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Theorem lplnnlelln 38414
Description: A lattice plane is not less than or equal to a lattice line. (Contributed by NM, 14-Jul-2012.)
Hypotheses
Ref Expression
lplnnlelln.l ≀ = (leβ€˜πΎ)
lplnnlelln.n 𝑁 = (LLinesβ€˜πΎ)
lplnnlelln.p 𝑃 = (LPlanesβ€˜πΎ)
Assertion
Ref Expression
lplnnlelln ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) β†’ Β¬ 𝑋 ≀ π‘Œ)
Proof of Theorem lplnnlelln
Dummy variables π‘Ÿ π‘ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1139 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) β†’ π‘Œ ∈ 𝑁)
2 eqid 2733 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3 eqid 2733 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
4 eqid 2733 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
5 lplnnlelln.n . . . . 5 𝑁 = (LLinesβ€˜πΎ)
62, 3, 4, 5islln2 38382 . . . 4 (𝐾 ∈ HL β†’ (π‘Œ ∈ 𝑁 ↔ (π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)(π‘ž β‰  π‘Ÿ ∧ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
763ad2ant1 1134 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) β†’ (π‘Œ ∈ 𝑁 ↔ (π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)(π‘ž β‰  π‘Ÿ ∧ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
81, 7mpbid 231 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) β†’ (π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)(π‘ž β‰  π‘Ÿ ∧ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ))))
9 simp11 1204 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) ∧ (π‘ž ∈ (Atomsβ€˜πΎ) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ 𝐾 ∈ HL)
10 simp12 1205 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) ∧ (π‘ž ∈ (Atomsβ€˜πΎ) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ 𝑋 ∈ 𝑃)
11 simp2l 1200 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) ∧ (π‘ž ∈ (Atomsβ€˜πΎ) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ π‘ž ∈ (Atomsβ€˜πΎ))
12 simp2r 1201 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) ∧ (π‘ž ∈ (Atomsβ€˜πΎ) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ π‘Ÿ ∈ (Atomsβ€˜πΎ))
13 lplnnlelln.l . . . . . . . 8 ≀ = (leβ€˜πΎ)
14 lplnnlelln.p . . . . . . . 8 𝑃 = (LPlanesβ€˜πΎ)
1513, 3, 4, 14lplnnle2at 38412 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ π‘ž ∈ (Atomsβ€˜πΎ) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ))) β†’ Β¬ 𝑋 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ))
169, 10, 11, 12, 15syl13anc 1373 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) ∧ (π‘ž ∈ (Atomsβ€˜πΎ) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ Β¬ 𝑋 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ))
17 simp3r 1203 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) ∧ (π‘ž ∈ (Atomsβ€˜πΎ) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ))
1817breq2d 5161 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) ∧ (π‘ž ∈ (Atomsβ€˜πΎ) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ)))
1916, 18mtbird 325 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) ∧ (π‘ž ∈ (Atomsβ€˜πΎ) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) ∧ (π‘ž β‰  π‘Ÿ ∧ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ Β¬ 𝑋 ≀ π‘Œ)
20193exp 1120 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) β†’ ((π‘ž ∈ (Atomsβ€˜πΎ) ∧ π‘Ÿ ∈ (Atomsβ€˜πΎ)) β†’ ((π‘ž β‰  π‘Ÿ ∧ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ)) β†’ Β¬ 𝑋 ≀ π‘Œ)))
2120rexlimdvv 3211 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) β†’ (βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)(π‘ž β‰  π‘Ÿ ∧ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ)) β†’ Β¬ 𝑋 ≀ π‘Œ))
2221adantld 492 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) β†’ ((π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)βˆƒπ‘Ÿ ∈ (Atomsβ€˜πΎ)(π‘ž β‰  π‘Ÿ ∧ π‘Œ = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ Β¬ 𝑋 ≀ π‘Œ))
238, 22mpd 15 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) β†’ Β¬ 𝑋 ≀ π‘Œ)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆƒwrex 3071   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  joincjn 18264  Atomscatm 38133  HLchlt 38220  LLinesclln 38362  LPlanesclpl 38363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-llines 38369  df-lplanes 38370
This theorem is referenced by:  lplnnelln  38417
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