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Theorem lplnn0N 35617
Description: A lattice plane is nonzero. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lplnn0.z 0 = (0.‘𝐾)
lplnn0.p 𝑃 = (LPlanes‘𝐾)
Assertion
Ref Expression
lplnn0N ((𝐾 ∈ HL ∧ 𝑋𝑃) → 𝑋0 )

Proof of Theorem lplnn0N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 eqid 2825 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
21atex 35476 . . . 4 (𝐾 ∈ HL → (Atoms‘𝐾) ≠ ∅)
3 n0 4162 . . . 4 ((Atoms‘𝐾) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (Atoms‘𝐾))
42, 3sylib 210 . . 3 (𝐾 ∈ HL → ∃𝑝 𝑝 ∈ (Atoms‘𝐾))
54adantr 474 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑃) → ∃𝑝 𝑝 ∈ (Atoms‘𝐾))
6 eqid 2825 . . . . 5 (le‘𝐾) = (le‘𝐾)
7 lplnn0.p . . . . 5 𝑃 = (LPlanes‘𝐾)
86, 1, 7lplnnleat 35612 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑝 ∈ (Atoms‘𝐾)) → ¬ 𝑋(le‘𝐾)𝑝)
983expa 1151 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ¬ 𝑋(le‘𝐾)𝑝)
10 hlop 35432 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ OP)
1110ad2antrr 717 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP)
12 eqid 2825 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
1312, 1atbase 35359 . . . . . . 7 (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
1413adantl 475 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Base‘𝐾))
15 lplnn0.z . . . . . . 7 0 = (0.‘𝐾)
1612, 6, 15op0le 35256 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑝)
1711, 14, 16syl2anc 579 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 0 (le‘𝐾)𝑝)
18 breq1 4878 . . . . 5 (𝑋 = 0 → (𝑋(le‘𝐾)𝑝0 (le‘𝐾)𝑝))
1917, 18syl5ibrcom 239 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑋 = 0𝑋(le‘𝐾)𝑝))
2019necon3bd 3013 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (¬ 𝑋(le‘𝐾)𝑝𝑋0 ))
219, 20mpd 15 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑋0 )
225, 21exlimddv 2034 1 ((𝐾 ∈ HL ∧ 𝑋𝑃) → 𝑋0 )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1656  wex 1878  wcel 2164  wne 2999  c0 4146   class class class wbr 4875  cfv 6127  Basecbs 16229  lecple 16319  0.cp0 17397  OPcops 35242  Atomscatm 35333  HLchlt 35420  LPlanesclpl 35562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-proset 17288  df-poset 17306  df-plt 17318  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-p0 17399  df-p1 17400  df-lat 17406  df-clat 17468  df-oposet 35246  df-ol 35248  df-oml 35249  df-covers 35336  df-ats 35337  df-atl 35368  df-cvlat 35392  df-hlat 35421  df-llines 35568  df-lplanes 35569
This theorem is referenced by: (None)
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