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Theorem llncvrlpln2 38416
Description: A lattice line under a lattice plane is covered by it. (Contributed by NM, 24-Jun-2012.)
Hypotheses
Ref Expression
llncvrlpln2.l ≀ = (leβ€˜πΎ)
llncvrlpln2.c 𝐢 = ( β‹– β€˜πΎ)
llncvrlpln2.n 𝑁 = (LLinesβ€˜πΎ)
llncvrlpln2.p 𝑃 = (LPlanesβ€˜πΎ)
Assertion
Ref Expression
llncvrlpln2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋 ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)
Proof of Theorem llncvrlpln2
Dummy variables π‘ž 𝑝 π‘Ÿ 𝑠 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 485 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋 ≀ π‘Œ) β†’ 𝑋 ≀ π‘Œ)
2 simpl1 1191 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋 ≀ π‘Œ) β†’ 𝐾 ∈ HL)
3 simpl3 1193 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋 ≀ π‘Œ) β†’ π‘Œ ∈ 𝑃)
4 llncvrlpln2.n . . . . . 6 𝑁 = (LLinesβ€˜πΎ)
5 llncvrlpln2.p . . . . . 6 𝑃 = (LPlanesβ€˜πΎ)
64, 5lplnnelln 38405 . . . . 5 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝑃) β†’ Β¬ π‘Œ ∈ 𝑁)
72, 3, 6syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋 ≀ π‘Œ) β†’ Β¬ π‘Œ ∈ 𝑁)
8 simpl2 1192 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋 ≀ π‘Œ) β†’ 𝑋 ∈ 𝑁)
9 eleq1 2821 . . . . . 6 (𝑋 = π‘Œ β†’ (𝑋 ∈ 𝑁 ↔ π‘Œ ∈ 𝑁))
108, 9syl5ibcom 244 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋 ≀ π‘Œ) β†’ (𝑋 = π‘Œ β†’ π‘Œ ∈ 𝑁))
1110necon3bd 2954 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋 ≀ π‘Œ) β†’ (Β¬ π‘Œ ∈ 𝑁 β†’ 𝑋 β‰  π‘Œ))
127, 11mpd 15 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋 ≀ π‘Œ) β†’ 𝑋 β‰  π‘Œ)
13 llncvrlpln2.l . . . . 5 ≀ = (leβ€˜πΎ)
14 eqid 2732 . . . . 5 (ltβ€˜πΎ) = (ltβ€˜πΎ)
1513, 14pltval 18281 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) β†’ (𝑋(ltβ€˜πΎ)π‘Œ ↔ (𝑋 ≀ π‘Œ ∧ 𝑋 β‰  π‘Œ)))
1615adantr 481 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 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 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋 ≀ π‘Œ) β†’ 𝑋(ltβ€˜πΎ)π‘Œ)
18 simpl1 1191 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ 𝐾 ∈ HL)
19 simpl2 1192 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ 𝑋 ∈ 𝑁)
20 eqid 2732 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2120, 4llnbase 38368 . . . . 5 (𝑋 ∈ 𝑁 β†’ 𝑋 ∈ (Baseβ€˜πΎ))
2219, 21syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ 𝑋 ∈ (Baseβ€˜πΎ))
23 simpl3 1193 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ π‘Œ ∈ 𝑃)
2420, 5lplnbase 38393 . . . . 5 (π‘Œ ∈ 𝑃 β†’ π‘Œ ∈ (Baseβ€˜πΎ))
2523, 24syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ π‘Œ ∈ (Baseβ€˜πΎ))
26 simpr 485 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ 𝑋(ltβ€˜πΎ)π‘Œ)
27 eqid 2732 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
28 llncvrlpln2.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‘Ÿ ∈ (Atomsβ€˜πΎ)(𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ))
3118, 22, 25, 26, 30syl31anc 1373 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)(𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ))
3220, 13, 27, 29, 5islpln2 38395 . . . . . . . 8 (𝐾 ∈ HL β†’ (π‘Œ ∈ 𝑃 ↔ (π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘  ∈ (Atomsβ€˜πΎ)βˆƒπ‘‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘’ ∈ (Atomsβ€˜πΎ)(𝑠 β‰  𝑑 ∧ Β¬ 𝑒 ≀ (𝑠(joinβ€˜πΎ)𝑑) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒)))))
3332adantr 481 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) β†’ (π‘Œ ∈ 𝑃 ↔ (π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘  ∈ (Atomsβ€˜πΎ)βˆƒπ‘‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘’ ∈ (Atomsβ€˜πΎ)(𝑠 β‰  𝑑 ∧ Β¬ 𝑒 ≀ (𝑠(joinβ€˜πΎ)𝑑) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒)))))
34 simp3 1138 . . . . . . . . . . 11 ((𝑠 β‰  𝑑 ∧ Β¬ 𝑒 ≀ (𝑠(joinβ€˜πΎ)𝑑) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒)) β†’ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒))
3520, 27, 29, 4islln2 38370 . . . . . . . . . . . . 13 (𝐾 ∈ HL β†’ (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)))))
36 simp3l 1201 . . . . . . . . . . . . . . . . . . . 20 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ 𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ))
