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Theorem lcvbr2 38403
Description: The covers relation for a left vector space (or a left module). (cvbr2 32041 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 38402 . 2 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
7 iman 401 . . . . . 6 (((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ¬ ((𝑇𝑠𝑠𝑈) ∧ ¬ 𝑠 = 𝑈))
8 anass 468 . . . . . . 7 (((𝑇𝑠𝑠𝑈) ∧ ¬ 𝑠 = 𝑈) ↔ (𝑇𝑠 ∧ (𝑠𝑈 ∧ ¬ 𝑠 = 𝑈)))
9 dfpss2 4080 . . . . . . . 8 (𝑠𝑈 ↔ (𝑠𝑈 ∧ ¬ 𝑠 = 𝑈))
109anbi2i 622 . . . . . . 7 ((𝑇𝑠𝑠𝑈) ↔ (𝑇𝑠 ∧ (𝑠𝑈 ∧ ¬ 𝑠 = 𝑈)))
118, 10bitr4i 278 . . . . . 6 (((𝑇𝑠𝑠𝑈) ∧ ¬ 𝑠 = 𝑈) ↔ (𝑇𝑠𝑠𝑈))
127, 11xchbinx 334 . . . . 5 (((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ¬ (𝑇𝑠𝑠𝑈))
1312ralbii 3087 . . . 4 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈))
14 ralnex 3066 . . . 4 (∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
1513, 14bitri 275 . . 3 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
1615anbi2i 622 . 2 ((𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈)) ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
176, 16bitr4di 289 1 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3055  wrex 3064  wss 3943  wpss 3944   class class class wbr 5141  cfv 6536  LSubSpclss 20776  L clcv 38399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-lcv 38400
This theorem is referenced by:  lsmcv2  38410  lsat0cv  38414
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