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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lplncvrlvol2 Structured version   Visualization version   GIF version
Theorem lplncvrlvol2 38474
Description: A lattice line under a lattice plane is covered by it. (Contributed by NM, 12-Jul-2012.)
Hypotheses
Ref Expression
lplncvrlvol2.l ≀ = (leβ€˜πΎ)
lplncvrlvol2.c 𝐢 = ( β‹– β€˜πΎ)
lplncvrlvol2.p 𝑃 = (LPlanesβ€˜πΎ)
lplncvrlvol2.v 𝑉 = (LVolsβ€˜πΎ)
Assertion
Ref Expression
lplncvrlvol2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)
Proof of Theorem lplncvrlvol2
Dummy variables π‘ž 𝑝 π‘Ÿ 𝑠 𝑑 𝑒 𝑣 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 485 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ 𝑋 ≀ π‘Œ)
2 simpl1 1191 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ 𝐾 ∈ HL)
3 simpl3 1193 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ π‘Œ ∈ 𝑉)
4 lplncvrlvol2.p . . . . . 6 𝑃 = (LPlanesβ€˜πΎ)
5 lplncvrlvol2.v . . . . . 6 𝑉 = (LVolsβ€˜πΎ)
64, 5lvolnelpln 38449 . . . . 5 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝑉) β†’ Β¬ π‘Œ ∈ 𝑃)
72, 3, 6syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ Β¬ π‘Œ ∈ 𝑃)
8 simpl2 1192 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ 𝑋 ∈ 𝑃)
9 eleq1 2821 . . . . . 6 (𝑋 = π‘Œ β†’ (𝑋 ∈ 𝑃 ↔ π‘Œ ∈ 𝑃))
108, 9syl5ibcom 244 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ (𝑋 = π‘Œ β†’ π‘Œ ∈ 𝑃))
1110necon3bd 2954 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ (Β¬ π‘Œ ∈ 𝑃 β†’ 𝑋 β‰  π‘Œ))
127, 11mpd 15 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ 𝑋 β‰  π‘Œ)
13 lplncvrlvol2.l . . . . 5 ≀ = (leβ€˜πΎ)
14 eqid 2732 . . . . 5 (ltβ€˜πΎ) = (ltβ€˜πΎ)
1513, 14pltval 18281 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋(ltβ€˜πΎ)π‘Œ ↔ (𝑋 ≀ π‘Œ ∧ 𝑋 β‰  π‘Œ)))
1615adantr 481 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ (𝑋(ltβ€˜πΎ)π‘Œ ↔ (𝑋 ≀ π‘Œ ∧ 𝑋 β‰  π‘Œ)))
171, 12, 16mpbir2and 711 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ 𝑋(ltβ€˜πΎ)π‘Œ)
18 simpl1 1191 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ 𝐾 ∈ HL)
19 simpl2 1192 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ 𝑋 ∈ 𝑃)
20 eqid 2732 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2120, 4lplnbase 38393 . . . . 5 (𝑋 ∈ 𝑃 β†’ 𝑋 ∈ (Baseβ€˜πΎ))
2219, 21syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ 𝑋 ∈ (Baseβ€˜πΎ))
23 simpl3 1193 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ π‘Œ ∈ 𝑉)
2420, 5lvolbase 38437 . . . . 5 (π‘Œ ∈ 𝑉 β†’ π‘Œ ∈ (Baseβ€˜πΎ))
2523, 24syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ π‘Œ ∈ (Baseβ€˜πΎ))
26 simpr 485 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ 𝑋(ltβ€˜πΎ)π‘Œ)
27 eqid 2732 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
28 lplncvrlvol2.c . . . . 5 𝐢 = ( β‹– β€˜πΎ)
29 eqid 2732 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
3020, 13, 14, 27, 28, 29hlrelat3 38271 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ (Baseβ€˜πΎ) ∧ π‘Œ ∈ (Baseβ€˜πΎ)) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ βˆƒπ‘  ∈ (Atomsβ€˜πΎ)(𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))
3118, 22, 25, 26, 30syl31anc 1373 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ βˆƒπ‘  ∈ (Atomsβ€˜πΎ)(𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))
3220, 13, 27, 29, 5islvol2 38439 . . . . . . . 8 (𝐾 ∈ HL β†’ (π‘Œ ∈ 𝑉 ↔ (π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘’ ∈ (Atomsβ€˜πΎ)βˆƒπ‘£ ∈ (Atomsβ€˜πΎ)βˆƒπ‘€ ∈ (Atomsβ€˜πΎ)((𝑑 β‰  𝑒 ∧ Β¬ 𝑣 ≀ (𝑑(joinβ€˜πΎ)𝑒) ∧ Β¬ 𝑀 ≀ ((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)))))
3332adantr 481 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) β†’ (π‘Œ ∈ 𝑉 ↔ (π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘’ ∈ (Atomsβ€˜πΎ)βˆƒπ‘£ ∈ (Atomsβ€˜πΎ)βˆƒπ‘€ ∈ (Atomsβ€˜πΎ)((𝑑 β‰  𝑒 ∧ Β¬ 𝑣 ≀ (𝑑(joinβ€˜πΎ)𝑒) ∧ Β¬ 𝑀 ≀ ((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)))))
34 simpr 485 . . . . . . . . . . 11 (((𝑑 β‰  𝑒 ∧ Β¬ 𝑣 ≀ (𝑑(joinβ€˜πΎ)𝑒) ∧ Β¬ 𝑀 ≀ ((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) β†’ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀))
3520, 13, 27, 29, 4islpln2 38395 . . . . . . . . . . . . 13 (𝐾 ∈ HL β†’ (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)))))
36 simp3rl 1246 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ 𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠))
37 simp3rr 1247 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ)
38 simp133 1310 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))
3938oveq1d 7420 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ (𝑋(joinβ€˜πΎ)𝑠) = (((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))
40 simp23 1208 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀))
4137, 39, 403brtr3d 5178 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ (((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠) ≀ (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀))
42 simp11 1203 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ (𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)))