37 simp3r 1202 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)
38 simp12r 1287 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž))
3938oveq1d 7420 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ (𝑋(joinβ€˜πΎ)π‘Ÿ) = ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))
40 simp22 1207 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒))
4137, 39, 403brtr3d 5178 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ) ≀ ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒))
42 simp111 1302 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ 𝐾 ∈ HL)
43 simp112 1303 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ 𝑝 ∈ (Atomsβ€˜πΎ))
44 simp113 1304 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ π‘ž ∈ (Atomsβ€˜πΎ))
45 simp23 1208 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ π‘Ÿ ∈ (Atomsβ€˜πΎ))
4643, 44, 453jca 1128 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)))
47 simp13l 1288 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ 𝑠 ∈ (Atomsβ€˜πΎ))
48 simp13r 1289 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ 𝑑 ∈ (Atomsβ€˜πΎ))
49 simp21 1206 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ 𝑒 ∈ (Atomsβ€˜πΎ))
5047, 48, 493jca 1128 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ) ∧ 𝑒 ∈ (Atomsβ€˜πΎ)))
5136, 38, 393brtr3d 5178 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ (𝑝(joinβ€˜πΎ)π‘ž)𝐢((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ))
5220, 27, 29hlatjcl 38225 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) β†’ (𝑝(joinβ€˜πΎ)π‘ž) ∈ (Baseβ€˜πΎ))
5342, 43, 44, 52syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ (𝑝(joinβ€˜πΎ)π‘ž) ∈ (Baseβ€˜πΎ))
5420, 13, 27, 28, 29cvr1 38269 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐾 ∈ HL ∧ (𝑝(joinβ€˜πΎ)π‘ž) ∈ (Baseβ€˜πΎ) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) β†’ (Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ↔ (𝑝(joinβ€˜πΎ)π‘ž)𝐢((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)))
5542, 53, 45, 54syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ (Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ↔ (𝑝(joinβ€˜πΎ)π‘ž)𝐢((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ)))
5651, 55mpbird 256 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž))
57 simp12l 1286 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ 𝑝 β‰  π‘ž)
5813, 27, 293at 38349 . . . . . . . . . . . . . . . . . . . . . . 23 (((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))) ∧ (Β¬ π‘Ÿ ≀ (𝑝(joinβ€˜πΎ)π‘ž) ∧ 𝑝 β‰  π‘ž)) β†’ (((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ) ≀ ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ↔ ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ) = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒)))
5942, 46, 50, 56, 57, 58syl32anc 1378 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ (((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ) ≀ ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ↔ ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ) = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒)))
6041, 59mpbid 231 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ ((𝑝(joinβ€˜πΎ)π‘ž)(joinβ€˜πΎ)π‘Ÿ) = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒))
6160, 39, 403eqtr4d 2782 . . . . . . . . . . . . . . . . . . . 20 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ (𝑋(joinβ€˜πΎ)π‘Ÿ) = π‘Œ)
6236, 61breqtrd 5173 . . . . . . . . . . . . . . . . . . 19 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) ∧ (𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ π‘‹πΆπ‘Œ)
63623exp 1119 . . . . . . . . . . . . . . . . . 18 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) β†’ ((𝑒 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))
64633expd 1353 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) β†’ (𝑒 ∈ (Atomsβ€˜πΎ) β†’ (π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) β†’ (π‘Ÿ ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))))
65643exp 1119 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) β†’ ((𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) β†’ ((𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ)) β†’ (𝑒 ∈ (Atomsβ€˜πΎ) β†’ (π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) β†’ (π‘Ÿ ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))))))
