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Theorem lplnnlelln 38500
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 1138 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑁) β†’ π‘Œ ∈ 𝑁)
2 eqid 2732 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3 eqid 2732 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
4 eqid 2732 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
5 lplnnlelln.n . . . . 5 𝑁 = (LLinesβ€˜πΎ)
62, 3, 4, 5islln2 38468 . . . 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 1133 . . 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 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†’ 𝐾 ∈ HL)
10 simp12 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†’ 𝑋 ∈ 𝑃)
11 simp2l 1199 . . . . . . 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 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‘Ÿ ∈ (Atomsβ€˜πΎ))
13 lplnnlelln.l . . . . . . . 8 ≀ = (leβ€˜πΎ)
14 lplnnlelln.p . . . . . . . 8 𝑃 = (LPlanesβ€˜πΎ)
1513, 3, 4, 14lplnnle2at 38498 . . . . . . 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 1372 . . . . . 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 1202 . . . . . . 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 5160 . . . . . 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 324 . . . . 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 1119 . . . 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 3210 . . 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 491 . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  lecple 17206  joincjn 18266  Atomscatm 38219  HLchlt 38306  LLinesclln 38448  LPlanesclpl 38449
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7367  df-ov 7414  df-oprab 7415  df-proset 18250  df-poset 18268  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-lat 18387  df-clat 18454  df-oposet 38132  df-ol 38134  df-oml 38135  df-covers 38222  df-ats 38223  df-atl 38254  df-cvlat 38278  df-hlat 38307  df-llines 38455  df-lplanes 38456
This theorem is referenced by:  lplnnelln  38503
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