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Theorem lplnexllnN 38073
Description: Given an atom on a lattice plane, there is a lattice line whose join with the atom equals the plane. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lplnexat.l ≀ = (leβ€˜πΎ)
lplnexat.j ∨ = (joinβ€˜πΎ)
lplnexat.a 𝐴 = (Atomsβ€˜πΎ)
lplnexat.n 𝑁 = (LLinesβ€˜πΎ)
lplnexat.p 𝑃 = (LPlanesβ€˜πΎ)
Assertion
Ref Expression
lplnexllnN (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
Distinct variable groups:   𝑦, ∨   𝑦, ≀   𝑦,𝑁   𝑦,𝑄   𝑦,𝑋
Allowed substitution hints:   𝐴(𝑦)   𝑃(𝑦)   𝐾(𝑦)

Proof of Theorem lplnexllnN
Dummy variables 𝑠 π‘Ÿ 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 1193 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) β†’ 𝑋 ∈ 𝑃)
2 simpl1 1192 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) β†’ 𝐾 ∈ HL)
3 eqid 2733 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4 lplnexat.p . . . . . 6 𝑃 = (LPlanesβ€˜πΎ)
53, 4lplnbase 38043 . . . . 5 (𝑋 ∈ 𝑃 β†’ 𝑋 ∈ (Baseβ€˜πΎ))
61, 5syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) β†’ 𝑋 ∈ (Baseβ€˜πΎ))
7 lplnexat.l . . . . 5 ≀ = (leβ€˜πΎ)
8 lplnexat.j . . . . 5 ∨ = (joinβ€˜πΎ)
9 lplnexat.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
10 lplnexat.n . . . . 5 𝑁 = (LLinesβ€˜πΎ)
113, 7, 8, 9, 10, 4islpln3 38042 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ (Baseβ€˜πΎ)) β†’ (𝑋 ∈ 𝑃 ↔ βˆƒπ‘§ ∈ 𝑁 βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))))
122, 6, 11syl2anc 585 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) β†’ (𝑋 ∈ 𝑃 ↔ βˆƒπ‘§ ∈ 𝑁 βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))))
131, 12mpbid 231 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) β†’ βˆƒπ‘§ ∈ 𝑁 βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))
14 simpll1 1213 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝐾 ∈ HL)
15 simpr2l 1233 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑧 ∈ 𝑁)
16 simpll3 1215 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑄 ∈ 𝐴)
17 simpr1 1195 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑄 ≀ 𝑧)
187, 8, 9, 10llnexatN 38030 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑧 ∈ 𝑁 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑧) β†’ βˆƒπ‘  ∈ 𝐴 (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))
1914, 15, 16, 17, 18syl31anc 1374 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ βˆƒπ‘  ∈ 𝐴 (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))
20 simp1l1 1267 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ 𝐾 ∈ HL)
21 simp22r 1294 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ π‘Ÿ ∈ 𝐴)
22 simp3l 1202 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ 𝑠 ∈ 𝐴)
23 simp1l3 1269 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ 𝑄 ∈ 𝐴)
24 simp23l 1295 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ Β¬ π‘Ÿ ≀ 𝑧)
25 simp3rr 1248 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ 𝑧 = (𝑄 ∨ 𝑠))
2625breq2d 5118 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ (π‘Ÿ ≀ 𝑧 ↔ π‘Ÿ ≀ (𝑄 ∨ 𝑠)))
2724, 26mtbid 324 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ Β¬ π‘Ÿ ≀ (𝑄 ∨ 𝑠))
287, 8, 9atnlej2 37889 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ Β¬ π‘Ÿ ≀ (𝑄 ∨ 𝑠)) β†’ π‘Ÿ β‰  𝑠)
2920, 21, 23, 22, 27, 28syl131anc 1384 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ π‘Ÿ β‰  𝑠)
308, 9, 10llni2 38021 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ π‘Ÿ β‰  𝑠) β†’ (π‘Ÿ ∨ 𝑠) ∈ 𝑁)
3120, 21, 22, 29, 30syl31anc 1374 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ (π‘Ÿ ∨ 𝑠) ∈ 𝑁)
32 simp3rl 1247 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ 𝑄 β‰  𝑠)
337, 8, 9hlatcon2 37961 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) ∧ (𝑄 β‰  𝑠 ∧ Β¬ π‘Ÿ ≀ (𝑄 ∨ 𝑠))) β†’ Β¬ 𝑄 ≀ (π‘Ÿ ∨ 𝑠))
3420, 23, 22, 21, 32, 27, 33syl132anc 1389 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ Β¬ 𝑄 ≀ (π‘Ÿ ∨ 𝑠))
35 simp23r 1296 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ 𝑋 = (𝑧 ∨ π‘Ÿ))
