Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  llni Structured version   Visualization version   GIF version

Theorem llni 39491
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 5151 . . . 4 (𝑝 = 𝑃 → (𝑝𝐶𝑋𝑃𝐶𝑋))
32rspcev 3622 . . 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 39489 . . 3 (𝐾𝐷 → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋)))
115, 10syl 17 . 2 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋)))
121, 4, 11mpbir2and 713 1 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → 𝑋𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068   class class class wbr 5148  cfv 6563  Basecbs 17245  ccvr 39244  Atomscatm 39245  LLinesclln 39474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-llines 39481
This theorem is referenced by:  llnle  39501  atcvrlln  39503  lplncvrlvol  39599
  Copyright terms: Public domain W3C validator