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

Theorem lcvntr 39027
Description: The covers relation is not transitive. (cvntr 32311 analog.) (Contributed by NM, 10-Jan-2015.)
Hypotheses
Ref Expression
lcvnbtwn.s 𝑆 = (LSubSp‘𝑊)
lcvnbtwn.c 𝐶 = ( ⋖L𝑊)
lcvnbtwn.w (𝜑𝑊𝑋)
lcvnbtwn.r (𝜑𝑅𝑆)
lcvnbtwn.t (𝜑𝑇𝑆)
lcvnbtwn.u (𝜑𝑈𝑆)
lcvnbtwn.d (𝜑𝑅𝐶𝑇)
lcvntr.p (𝜑𝑇𝐶𝑈)
Assertion
Ref Expression
lcvntr (𝜑 → ¬ 𝑅𝐶𝑈)

Proof of Theorem lcvntr
StepHypRef Expression
1 lcvnbtwn.s . . . 4 𝑆 = (LSubSp‘𝑊)
2 lcvnbtwn.c . . . 4 𝐶 = ( ⋖L𝑊)
3 lcvnbtwn.w . . . 4 (𝜑𝑊𝑋)
4 lcvnbtwn.r . . . 4 (𝜑𝑅𝑆)
5 lcvnbtwn.t . . . 4 (𝜑𝑇𝑆)
6 lcvnbtwn.d . . . 4 (𝜑𝑅𝐶𝑇)
71, 2, 3, 4, 5, 6lcvpss 39025 . . 3 (𝜑𝑅𝑇)
8 lcvnbtwn.u . . . 4 (𝜑𝑈𝑆)
9 lcvntr.p . . . 4 (𝜑𝑇𝐶𝑈)
101, 2, 3, 5, 8, 9lcvpss 39025 . . 3 (𝜑𝑇𝑈)
117, 10jca 511 . 2 (𝜑 → (𝑅𝑇𝑇𝑈))
123adantr 480 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑊𝑋)
134adantr 480 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑅𝑆)
148adantr 480 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑈𝑆)
155adantr 480 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑇𝑆)
16 simpr 484 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑅𝐶𝑈)
171, 2, 12, 13, 14, 15, 16lcvnbtwn 39026 . . 3 ((𝜑𝑅𝐶𝑈) → ¬ (𝑅𝑇𝑇𝑈))
1817ex 412 . 2 (𝜑 → (𝑅𝐶𝑈 → ¬ (𝑅𝑇𝑇𝑈)))
1911, 18mt2d 136 1 (𝜑 → ¬ 𝑅𝐶𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wpss 3952   class class class wbr 5143  cfv 6561  LSubSpclss 20929  L clcv 39019
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-pow 5365  ax-pr 5432  ax-un 7755
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-pss 3971  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-lcv 39020
This theorem is referenced by:  lsatcv0eq  39048
  Copyright terms: Public domain W3C validator