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Theorem llni 37519
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 1191 . 2 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → 𝑋𝐵)
2 breq1 5079 . . . 4 (𝑝 = 𝑃 → (𝑝𝐶𝑋𝑃𝐶𝑋))
32rspcev 3561 . . 3 ((𝑃𝐴𝑃𝐶𝑋) → ∃𝑝𝐴 𝑝𝐶𝑋)
433ad2antl3 1186 . 2 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → ∃𝑝𝐴 𝑝𝐶𝑋)
5 simpl1 1190 . . 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 37517 . . 3 (𝐾𝐷 → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋)))
115, 10syl 17 . 2 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋)))
121, 4, 11mpbir2and 710 1 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → 𝑋𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wrex 3065   class class class wbr 5076  cfv 6435  Basecbs 16910  ccvr 37273  Atomscatm 37274  LLinesclln 37502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5225  ax-nul 5232  ax-pr 5354
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5077  df-opab 5139  df-mpt 5160  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-iota 6393  df-fun 6437  df-fv 6443  df-llines 37509
This theorem is referenced by:  llnle  37529  atcvrlln  37531  lplncvrlvol  37627
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