66653expib 1122 . . . . . . . . . . . . . . 15 (𝐾 ∈ HL β†’ ((𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) β†’ ((𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) β†’ ((𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ)) β†’ (𝑒 ∈ (Atomsβ€˜πΎ) β†’ (π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) β†’ (π‘Ÿ ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ))))))))
6766rexlimdvv 3210 . . . . . . . . . . . . . 14 (𝐾 ∈ HL β†’ (βˆƒπ‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)) β†’ ((𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ)) β†’ (𝑒 ∈ (Atomsβ€˜πΎ) β†’ (π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) β†’ (π‘Ÿ ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))))))
6867adantld 491 . . . . . . . . . . . . 13 (𝐾 ∈ HL β†’ ((𝑋 ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž))) β†’ ((𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ)) β†’ (𝑒 ∈ (Atomsβ€˜πΎ) β†’ (π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) β†’ (π‘Ÿ ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))))))
6935, 68sylbid 239 . . . . . . . . . . . 12 (𝐾 ∈ HL β†’ (𝑋 ∈ 𝑁 β†’ ((𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ)) β†’ (𝑒 ∈ (Atomsβ€˜πΎ) β†’ (π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) β†’ (π‘Ÿ ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))))))
7069imp31 418 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) β†’ (𝑒 ∈ (Atomsβ€˜πΎ) β†’ (π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒) β†’ (π‘Ÿ ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))))
7134, 70syl7 74 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) β†’ (𝑒 ∈ (Atomsβ€˜πΎ) β†’ ((𝑠 β‰  𝑑 ∧ Β¬ 𝑒 ≀ (𝑠(joinβ€˜πΎ)𝑑) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒)) β†’ (π‘Ÿ ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))))
7271rexlimdv 3153 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ (𝑠 ∈ (Atomsβ€˜πΎ) ∧ 𝑑 ∈ (Atomsβ€˜πΎ))) β†’ (βˆƒπ‘’ ∈ (Atomsβ€˜πΎ)(𝑠 β‰  𝑑 ∧ Β¬ 𝑒 ≀ (𝑠(joinβ€˜πΎ)𝑑) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒)) β†’ (π‘Ÿ ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ))))
7372rexlimdvva 3211 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) β†’ (βˆƒπ‘  ∈ (Atomsβ€˜πΎ)βˆƒπ‘‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘’ ∈ (Atomsβ€˜πΎ)(𝑠 β‰  𝑑 ∧ Β¬ 𝑒 ≀ (𝑠(joinβ€˜πΎ)𝑑) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒)) β†’ (π‘Ÿ ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ))))
7473adantld 491 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) β†’ ((π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘  ∈ (Atomsβ€˜πΎ)βˆƒπ‘‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘’ ∈ (Atomsβ€˜πΎ)(𝑠 β‰  𝑑 ∧ Β¬ 𝑒 ≀ (𝑠(joinβ€˜πΎ)𝑑) ∧ π‘Œ = ((𝑠(joinβ€˜πΎ)𝑑)(joinβ€˜πΎ)𝑒))) β†’ (π‘Ÿ ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ))))
7533, 74sylbid 239 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) β†’ (π‘Œ ∈ 𝑃 β†’ (π‘Ÿ ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ))))
76753impia 1117 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) β†’ (π‘Ÿ ∈ (Atomsβ€˜πΎ) β†’ ((𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ)))
7776rexlimdv 3153 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) β†’ (βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)(𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ) β†’ π‘‹πΆπ‘Œ))
7877imp 407 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)(𝑋𝐢(𝑋(joinβ€˜πΎ)π‘Ÿ) ∧ (𝑋(joinβ€˜πΎ)π‘Ÿ) ≀ π‘Œ)) β†’ π‘‹πΆπ‘Œ)
7931, 78syldan 591 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 𝑋(ltβ€˜πΎ)π‘Œ) β†’ π‘‹πΆπ‘Œ)
8017, 79syldan 591 1 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑃) ∧ 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   β‹– ccvr 38120  Atomscatm 38121  HLchlt 38208  LLinesclln 38350  LPlanesclpl 38351
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-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
This theorem is referenced by:  llncvrlpln  38417  2llnmj  38419  lplncmp  38421  lplnexatN  38422  2llnm2N  38427  2lplnmj  38481
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