43 simp12 1204 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ π‘Ÿ ∈ (Atomsβ€˜πΎ))
44 simp3l 1201 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ 𝑠 ∈ (Atomsβ€˜πΎ))
45 simp21l 1290 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ 𝑑 ∈ (Atomsβ€˜πΎ))
4643, 44, 453jca 1128 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ (π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ 𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ)))
47 simp21r 1291 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ 𝑒 ∈ (Atomsβ€˜πΎ))
48 simp22l 1292 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ 𝑣 ∈ (Atomsβ€˜πΎ))
49 simp22r 1293 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ 𝑀 ∈ (Atomsβ€˜πΎ))
5047, 48, 493jca 1128 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ 𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)))
51 simp131 1308 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ 𝑝 β‰  π‘ž)
52 simp132 1309 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž))
5336, 38, 393brtr3d 5178 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)𝐢(((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠))
54 simp111 1302 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ 𝐾 ∈ HL)
5554hllatd 38222 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ 𝐾 ∈ Lat)
5620, 27, 29hlatjcl 38225 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) β†’ (𝑝(joinβ€˜πΎ)π‘ž) ∈ (Baseβ€˜πΎ))
5742, 56syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ (𝑝(joinβ€˜πΎ)π‘ž) ∈ (Baseβ€˜πΎ))
5820, 29atbase 38147 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘Ÿ ∈ (Atomsβ€˜πΎ) β†’ π‘Ÿ ∈ (Baseβ€˜πΎ))
5943, 58syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ π‘Ÿ ∈ (Baseβ€˜πΎ))
6020, 27latjcl 18388 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐾 ∈ Lat ∧ (𝑝(joinβ€˜πΎ)π‘ž) ∈ (Baseβ€˜πΎ) ∧ π‘Ÿ ∈ (Baseβ€˜πΎ)) β†’ ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ) ∈ (Baseβ€˜πΎ))
6155, 57, 59, 60syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ) ∈ (Baseβ€˜πΎ))
6220, 13, 27, 28, 29cvr1 38269 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐾 ∈ HL ∧ ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ) ∈ (Baseβ€˜πΎ) ∧ 𝑠 ∈ (Atomsβ€˜πΎ)) β†’ (Β¬ 𝑠 ≀ ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ) ↔ ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)𝐢(((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠)))
6354, 61, 44, 62syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ (Β¬ 𝑠 ≀ ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ) ↔ ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)𝐢(((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠)))
6453, 63mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ Β¬ 𝑠 ≀ ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))
6513, 27, 294at2 38473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ 𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ)) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ 𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ))) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ Β¬ 𝑠 ≀ ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) β†’ ((((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠) ≀ (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀) ↔ (((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠) = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)))
6642, 46, 50, 51, 52, 64, 65syl33anc 1385 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ ((((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠) ≀ (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀) ↔ (((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠) = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)))
6741, 66mpbid 231 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ (((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)(joinβ€˜πΎ)𝑠) = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀))
6867, 39, 403eqtr4d 2782 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ (𝑋(joinβ€˜πΎ)𝑠) = π‘Œ)
6936, 68breqtrd 5173 . . . . . . . . . . . . . . . . . . . 20 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) ∧ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ))) β†’ π‘‹πΆπ‘Œ)
70693exp 1119 . . . . . . . . . . . . . . . . . . 19 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) β†’ (((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) β†’ ((𝑠 ∈ (Atomsβ€˜πΎ) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ)) β†’ π‘‹πΆπ‘Œ)))
7170exp4a 432 . . . . . . . . . . . . . . . . . 18 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) β†’ (((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) ∧ (𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) β†’ (𝑠 ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ))))
72713expd 1353 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ) ∧ (𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) β†’ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) β†’ ((𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) β†’ (π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀) β†’ (𝑠 ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ))))))
