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Theorem lcvntr 34139
Description: The covers relation is not transitive. (cvntr 29135 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 34137 . . 3 (𝜑𝑅𝑇)
8 lcvnbtwn.u . . . 4 (𝜑𝑈𝑆)
9 lcvntr.p . . . 4 (𝜑𝑇𝐶𝑈)
101, 2, 3, 5, 8, 9lcvpss 34137 . . 3 (𝜑𝑇𝑈)
117, 10jca 554 . 2 (𝜑 → (𝑅𝑇𝑇𝑈))
123adantr 481 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑊𝑋)
134adantr 481 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑅𝑆)
148adantr 481 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑈𝑆)
155adantr 481 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑇𝑆)
16 simpr 477 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑅𝐶𝑈)
171, 2, 12, 13, 14, 15, 16lcvnbtwn 34138 . . 3 ((𝜑𝑅𝐶𝑈) → ¬ (𝑅𝑇𝑇𝑈))
1817ex 450 . 2 (𝜑 → (𝑅𝐶𝑈 → ¬ (𝑅𝑇𝑇𝑈)))
1911, 18mt2d 131 1 (𝜑 → ¬ 𝑅𝐶𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1482  wcel 1989  wpss 3573   class class class wbr 4651  cfv 5886  LSubSpclss 18926  L clcv 34131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fv 5894  df-lcv 34132
This theorem is referenced by:  lsatcv0eq  34160
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