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Theorem llni 35584
Description: Condition implying a lattice line. (Contributed by NM, 17-Jun-2012.)
Hypotheses
Ref Expression
llnset.b 𝐵 = (Base‘𝐾)
llnset.c 𝐶 = ( ⋖ ‘𝐾)
llnset.a 𝐴 = (Atoms‘𝐾)
llnset.n 𝑁 = (LLines‘𝐾)
Assertion
Ref Expression
llni (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → 𝑋𝑁)

Proof of Theorem llni
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simpl2 1250 . 2 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → 𝑋𝐵)
2 breq1 4877 . . . 4 (𝑝 = 𝑃 → (𝑝𝐶𝑋𝑃𝐶𝑋))
32rspcev 3527 . . 3 ((𝑃𝐴𝑃𝐶𝑋) → ∃𝑝𝐴 𝑝𝐶𝑋)
433ad2antl3 1244 . 2 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → ∃𝑝𝐴 𝑝𝐶𝑋)
5 simpl1 1248 . . 3 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → 𝐾𝐷)
6 llnset.b . . . 4 𝐵 = (Base‘𝐾)
7 llnset.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
8 llnset.a . . . 4 𝐴 = (Atoms‘𝐾)
9 llnset.n . . . 4 𝑁 = (LLines‘𝐾)
106, 7, 8, 9islln 35582 . . 3 (𝐾𝐷 → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋)))
115, 10syl 17 . 2 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋)))
121, 4, 11mpbir2and 706 1 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → 𝑋𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wrex 3119   class class class wbr 4874  cfv 6124  Basecbs 16223  ccvr 35338  Atomscatm 35339  LLinesclln 35567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-iota 6087  df-fun 6126  df-fv 6132  df-llines 35574
This theorem is referenced by:  llnle  35594  atcvrlln  35596  lplncvrlvol  35692
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