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Theorem lplnnlelln 37553
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 = (le‘𝐾)
lplnnlelln.n 𝑁 = (LLines‘𝐾)
lplnnlelln.p 𝑃 = (LPlanes‘𝐾)
Assertion
Ref Expression
lplnnlelln ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) → ¬ 𝑋 𝑌)

Proof of Theorem lplnnlelln
Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1137 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) → 𝑌𝑁)
2 eqid 2740 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2740 . . . . 5 (join‘𝐾) = (join‘𝐾)
4 eqid 2740 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
5 lplnnlelln.n . . . . 5 𝑁 = (LLines‘𝐾)
62, 3, 4, 5islln2 37521 . . . 4 (𝐾 ∈ HL → (𝑌𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟)))))
763ad2ant1 1132 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) → (𝑌𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟)))))
81, 7mpbid 231 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) → (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))))
9 simp11 1202 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → 𝐾 ∈ HL)
10 simp12 1203 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → 𝑋𝑃)
11 simp2l 1198 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → 𝑞 ∈ (Atoms‘𝐾))
12 simp2r 1199 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → 𝑟 ∈ (Atoms‘𝐾))
13 lplnnlelln.l . . . . . . . 8 = (le‘𝐾)
14 lplnnlelln.p . . . . . . . 8 𝑃 = (LPlanes‘𝐾)
1513, 3, 4, 14lplnnle2at 37551 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾))) → ¬ 𝑋 (𝑞(join‘𝐾)𝑟))
169, 10, 11, 12, 15syl13anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → ¬ 𝑋 (𝑞(join‘𝐾)𝑟))
17 simp3r 1201 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → 𝑌 = (𝑞(join‘𝐾)𝑟))
1817breq2d 5091 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → (𝑋 𝑌𝑋 (𝑞(join‘𝐾)𝑟)))
1916, 18mtbird 325 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → ¬ 𝑋 𝑌)
20193exp 1118 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) → ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) → ((𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟)) → ¬ 𝑋 𝑌)))
2120rexlimdvv 3224 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟)) → ¬ 𝑋 𝑌))
2221adantld 491 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) → ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → ¬ 𝑋 𝑌))
238, 22mpd 15 1 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) → ¬ 𝑋 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086   = wceq 1542  wcel 2110  wne 2945  wrex 3067   class class class wbr 5079  cfv 6432  (class class class)co 7271  Basecbs 16910  lecple 16967  joincjn 18027  Atomscatm 37273  HLchlt 37360  LLinesclln 37501  LPlanesclpl 37502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-proset 18011  df-poset 18029  df-plt 18046  df-lub 18062  df-glb 18063  df-join 18064  df-meet 18065  df-p0 18141  df-lat 18148  df-clat 18215  df-oposet 37186  df-ol 37188  df-oml 37189  df-covers 37276  df-ats 37277  df-atl 37308  df-cvlat 37332  df-hlat 37361  df-llines 37508  df-lplanes 37509
This theorem is referenced by:  lplnnelln  37556
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