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Theorem lcvbr2 34128
 Description: The covers relation for a left vector space (or a left module). (cvbr2 29112 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 34127 . 2 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
7 iman 440 . . . . . 6 (((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ¬ ((𝑇𝑠𝑠𝑈) ∧ ¬ 𝑠 = 𝑈))
8 anass 680 . . . . . . 7 (((𝑇𝑠𝑠𝑈) ∧ ¬ 𝑠 = 𝑈) ↔ (𝑇𝑠 ∧ (𝑠𝑈 ∧ ¬ 𝑠 = 𝑈)))
9 dfpss2 3684 . . . . . . . 8 (𝑠𝑈 ↔ (𝑠𝑈 ∧ ¬ 𝑠 = 𝑈))
109anbi2i 729 . . . . . . 7 ((𝑇𝑠𝑠𝑈) ↔ (𝑇𝑠 ∧ (𝑠𝑈 ∧ ¬ 𝑠 = 𝑈)))
118, 10bitr4i 267 . . . . . 6 (((𝑇𝑠𝑠𝑈) ∧ ¬ 𝑠 = 𝑈) ↔ (𝑇𝑠𝑠𝑈))
127, 11xchbinx 324 . . . . 5 (((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ¬ (𝑇𝑠𝑠𝑈))
1312ralbii 2977 . . . 4 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈))
14 ralnex 2989 . . . 4 (∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
1513, 14bitri 264 . . 3 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
1615anbi2i 729 . 2 ((𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈)) ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
176, 16syl6bbr 278 1 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1481   ∈ wcel 1988  ∀wral 2909  ∃wrex 2910   ⊆ wss 3567   ⊊ wpss 3568   class class class wbr 4644  ‘cfv 5876  LSubSpclss 18913   ⋖L clcv 34124 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-lcv 34125 This theorem is referenced by:  lsmcv2  34135  lsat0cv  34139
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