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Theorem lplnexllnN 39092
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 1189 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) β†’ 𝑋 ∈ 𝑃)
2 simpl1 1188 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑋) β†’ 𝐾 ∈ HL)
3 eqid 2725 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4 lplnexat.p . . . . . 6 𝑃 = (LPlanesβ€˜πΎ)
53, 4lplnbase 39062 . . . . 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 39061 . . . 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 582 . . 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 1209 . . . . . . 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 1229 . . . . . . 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 1211 . . . . . . 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 1191 . . . . . . 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 39049 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑧 ∈ 𝑁 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≀ 𝑧) β†’ βˆƒπ‘  ∈ 𝐴 (𝑄 β‰  𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))
1914, 15, 16, 17, 18syl31anc 1370 . . . . . 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 1263 . . . . . . . . . . 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 1290 . . . . . . . . . . 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 1198 . . . . . . . . . . 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 1265 . . . . . . . . . . . 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 1291 . . . . . . . . . . . . 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 1244 . . . . . . . . . . . . . 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 5155 . . . . . . . . . . . . 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 323 . . . . . . . . . . . 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 38908 . . . . . . . . . . . 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 1380 . . . . . . . . . . 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 39040 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ π‘Ÿ β‰  𝑠) β†’ (π‘Ÿ ∨ 𝑠) ∈ 𝑁)
3120, 21, 22, 29, 30syl31anc 1370 . . . . . . . . . 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 1243 . . . . . . . . . . 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 38980 . . . . . . . . . . 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 1385 . . . . . . . . . 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 1292 . . . . . . . . . . 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 7430 . . . . . . . . . . 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 38891 . . . . . . . . . . . 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 38816 . . . . . . . . . . . . 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 38816 . . . . . . . . . . . . 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 38816 . . . . . . . . . . . . 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 18476 . . . . . . . . . . . 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 1369 . . . . . . . . . . 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 2769 . . . . . . . . . 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 5147 . . . . . . . . . . . . 13 (𝑦 = (π‘Ÿ ∨ 𝑠) β†’ (𝑄 ≀ 𝑦 ↔ 𝑄 ≀ (π‘Ÿ ∨ 𝑠)))
4847notbid 317 . . . . . . . . . . . 12 (𝑦 = (π‘Ÿ ∨ 𝑠) β†’ (Β¬ 𝑄 ≀ 𝑦 ↔ Β¬ 𝑄 ≀ (π‘Ÿ ∨ 𝑠)))
49 oveq1 7422 . . . . . . . . . . . . 13 (𝑦 = (π‘Ÿ ∨ 𝑠) β†’ (𝑦 ∨ 𝑄) = ((π‘Ÿ ∨ 𝑠) ∨ 𝑄))
5049eqeq2d 2736 . . . . . . . . . . . 12 (𝑦 = (π‘Ÿ ∨ 𝑠) β†’ (𝑋 = (𝑦 ∨ 𝑄) ↔ 𝑋 = ((π‘Ÿ ∨ 𝑠) ∨ 𝑄)))
5148, 50anbi12d 630 . . . . . . . . . . 11 (𝑦 = (π‘Ÿ ∨ 𝑠) β†’ ((Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)) ↔ (Β¬ 𝑄 ≀ (π‘Ÿ ∨ 𝑠) ∧ 𝑋 = ((π‘Ÿ ∨ 𝑠) ∨ 𝑄))))
5251rspcev 3602 . . . . . . . . . 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 835 . . . . . . . . 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 1118 . . . . . . . 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 414 . . . . . . 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 3143 . . . . . 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 1351 . . . 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 1229 . . . . . 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 1191 . . . . . 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 1209 . . . . . . . . . . . 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 38891 . . . . . . . . . . 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 39037 . . . . . . . . . . . 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 1230 . . . . . . . . . . . 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 18437 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Baseβ€˜πΎ) ∧ π‘Ÿ ∈ (Baseβ€˜πΎ)) β†’ 𝑧 ≀ (𝑧 ∨ π‘Ÿ))
6862, 64, 66, 67syl3anc 1368 . . . . . . . . . 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 1232 . . . . . . . . . 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 5171 . . . . . . . . 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 767 . . . . . . . . 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 1211 . . . . . . . . . . 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 1210 . . . . . . . . . . 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 18439 . . . . . . . . . 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 1369 . . . . . . . . 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 710 . . . . . . . 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 18428 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (𝑧 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
8062, 64, 73, 79syl3anc 1368 . . . . . . . . . 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 2725 . . . . . . . . . . . . 13 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
823, 7, 8, 81, 9cvr1 38938 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑧 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ 𝐴) β†’ (Β¬ 𝑄 ≀ 𝑧 ↔ 𝑧( β‹– β€˜πΎ)(𝑧 ∨ 𝑄)))
8361, 64, 72, 82syl3anc 1368 . . . . . . . . . . 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 39060 . . . . . . . . . 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 1370 . . . . . . . . 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 39090 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑧 ∨ 𝑄) ∈ 𝑃 ∧ 𝑋 ∈ 𝑃) β†’ ((𝑧 ∨ 𝑄) ≀ 𝑋 ↔ (𝑧 ∨ 𝑄) = 𝑋))
8861, 86, 74, 87syl3anc 1368 . . . . . . . 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 2731 . . . . . 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 5147 . . . . . . . . 9 (𝑦 = 𝑧 β†’ (𝑄 ≀ 𝑦 ↔ 𝑄 ≀ 𝑧))
9291notbid 317 . . . . . . . 8 (𝑦 = 𝑧 β†’ (Β¬ 𝑄 ≀ 𝑦 ↔ Β¬ 𝑄 ≀ 𝑧))
93 oveq1 7422 . . . . . . . . 9 (𝑦 = 𝑧 β†’ (𝑦 ∨ 𝑄) = (𝑧 ∨ 𝑄))
9493eqeq2d 2736 . . . . . . . 8 (𝑦 = 𝑧 β†’ (𝑋 = (𝑦 ∨ 𝑄) ↔ 𝑋 = (𝑧 ∨ 𝑄)))
9592, 94anbi12d 630 . . . . . . 7 (𝑦 = 𝑧 β†’ ((Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)) ↔ (Β¬ 𝑄 ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑄))))
9695rspcev 3602 . . . . . 6 ((𝑧 ∈ 𝑁 ∧ (Β¬ 𝑄 ≀ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑄))) β†’ βˆƒπ‘¦ ∈ 𝑁 (Β¬ 𝑄 ≀ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
9759, 60, 90, 96syl12anc 835 . . . . 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 1351 . . . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆƒwrex 3060   class class class wbr 5143  β€˜cfv 6542  (class class class)co 7415  Basecbs 17177  lecple 17237  joincjn 18300  Latclat 18420   β‹– ccvr 38789  Atomscatm 38790  HLchlt 38877  LLinesclln 39019  LPlanesclpl 39020
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 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-proset 18284  df-poset 18302  df-plt 18319  df-lub 18335  df-glb 18336  df-join 18337  df-meet 18338  df-p0 18414  df-lat 18421  df-clat 18488  df-oposet 38703  df-ol 38705  df-oml 38706  df-covers 38793  df-ats 38794  df-atl 38825  df-cvlat 38849  df-hlat 38878  df-llines 39026  df-lplanes 39027
This theorem is referenced by: (None)
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