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Theorem lcvbr2 35170
Description: The covers relation for a left vector space (or a left module). (cvbr2 29728 analog.) (Contributed by NM, 9-Jan-2015.)
Hypotheses
Ref Expression
lcvfbr.s 𝑆 = (LSubSp‘𝑊)
lcvfbr.c 𝐶 = ( ⋖L𝑊)
lcvfbr.w (𝜑𝑊𝑋)
lcvfbr.t (𝜑𝑇𝑆)
lcvfbr.u (𝜑𝑈𝑆)
Assertion
Ref Expression
lcvbr2 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈))))
Distinct variable groups:   𝑆,𝑠   𝑊,𝑠   𝑇,𝑠   𝑈,𝑠
Allowed substitution hints:   𝜑(𝑠)   𝐶(𝑠)   𝑋(𝑠)

Proof of Theorem lcvbr2
StepHypRef Expression
1 lcvfbr.s . . 3 𝑆 = (LSubSp‘𝑊)
2 lcvfbr.c . . 3 𝐶 = ( ⋖L𝑊)
3 lcvfbr.w . . 3 (𝜑𝑊𝑋)
4 lcvfbr.t . . 3 (𝜑𝑇𝑆)
5 lcvfbr.u . . 3 (𝜑𝑈𝑆)
61, 2, 3, 4, 5lcvbr 35169 . 2 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
7 iman 392 . . . . . 6 (((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ¬ ((𝑇𝑠𝑠𝑈) ∧ ¬ 𝑠 = 𝑈))
8 anass 462 . . . . . . 7 (((𝑇𝑠𝑠𝑈) ∧ ¬ 𝑠 = 𝑈) ↔ (𝑇𝑠 ∧ (𝑠𝑈 ∧ ¬ 𝑠 = 𝑈)))
9 dfpss2 3913 . . . . . . . 8 (𝑠𝑈 ↔ (𝑠𝑈 ∧ ¬ 𝑠 = 𝑈))
109anbi2i 616 . . . . . . 7 ((𝑇𝑠𝑠𝑈) ↔ (𝑇𝑠 ∧ (𝑠𝑈 ∧ ¬ 𝑠 = 𝑈)))
118, 10bitr4i 270 . . . . . 6 (((𝑇𝑠𝑠𝑈) ∧ ¬ 𝑠 = 𝑈) ↔ (𝑇𝑠𝑠𝑈))
127, 11xchbinx 326 . . . . 5 (((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ¬ (𝑇𝑠𝑠𝑈))
1312ralbii 3161 . . . 4 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈))
14 ralnex 3173 . . . 4 (∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
1513, 14bitri 267 . . 3 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
1615anbi2i 616 . 2 ((𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈)) ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
176, 16syl6bbr 281 1 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386   = wceq 1601  wcel 2106  wral 3089  wrex 3090  wss 3791  wpss 3792   class class class wbr 4886  cfv 6135  LSubSpclss 19324  L clcv 35166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-lcv 35167
This theorem is referenced by:  lsmcv2  35177  lsat0cv  35181
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