3625oveq1d 7373 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ (𝑧 ∨ π‘Ÿ) = ((𝑄 ∨ 𝑠) ∨ π‘Ÿ))
3720hllatd 37872 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ 𝐾 ∈ Lat)
383, 9atbase 37797 . . . . . . . . . . . . 13 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
3923, 38syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
403, 9atbase 37797 . . . . . . . . . . . . 13 (𝑠 ∈ 𝐴 β†’ 𝑠 ∈ (Baseβ€˜πΎ))
4122, 40syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ 𝑠 ∈ (Baseβ€˜πΎ))
423, 9atbase 37797 . . . . . . . . . . . . 13 (π‘Ÿ ∈ 𝐴 β†’ π‘Ÿ ∈ (Baseβ€˜πΎ))
4321, 42syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ π‘Ÿ ∈ (Baseβ€˜πΎ))
443, 8latj31 18381 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Baseβ€˜πΎ) ∧ 𝑠 ∈ (Baseβ€˜πΎ) ∧ π‘Ÿ ∈ (Baseβ€˜πΎ))) β†’ ((𝑄 ∨ 𝑠) ∨ π‘Ÿ) = ((π‘Ÿ ∨ 𝑠) ∨ 𝑄))
4537, 39, 41, 43, 44syl13anc 1373 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ ((𝑄 ∨ 𝑠) ∨ π‘Ÿ) = ((π‘Ÿ ∨ 𝑠) ∨ 𝑄))
4635, 36, 453eqtrd 2777 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ 𝑋 = ((π‘Ÿ ∨ 𝑠) ∨ 𝑄))
47 breq2 5110 . . . . . . . . . . . . 13 (𝑦 = (π‘Ÿ ∨ 𝑠) β†’ (𝑄 ≀ 𝑦 ↔ 𝑄 ≀ (π‘Ÿ ∨ 𝑠)))
4847notbid 318 . . . . . . . . . . . 12 (𝑦 = (π‘Ÿ ∨ 𝑠) β†’ (Β¬ 𝑄 ≀ 𝑦 ↔ Β¬ 𝑄 ≀ (π‘Ÿ ∨ 𝑠)))
49 oveq1 7365 . . . . . . . . . . . . 13 (𝑦 = (π‘Ÿ ∨ 𝑠) β†’ (𝑦 ∨ 𝑄) = ((π‘Ÿ ∨ 𝑠) ∨ 𝑄))
5049eqeq2d 2744 . . . . . . . . . . . 12 (𝑦 = (π‘Ÿ ∨ 𝑠) β†’ (𝑋 = (𝑦 ∨ 𝑄) ↔ 𝑋 = ((π‘Ÿ ∨ 𝑠) ∨ 𝑄)))
5148, 50anbi12d 632 . . . . . . . . . . 11 (𝑦 = (π‘Ÿ ∨ 𝑠) β†’ ((Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)) ↔ (Β¬ 𝑄 ≀ (π‘Ÿ ∨ 𝑠) ∧ 𝑋 = ((π‘Ÿ ∨ 𝑠) ∨ 𝑄))))
5251rspcev 3580 . . . . . . . . . 10 (((π‘Ÿ ∨ 𝑠) ∈ 𝑁 ∧ (Β¬ 𝑄 ≀ (π‘Ÿ ∨ 𝑠) ∧ 𝑋 = ((π‘Ÿ ∨ 𝑠) ∨ 𝑄))) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
5331, 34, 46, 52syl12anc 836 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
54533expia 1122 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ ((𝑠 ∈ 𝐴 ∧ (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠))) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄))))
5554expd 417 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ (𝑠 ∈ 𝐴 β†’ ((𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))))
5655rexlimdv 3147 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ (βˆƒπ‘  ∈ 𝐴 (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄))))
5719, 56mpd 15 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
58573exp2 1355 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) β†’ (𝑄 ≀ 𝑧 β†’ ((𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) β†’ ((Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄))))))
59 simpr2l 1233 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑧 ∈ 𝑁)
60 simpr1 1195 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ Β¬ 𝑄 ≀ 𝑧)
61 simpll1 1213 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝐾 ∈ HL)
6261hllatd 37872 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝐾 ∈ Lat)
633, 10llnbase 38018 . . . . . . . . . . . 12 (𝑧 ∈ 𝑁 β†’ 𝑧 ∈ (Baseβ€˜πΎ))
6459, 63syl 17 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑧 ∈ (Baseβ€˜πΎ))
65 simpr2r 1234 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ π‘Ÿ ∈ 𝐴)
6665, 42syl 17 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ π‘Ÿ ∈ (Baseβ€˜πΎ))
673, 7, 8latlej1 18342 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Baseβ€˜πΎ) ∧ π‘Ÿ ∈ (Baseβ€˜πΎ)) β†’ 𝑧 ≀ (𝑧 ∨ π‘Ÿ))
6862, 64, 66, 67syl3anc 1372 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑧 ≀ (𝑧 ∨ π‘Ÿ))
69 simpr3r 1236 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑋 = (𝑧 ∨ π‘Ÿ))
7068, 69breqtrrd 5134 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑧 ≀ 𝑋)
71 simplr 768 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑄 ≀ 𝑋)
72 simpll3 1215 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑄 ∈ 𝐴)
7372, 38syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
74 simpll2 1214 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑋 ∈ 𝑃)
7574, 5syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑋 ∈ (Baseβ€˜πΎ))
