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 35316
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 1229 . 2 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → 𝑋𝐵)
2 breq1 4789 . . . 4 (𝑝 = 𝑃 → (𝑝𝐶𝑋𝑃𝐶𝑋))
32rspcev 3460 . . 3 ((𝑃𝐴𝑃𝐶𝑋) → ∃𝑝𝐴 𝑝𝐶𝑋)
433ad2antl3 1202 . 2 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → ∃𝑝𝐴 𝑝𝐶𝑋)
5 simpl1 1227 . . 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 35314 . . 3 (𝐾𝐷 → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋)))
115, 10syl 17 . 2 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋)))
121, 4, 11mpbir2and 692 1 (((𝐾𝐷𝑋𝐵𝑃𝐴) ∧ 𝑃𝐶𝑋) → 𝑋𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wrex 3062   class class class wbr 4786  cfv 6031  Basecbs 16064  ccvr 35071  Atomscatm 35072  LLinesclln 35299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-llines 35306
This theorem is referenced by:  llnle  35326  atcvrlln  35328  lplncvrlvol  35424
  Copyright terms: Public domain W3C validator