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Theorem lplnn0N 39052
Description: A lattice plane is nonzero. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lplnn0.z 0 = (0.β€˜πΎ)
lplnn0.p 𝑃 = (LPlanesβ€˜πΎ)
Assertion
Ref Expression
lplnn0N ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) β†’ 𝑋 β‰  0 )

Proof of Theorem lplnn0N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 eqid 2728 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
21atex 38911 . . . 4 (𝐾 ∈ HL β†’ (Atomsβ€˜πΎ) β‰  βˆ…)
3 n0 4350 . . . 4 ((Atomsβ€˜πΎ) β‰  βˆ… ↔ βˆƒπ‘ 𝑝 ∈ (Atomsβ€˜πΎ))
42, 3sylib 217 . . 3 (𝐾 ∈ HL β†’ βˆƒπ‘ 𝑝 ∈ (Atomsβ€˜πΎ))
54adantr 479 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) β†’ βˆƒπ‘ 𝑝 ∈ (Atomsβ€˜πΎ))
6 eqid 2728 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
7 lplnn0.p . . . . 5 𝑃 = (LPlanesβ€˜πΎ)
86, 1, 7lplnnleat 39047 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ Β¬ 𝑋(leβ€˜πΎ)𝑝)
983expa 1115 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ Β¬ 𝑋(leβ€˜πΎ)𝑝)
10 hlop 38866 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
1110ad2antrr 724 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝐾 ∈ OP)
12 eqid 2728 . . . . . . . 8 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1312, 1atbase 38793 . . . . . . 7 (𝑝 ∈ (Atomsβ€˜πΎ) β†’ 𝑝 ∈ (Baseβ€˜πΎ))
1413adantl 480 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝑝 ∈ (Baseβ€˜πΎ))
15 lplnn0.z . . . . . . 7 0 = (0.β€˜πΎ)
1612, 6, 15op0le 38690 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Baseβ€˜πΎ)) β†’ 0 (leβ€˜πΎ)𝑝)
1711, 14, 16syl2anc 582 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 0 (leβ€˜πΎ)𝑝)
18 breq1 5155 . . . . 5 (𝑋 = 0 β†’ (𝑋(leβ€˜πΎ)𝑝 ↔ 0 (leβ€˜πΎ)𝑝))
1917, 18syl5ibrcom 246 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (𝑋 = 0 β†’ 𝑋(leβ€˜πΎ)𝑝))
2019necon3bd 2951 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (Β¬ 𝑋(leβ€˜πΎ)𝑝 β†’ 𝑋 β‰  0 ))
219, 20mpd 15 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝑋 β‰  0 )
225, 21exlimddv 1930 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) β†’ 𝑋 β‰  0 )
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098   β‰  wne 2937  βˆ…c0 4326   class class class wbr 5152  β€˜cfv 6553  Basecbs 17187  lecple 17247  0.cp0 18422  OPcops 38676  Atomscatm 38767  HLchlt 38854  LPlanesclpl 38997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-p1 18425  df-lat 18431  df-clat 18498  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-llines 39003  df-lplanes 39004
This theorem is referenced by: (None)
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