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Theorem lcvntr 37517
Description: The covers relation is not transitive. (cvntr 31276 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 37515 . . 3 (𝜑𝑅𝑇)
8 lcvnbtwn.u . . . 4 (𝜑𝑈𝑆)
9 lcvntr.p . . . 4 (𝜑𝑇𝐶𝑈)
101, 2, 3, 5, 8, 9lcvpss 37515 . . 3 (𝜑𝑇𝑈)
117, 10jca 513 . 2 (𝜑 → (𝑅𝑇𝑇𝑈))
123adantr 482 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑊𝑋)
134adantr 482 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑅𝑆)
148adantr 482 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑈𝑆)
155adantr 482 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑇𝑆)
16 simpr 486 . . . 4 ((𝜑𝑅𝐶𝑈) → 𝑅𝐶𝑈)
171, 2, 12, 13, 14, 15, 16lcvnbtwn 37516 . . 3 ((𝜑𝑅𝐶𝑈) → ¬ (𝑅𝑇𝑇𝑈))
1817ex 414 . 2 (𝜑 → (𝑅𝐶𝑈 → ¬ (𝑅𝑇𝑇𝑈)))
1911, 18mt2d 136 1 (𝜑 → ¬ 𝑅𝐶𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wpss 3916   class class class wbr 5110  cfv 6501  LSubSpclss 20408  L clcv 37509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-lcv 37510
This theorem is referenced by:  lsatcv0eq  37538
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