7372rexlimdv3a 3159 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) β†’ (βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)) β†’ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) β†’ ((𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) β†’ (π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀) β†’ (𝑠 ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))))))
74733expib 1122 . . . . . . . . . . . . . . 15 (𝐾 ∈ HL β†’ ((𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) β†’ (βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)) β†’ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) β†’ ((𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) β†’ (π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀) β†’ (𝑠 ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ))))))))
7574rexlimdvv 3210 . . . . . . . . . . . . . 14 (𝐾 ∈ HL β†’ (βˆƒπ‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)) β†’ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) β†’ ((𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) β†’ (π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀) β†’ (𝑠 ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))))))
7675adantld 491 . . . . . . . . . . . . 13 (𝐾 ∈ HL β†’ ((𝑋 ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑋 = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))) β†’ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) β†’ ((𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) β†’ (π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀) β†’ (𝑠 ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))))))
7735, 76sylbid 239 . . . . . . . . . . . 12 (𝐾 ∈ HL β†’ (𝑋 ∈ 𝑃 β†’ ((𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)) β†’ ((𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) β†’ (π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀) β†’ (𝑠 ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))))))
7877imp31 418 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ (𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ))) β†’ ((𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) β†’ (π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀) β†’ (𝑠 ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))))
7934, 78syl7 74 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ (𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ))) β†’ ((𝑣 ∈ (Atomsβ€˜πΎ) ∧ 𝑀 ∈ (Atomsβ€˜πΎ)) β†’ (((𝑑 β‰  𝑒 ∧ Β¬ 𝑣 ≀ (𝑑(joinβ€˜πΎ)𝑒) ∧ Β¬ 𝑀 ≀ ((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) β†’ (𝑠 ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))))
8079rexlimdvv 3210 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ (𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ))) β†’ (βˆƒπ‘£ ∈ (Atomsβ€˜πΎ)βˆƒπ‘€ ∈ (Atomsβ€˜πΎ)((𝑑 β‰  𝑒 ∧ Β¬ 𝑣 ≀ (𝑑(joinβ€˜πΎ)𝑒) ∧ Β¬ 𝑀 ≀ ((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) β†’ (𝑠 ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ))))
8180rexlimdvva 3211 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) β†’ (βˆƒπ‘‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘’ ∈ (Atomsβ€˜πΎ)βˆƒπ‘£ ∈ (Atomsβ€˜πΎ)βˆƒπ‘€ ∈ (Atomsβ€˜πΎ)((𝑑 β‰  𝑒 ∧ Β¬ 𝑣 ≀ (𝑑(joinβ€˜πΎ)𝑒) ∧ Β¬ 𝑀 ≀ ((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀)) β†’ (𝑠 ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ))))
8281adantld 491 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) β†’ ((π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘’ ∈ (Atomsβ€˜πΎ)βˆƒπ‘£ ∈ (Atomsβ€˜πΎ)βˆƒπ‘€ ∈ (Atomsβ€˜πΎ)((𝑑 β‰  𝑒 ∧ Β¬ 𝑣 ≀ (𝑑(joinβ€˜πΎ)𝑒) ∧ Β¬ 𝑀 ≀ ((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)) ∧ π‘Œ = (((𝑑(joinβ€˜πΎ)𝑒)(joinβ€˜πΎ)𝑣)(joinβ€˜πΎ)𝑀))) β†’ (𝑠 ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ))))
8333, 82sylbid 239 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) β†’ (π‘Œ ∈ 𝑉 β†’ (𝑠 ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ))))
84833impia 1117 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) β†’ (𝑠 ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))
8584rexlimdv 3153 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) β†’ (βˆƒπ‘  ∈ (Atomsβ€˜πΎ)(𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ))
8685imp 407 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ βˆƒπ‘  ∈ (Atomsβ€˜πΎ)(𝑋𝐢(𝑋(joinβ€˜πΎ)𝑠) ∧ (𝑋(joinβ€˜πΎ)𝑠) ≀ π‘Œ)) β†’ π‘‹πΆπ‘Œ)
8731, 86syldan 591 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ π‘‹πΆπ‘Œ)
8817, 87syldan 591 1 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  ltcplt 18257  joincjn 18260  Latclat 18380   β‹– ccvr 38120  Atomscatm 38121  HLchlt 38208  LPlanesclpl 38351  LVolsclvol 38352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209  df-llines 38357  df-lplanes 38358  df-lvols 38359
This theorem is referenced by:  lplncvrlvol  38475  lvolcmp  38476  2lplnm2N  38480  2lplnmj  38481
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