763, 7, 8latjle12 18344 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑧 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ) ∧ 𝑋 ∈ (Baseβ€˜πΎ))) β†’ ((𝑧 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋) ↔ (𝑧 ∨ 𝑄) ≀ 𝑋))
7762, 64, 73, 75, 76syl13anc 1373 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ ((𝑧 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋) ↔ (𝑧 ∨ 𝑄) ≀ 𝑋))
7870, 71, 77mpbi2and 711 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ (𝑧 ∨ 𝑄) ≀ 𝑋)
793, 8latjcl 18333 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (𝑧 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
8062, 64, 73, 79syl3anc 1372 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ (𝑧 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
81 eqid 2733 . . . . . . . . . . . . 13 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
823, 7, 8, 81, 9cvr1 37919 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑧 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ 𝐴) β†’ (Β¬ 𝑄 ≀ 𝑧 ↔ 𝑧( β‹– β€˜πΎ)(𝑧 ∨ 𝑄)))
8361, 64, 72, 82syl3anc 1372 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ (Β¬ 𝑄 ≀ 𝑧 ↔ 𝑧( β‹– β€˜πΎ)(𝑧 ∨ 𝑄)))
8460, 83mpbid 231 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑧( β‹– β€˜πΎ)(𝑧 ∨ 𝑄))
853, 81, 10, 4lplni 38041 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑧 ∨ 𝑄) ∈ (Baseβ€˜πΎ) ∧ 𝑧 ∈ 𝑁) ∧ 𝑧( β‹– β€˜πΎ)(𝑧 ∨ 𝑄)) β†’ (𝑧 ∨ 𝑄) ∈ 𝑃)
8661, 80, 59, 84, 85syl31anc 1374 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ (𝑧 ∨ 𝑄) ∈ 𝑃)
877, 4lplncmp 38071 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑧 ∨ 𝑄) ∈ 𝑃 ∧ 𝑋 ∈ 𝑃) β†’ ((𝑧 ∨ 𝑄) ≀ 𝑋 ↔ (𝑧 ∨ 𝑄) = 𝑋))
8861, 86, 74, 87syl3anc 1372 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ ((𝑧 ∨ 𝑄) ≀ 𝑋 ↔ (𝑧 ∨ 𝑄) = 𝑋))
8978, 88mpbid 231 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ (𝑧 ∨ 𝑄) = 𝑋)
9089eqcomd 2739 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ 𝑋 = (𝑧 ∨ 𝑄))
91 breq2 5110 . . . . . . . . 9 (𝑦 = 𝑧 β†’ (𝑄 ≀ 𝑦 ↔ 𝑄 ≀ 𝑧))
9291notbid 318 . . . . . . . 8 (𝑦 = 𝑧 β†’ (Β¬ 𝑄 ≀ 𝑦 ↔ Β¬ 𝑄 ≀ 𝑧))
93 oveq1 7365 . . . . . . . . 9 (𝑦 = 𝑧 β†’ (𝑦 ∨ 𝑄) = (𝑧 ∨ 𝑄))
9493eqeq2d 2744 . . . . . . . 8 (𝑦 = 𝑧 β†’ (𝑋 = (𝑦 ∨ 𝑄) ↔ 𝑋 = (𝑧 ∨ 𝑄)))
9592, 94anbi12d 632 . . . . . . 7 (𝑦 = 𝑧 β†’ ((Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)) ↔ (Β¬ 𝑄 ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑄))))
9695rspcev 3580 . . . . . 6 ((𝑧 ∈ 𝑁 ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑄))) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
9759, 60, 90, 96syl12anc 836 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) ∧ (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)))) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
98973exp2 1355 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) β†’ (Β¬ 𝑄 ≀ 𝑧 β†’ ((𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) β†’ ((Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄))))))
9958, 98pm2.61d 179 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) β†’ ((𝑧 ∈ 𝑁 ∧ π‘Ÿ ∈ 𝐴) β†’ ((Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))))
10099rexlimdvv 3201 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) β†’ (βˆƒπ‘§ ∈ 𝑁 βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ π‘Ÿ)) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄))))
10113, 100mpd 15 1 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆƒwrex 3070   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145  joincjn 18205  Latclat 18325   β‹– ccvr 37770  Atomscatm 37771  HLchlt 37858  LLinesclln 38000  LPlanesclpl 38001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-proset 18189  df-poset 18207  df-plt 18224  df-lub 18240  df-glb 18241  df-join 18242  df-meet 18243  df-p0 18319  df-lat 18326  df-clat 18393  df-oposet 37684  df-ol 37686  df-oml 37687  df-covers 37774  df-ats 37775  df-atl 37806  df-cvlat 37830  df-hlat 37859  df-llines 38007  df-lplanes 38008
This theorem is referenced by: (None)
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