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Theorem lplnnlelln 36832
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 1135 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) → 𝑌𝑁)
2 eqid 2801 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2801 . . . . 5 (join‘𝐾) = (join‘𝐾)
4 eqid 2801 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
5 lplnnlelln.n . . . . 5 𝑁 = (LLines‘𝐾)
62, 3, 4, 5islln2 36800 . . . 4 (𝐾 ∈ HL → (𝑌𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟)))))
763ad2ant1 1130 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) → (𝑌𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟)))))
81, 7mpbid 235 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) → (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))))
9 simp11 1200 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → 𝐾 ∈ HL)
10 simp12 1201 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → 𝑋𝑃)
11 simp2l 1196 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → 𝑞 ∈ (Atoms‘𝐾))
12 simp2r 1197 . . . . . . 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 36830 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾))) → ¬ 𝑋 (𝑞(join‘𝐾)𝑟))
169, 10, 11, 12, 15syl13anc 1369 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → ¬ 𝑋 (𝑞(join‘𝐾)𝑟))
17 simp3r 1199 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → 𝑌 = (𝑞(join‘𝐾)𝑟))
1817breq2d 5045 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → (𝑋 𝑌𝑋 (𝑞(join‘𝐾)𝑟)))
1916, 18mtbird 328 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟))) → ¬ 𝑋 𝑌)
20193exp 1116 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) → ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) → ((𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟)) → ¬ 𝑋 𝑌)))
2120rexlimdvv 3255 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑁) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑌 = (𝑞(join‘𝐾)𝑟)) → ¬ 𝑋 𝑌))
2221adantld 494 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wrex 3110   class class class wbr 5033  cfv 6328  (class class class)co 7139  Basecbs 16478  lecple 16567  joincjn 17549  Atomscatm 36552  HLchlt 36639  LLinesclln 36780  LPlanesclpl 36781
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-lat 17651  df-clat 17713  df-oposet 36465  df-ol 36467  df-oml 36468  df-covers 36555  df-ats 36556  df-atl 36587  df-cvlat 36611  df-hlat 36640  df-llines 36787  df-lplanes 36788
This theorem is referenced by:  lplnnelln  36835
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