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 39510
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 1193 . 2 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → 𝑋𝐵)
2 breq1 5146 . . . 4 (𝑝 = 𝑃 → (𝑝𝐶𝑋𝑃𝐶𝑋))
32rspcev 3622 . . 3 ((𝑃𝐴𝑃𝐶𝑋) → ∃𝑝𝐴 𝑝𝐶𝑋)
433ad2antl3 1188 . 2 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → ∃𝑝𝐴 𝑝𝐶𝑋)
5 simpl1 1192 . . 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 39508 . . 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 1087   = wceq 1540  wcel 2108  wrex 3070   class class class wbr 5143  cfv 6561  Basecbs 17247  ccvr 39263  Atomscatm 39264  LLinesclln 39493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-llines 39500
This theorem is referenced by:  llnle  39520  atcvrlln  39522  lplncvrlvol  39618
  Copyright terms: Public domain W3C validator