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Theorem llni 38976
Description: Condition implying a lattice line. (Contributed by NM, 17-Jun-2012.)
Hypotheses
Ref Expression
llnset.b 𝐡 = (Baseβ€˜πΎ)
llnset.c 𝐢 = ( β‹– β€˜πΎ)
llnset.a 𝐴 = (Atomsβ€˜πΎ)
llnset.n 𝑁 = (LLinesβ€˜πΎ)
Assertion
Ref Expression
llni (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐢𝑋) β†’ 𝑋 ∈ 𝑁)

Proof of Theorem llni
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simpl2 1190 . 2 (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐢𝑋) β†’ 𝑋 ∈ 𝐡)
2 breq1 5146 . . . 4 (𝑝 = 𝑃 β†’ (𝑝𝐢𝑋 ↔ 𝑃𝐢𝑋))
32rspcev 3608 . . 3 ((𝑃 ∈ 𝐴 ∧ 𝑃𝐢𝑋) β†’ βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋)
433ad2antl3 1185 . 2 (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐢𝑋) β†’ βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋)
5 simpl1 1189 . . 3 (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐢𝑋) β†’ 𝐾 ∈ 𝐷)
6 llnset.b . . . 4 𝐡 = (Baseβ€˜πΎ)
7 llnset.c . . . 4 𝐢 = ( β‹– β€˜πΎ)
8 llnset.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
9 llnset.n . . . 4 𝑁 = (LLinesβ€˜πΎ)
106, 7, 8, 9islln 38974 . . 3 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋)))
115, 10syl 17 . 2 (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐢𝑋) β†’ (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋)))
121, 4, 11mpbir2and 712 1 (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐢𝑋) β†’ 𝑋 ∈ 𝑁)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3066   class class class wbr 5143  β€˜cfv 6543  Basecbs 17174   β‹– ccvr 38729  Atomscatm 38730  LLinesclln 38959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-llines 38966
This theorem is referenced by:  llnle  38986  atcvrlln  38988  lplncvrlvol  39084
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