42		cdleme26eALT.e	. . . . . 6 ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))
43	32, 17, 18, 19, 20, 21, 22, 24, 26, 42	cdleme25cl 41151	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) → 𝐸 ∈ 𝐵)
44	29, 4, 5, 3, 40, 6, 41, 43	syl322anc 1425	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝐸 ∈ 𝐵)
45		simp333 1347	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))
46	37, 42	riotasv 39753	. . . 4 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → 𝐸 = 𝑂)
47	44, 13, 14, 45, 46	syl112anc 1401	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝐸 = 𝑂)
48	47	oveq1d 7425	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → (𝐸 ∨ 𝑉) = (𝑂 ∨ 𝑉))
49	28, 39, 48	3brtr4d 5143	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝐼 ≤ (𝐸 ∨ 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079   class class class wbr 5109  ‘cfv 6536  ℩crio 7366  (class class class)co 7410  Basecbs 17264  lecple 17312  joincjn 18362  meetcmee 18363  Atomscatm 40057  HLchlt 40144  LHypclh 40778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-riotaBAD 39747

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-undef 8265  df-proset 18345  df-poset 18364  df-plt 18379  df-lub 18395  df-glb 18396  df-join 18397  df-meet 18398  df-p0 18474  df-p1 18475  df-lat 18483  df-clat 18550  df-oposet 39970  df-ol 39972  df-oml 39973  df-covers 40060  df-ats 40061  df-atl 40092  df-cvlat 40116  df-hlat 40145  df-llines 40292  df-lplanes 40293  df-lvols 40294  df-lines 40295  df-psubsp 40297  df-pmap 40298  df-padd 40590  df-lhyp 40782

This theorem is referenced by: (None)