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Theorem cdleme26eALTN 39886
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), fz(s), fz(t) respectively. When t ∨ v = p ∨ q, fz(s) ≀ fz(t) ∨ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme26.b 𝐡 = (Baseβ€˜πΎ)
cdleme26.l ≀ = (leβ€˜πΎ)
cdleme26.j ∨ = (joinβ€˜πΎ)
cdleme26.m ∧ = (meetβ€˜πΎ)
cdleme26.a 𝐴 = (Atomsβ€˜πΎ)
cdleme26.h 𝐻 = (LHypβ€˜πΎ)
cdleme26eALT.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdleme26eALT.f 𝐹 = ((𝑦 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑦) ∧ π‘Š)))
cdleme26eALT.g 𝐺 = ((𝑧 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ π‘Š)))
cdleme26eALT.n 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑦) ∧ π‘Š)))
cdleme26eALT.o 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ π‘Š)))
cdleme26eALT.i 𝐼 = (℩𝑒 ∈ 𝐡 βˆ€π‘¦ ∈ 𝐴 ((Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁))
cdleme26eALT.e 𝐸 = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑂))
Assertion
Ref Expression
cdleme26eALTN ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝐼 ≀ (𝐸 ∨ 𝑉))
Distinct variable groups:   𝑦,𝑧,𝑒,𝐴   𝑦,𝐡,𝑧,𝑒   𝑦,𝐻,𝑧   𝑦, ∨ ,𝑧,𝑒   𝑦,𝐾,𝑧   𝑦, ≀ ,𝑧,𝑒   𝑦, ∧ ,𝑧,𝑒   𝑒,𝑁   𝑒,𝑂   𝑦,𝑃,𝑧,𝑒   𝑦,𝑄,𝑧,𝑒   𝑦,𝑆,𝑒   𝑧,𝑇,𝑒   𝑦,π‘ˆ,𝑧,𝑒   𝑦,π‘Š,𝑧,𝑒
Allowed substitution hints:   𝑆(𝑧)   𝑇(𝑦)   𝐸(𝑦,𝑧,𝑒)   𝐹(𝑦,𝑧,𝑒)   𝐺(𝑦,𝑧,𝑒)   𝐻(𝑒)   𝐼(𝑦,𝑧,𝑒)   𝐾(𝑒)   𝑁(𝑦,𝑧)   𝑂(𝑦,𝑧)   𝑉(𝑦,𝑧,𝑒)

Proof of Theorem cdleme26eALTN
StepHypRef Expression
1 simp11l 1281 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝐾 ∈ HL)
2 simp11r 1282 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ π‘Š ∈ 𝐻)
3 simp231 1314 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝑇 ∈ 𝐴)
4 simp12 1201 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
5 simp13 1202 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
6 simp21 1203 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝑃 β‰  𝑄)
7 simp221 1311 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝑆 ∈ 𝐴)
8 simp31 1206 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ (𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)))
9 simp21 1203 . . . . 5 (((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑦 ∈ 𝐴)
1093ad2ant3 1132 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝑦 ∈ 𝐴)
11 simp322 1321 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ Β¬ 𝑦 ≀ π‘Š)
12 simp31 1206 . . . . . 6 (((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑧 ∈ 𝐴)
13123ad2ant3 1132 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝑧 ∈ 𝐴)
14 simp332 1324 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ Β¬ 𝑧 ≀ π‘Š)
1513, 14jca 510 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š))
1610, 11, 15jca31 513 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ ((𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š)))
17 cdleme26.l . . . 4 ≀ = (leβ€˜πΎ)
18 cdleme26.j . . . 4 ∨ = (joinβ€˜πΎ)
19 cdleme26.m . . . 4 ∧ = (meetβ€˜πΎ)
20 cdleme26.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
21 cdleme26.h . . . 4 𝐻 = (LHypβ€˜πΎ)
22 cdleme26eALT.u . . . 4 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
23 cdleme26eALT.f . . . 4 𝐹 = ((𝑦 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑦) ∧ π‘Š)))
24 cdleme26eALT.g . . . 4 𝐺 = ((𝑧 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ π‘Š)))
25 cdleme26eALT.n . . . 4 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑦) ∧ π‘Š)))
26 cdleme26eALT.o . . . 4 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ π‘Š)))
