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Theorem lcvbr2 36276
Description: The covers relation for a left vector space (or a left module). (cvbr2 30064 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 36275 . 2 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
7 iman 405 . . . . . 6 (((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ¬ ((𝑇𝑠𝑠𝑈) ∧ ¬ 𝑠 = 𝑈))
8 anass 472 . . . . . . 7 (((𝑇𝑠𝑠𝑈) ∧ ¬ 𝑠 = 𝑈) ↔ (𝑇𝑠 ∧ (𝑠𝑈 ∧ ¬ 𝑠 = 𝑈)))
9 dfpss2 4037 . . . . . . . 8 (𝑠𝑈 ↔ (𝑠𝑈 ∧ ¬ 𝑠 = 𝑈))
109anbi2i 625 . . . . . . 7 ((𝑇𝑠𝑠𝑈) ↔ (𝑇𝑠 ∧ (𝑠𝑈 ∧ ¬ 𝑠 = 𝑈)))
118, 10bitr4i 281 . . . . . 6 (((𝑇𝑠𝑠𝑈) ∧ ¬ 𝑠 = 𝑈) ↔ (𝑇𝑠𝑠𝑈))
127, 11xchbinx 337 . . . . 5 (((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ¬ (𝑇𝑠𝑠𝑈))
1312ralbii 3157 . . . 4 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈))
14 ralnex 3224 . . . 4 (∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
1513, 14bitri 278 . . 3 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
1615anbi2i 625 . 2 ((𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈)) ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
176, 16syl6bbr 292 1 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wral 3130  wrex 3131  wss 3908  wpss 3909   class class class wbr 5042  cfv 6334  LSubSpclss 19694  L clcv 36272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-lcv 36273
This theorem is referenced by:  lsmcv2  36283  lsat0cv  36287
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