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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lplncvrlvol2 Structured version   Visualization version   GIF version
Theorem lplncvrlvol2 39140
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 483 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ 𝑋 ≀ π‘Œ)
2 simpl1 1188 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ 𝐾 ∈ HL)
3 simpl3 1190 . . . . 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 39115 . . . . 5 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝑉) β†’ Β¬ π‘Œ ∈ 𝑃)
72, 3, 6syl2anc 582 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ Β¬ π‘Œ ∈ 𝑃)
8 simpl2 1189 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 ≀ π‘Œ) β†’ 𝑋 ∈ 𝑃)
9 eleq1 2813 . . . . . 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 2944 . . . 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 2725 . . . . 5 (ltβ€˜πΎ) = (ltβ€˜πΎ)
1513, 14pltval 18318 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋(ltβ€˜πΎ)π‘Œ ↔ (𝑋 ≀ π‘Œ ∧ 𝑋 β‰  π‘Œ)))
1615adantr 479 . . 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 1188 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ 𝐾 ∈ HL)
19 simpl2 1189 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ 𝑋 ∈ 𝑃)
20 eqid 2725 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2120, 4lplnbase 39059 . . . . 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 1190 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ π‘Œ ∈ 𝑉)
2420, 5lvolbase 39103 . . . . 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 483 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ 𝑋(ltβ€˜πΎ)π‘Œ)
27 eqid 2725 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
28 lplncvrlvol2.c . . . . 5 𝐢 = ( β‹– β€˜πΎ)
29 eqid 2725 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
3020, 13, 14, 27, 28, 29hlrelat3 38937 . . . 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 1370 . . 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 39105 . . . . . . . 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 479 . . . . . . 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 483 . . . . . . . . . . 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 39061 . . . . . . . . . . . . 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 1243 . . . . . . . . . . . . . . . . . . . . 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 1244 . . . . . . . . . . . . . . . . . . . . . . . 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 1307 . . . . . . . . . . . . . . . . . . . . . . . . 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 7428 . . . . . . . . . . . . . . . . . . . . . . . 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 1205 . . . . . . . . . . . . . . . . . . . . . . . 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 5175 . . . . . . . . . . . . . . . . . . . . . . 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 1200 . . . . . . . . . . . . . . . . . . . . . . . 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 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†’ π‘Ÿ ∈ (Atomsβ€˜πΎ))
44 simp3l 1198 . . . . . . . . . . . . . . . . . . . . . . . . 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 1287 . . . . . . . . . . . . . . . . . . . . . . . . 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 1125 . . . . . . . . . . . . . . . . . . . . . . . 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 1288 . . . . . . . . . . . . . . . . . . . . . . . . 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 1289 . . . . . . . . . . . . . . . . . . . . . . . . 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 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†’ 𝑀 ∈ (Atomsβ€˜πΎ))
5047, 48, 493jca 1125 . . . . . . . . . . . . . . . . . . . . . . . 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 1305 . . . . . . . . . . . . . . . . . . . . . . . 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 1306 . . . . . . . . . . . . . . . . . . . . . . . 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 5175 . . . . . . . . . . . . . . . . . . . . . . . . 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 1299 . . . . . . . . . . . . . . . . . . . . . . . . . 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 38888 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 38891 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 38813 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18425 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . . . . . . . . . . . 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 38935 . . . . . . . . . . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . . . . . . . . . . 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 39139 . . . . . . . . . . . . . . . . . . . . . . . 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 1382 . . . . . . . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . . . . . . . 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 5170 . . . . . . . . . . . . . . . . . . . 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 1116 . . . . . . . . . . . . . . . . . . 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 430 . . . . . . . . . . . . . . . . . 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 1350 . . . . . . . . . . . . . . . . 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 3149 . . . . . . . . . . . . . . . 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 1119 . . . . . . . . . . . . . . 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 3201 . . . . . . . . . . . . . 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 489 . . . . . . . . . . . . 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 416 . . . . . . . . . . 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 3201 . . . . . . . . 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 3202 . . . . . . . 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 489 . . . . . . 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 1114 . . . . 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 3143 . . . 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 405 . . 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 589 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ π‘‹πΆπ‘Œ)
8817, 87syldan 589 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆƒwrex 3060   class class class wbr 5144  β€˜cfv 6543  (class class class)co 7413  Basecbs 17174  lecple 17234  ltcplt 18294  joincjn 18297  Latclat 18417   β‹– ccvr 38786  Atomscatm 38787  HLchlt 38874  LPlanesclpl 39017  LVolsclvol 39018
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
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-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-proset 18281  df-poset 18299  df-plt 18316  df-lub 18332  df-glb 18333  df-join 18334  df-meet 18335  df-p0 18411  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
This theorem is referenced by:  lplncvrlvol  39141  lvolcmp  39142  2lplnm2N  39146  2lplnmj  39147
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