2717, 18, 19, 20, 21, 22, 23, 24, 25, 26cdleme22eALTN 39870 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑃 β‰  𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š)))) β†’ 𝑁 ≀ (𝑂 ∨ 𝑉))
281, 2, 3, 4, 5, 6, 7, 8, 16, 27syl333anc 1399 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝑁 ≀ (𝑂 ∨ 𝑉))
29 simp11 1200 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
30 simp222 1312 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ Β¬ 𝑆 ≀ π‘Š)
31 simp223 1313 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝑆 ≀ (𝑃 ∨ 𝑄))
32 cdleme26.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
33 cdleme26eALT.i . . . . 5 𝐼 = (℩𝑒 ∈ 𝐡 βˆ€π‘¦ ∈ 𝐴 ((Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁))
3432, 17, 18, 19, 20, 21, 22, 23, 25, 33cdleme25cl 39882 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐼 ∈ 𝐡)
3529, 4, 5, 7, 30, 6, 31, 34syl322anc 1395 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝐼 ∈ 𝐡)
36 simp323 1322 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄))
3732fvexi 6904 . . . 4 𝐡 ∈ V
3837, 33riotasv 38483 . . 3 ((𝐼 ∈ 𝐡 ∧ 𝑦 ∈ 𝐴 ∧ (Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐼 = 𝑁)
3935, 10, 11, 36, 38syl112anc 1371 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝐼 = 𝑁)
40 simp232 1315 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ Β¬ 𝑇 ≀ π‘Š)
41 simp233 1316 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝑇 ≀ (𝑃 ∨ 𝑄))
42 cdleme26eALT.e . . . . . 6 𝐸 = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑂))
4332, 17, 18, 19, 20, 21, 22, 24, 26, 42cdleme25cl 39882 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐸 ∈ 𝐡)
4429, 4, 5, 3, 40, 6, 41, 43syl322anc 1395 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝐸 ∈ 𝐡)
45 simp333 1325 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄))
4637, 42riotasv 38483 . . . 4 ((𝐸 ∈ 𝐡 ∧ 𝑧 ∈ 𝐴 ∧ (Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐸 = 𝑂)
4744, 13, 14, 45, 46syl112anc 1371 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝐸 = 𝑂)
4847oveq1d 7428 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ (𝐸 ∨ 𝑉) = (𝑂 ∨ 𝑉))
4928, 39, 483brtr4d 5176 1 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š ∧ 𝑆 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ Β¬ 𝑇 ≀ π‘Š ∧ 𝑇 ≀ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≀ π‘Š ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ Β¬ 𝑦 ≀ π‘Š ∧ Β¬ 𝑦 ≀ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝐼 ≀ (𝐸 ∨ 𝑉))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆ€wral 3051   class class class wbr 5144  β€˜cfv 6543  β„©crio 7368  (class class class)co 7413  Basecbs 17174  lecple 17234  joincjn 18297  meetcmee 18298  Atomscatm 38787  HLchlt 38874  LHypclh 39509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-riotaBAD 38477
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-iin 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7987  df-2nd 7988  df-undef 8272  df-proset 18281  df-poset 18299  df-plt 18316  df-lub 18332  df-glb 18333  df-join 18334  df-meet 18335  df-p0 18411  df-p1 18412  df-lat 18418  df-clat 18485  df-oposet 38700  df-ol 38702  df-oml 38703  df-covers 38790  df-ats 38791  df-atl 38822  df-cvlat 38846  df-hlat 38875  df-llines 39023  df-lplanes 39024  df-lvols 39025  df-lines 39026  df-psubsp 39028  df-pmap 39029  df-padd 39321  df-lhyp 39513
This theorem is referenced by: (None)
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