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Theorem lcvbr3 33790
Description: The covers relation for a left vector space (or a left module). (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
lcvbr3 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)))))
Distinct variable groups:   𝑆,𝑠   𝑊,𝑠   𝑇,𝑠   𝑈,𝑠
Allowed substitution hints:   𝜑(𝑠)   𝐶(𝑠)   𝑋(𝑠)

Proof of Theorem lcvbr3
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 33788 . 2 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
7 iman 440 . . . . . 6 (((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ¬ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
8 df-pss 3571 . . . . . . . . 9 (𝑇𝑠 ↔ (𝑇𝑠𝑇𝑠))
9 necom 2843 . . . . . . . . . 10 (𝑇𝑠𝑠𝑇)
109anbi2i 729 . . . . . . . . 9 ((𝑇𝑠𝑇𝑠) ↔ (𝑇𝑠𝑠𝑇))
118, 10bitri 264 . . . . . . . 8 (𝑇𝑠 ↔ (𝑇𝑠𝑠𝑇))
12 df-pss 3571 . . . . . . . 8 (𝑠𝑈 ↔ (𝑠𝑈𝑠𝑈))
1311, 12anbi12i 732 . . . . . . 7 ((𝑇𝑠𝑠𝑈) ↔ ((𝑇𝑠𝑠𝑇) ∧ (𝑠𝑈𝑠𝑈)))
14 an4 864 . . . . . . . 8 (((𝑇𝑠𝑠𝑇) ∧ (𝑠𝑈𝑠𝑈)) ↔ ((𝑇𝑠𝑠𝑈) ∧ (𝑠𝑇𝑠𝑈)))
15 neanior 2882 . . . . . . . . 9 ((𝑠𝑇𝑠𝑈) ↔ ¬ (𝑠 = 𝑇𝑠 = 𝑈))
1615anbi2i 729 . . . . . . . 8 (((𝑇𝑠𝑠𝑈) ∧ (𝑠𝑇𝑠𝑈)) ↔ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
1714, 16bitri 264 . . . . . . 7 (((𝑇𝑠𝑠𝑇) ∧ (𝑠𝑈𝑠𝑈)) ↔ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
1813, 17bitri 264 . . . . . 6 ((𝑇𝑠𝑠𝑈) ↔ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
197, 18xchbinxr 325 . . . . 5 (((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ¬ (𝑇𝑠𝑠𝑈))
2019ralbii 2974 . . . 4 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈))
21 ralnex 2986 . . . 4 (∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
2220, 21bitri 264 . . 3 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
2322anbi2i 729 . 2 ((𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈))) ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
246, 23syl6bbr 278 1 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  wss 3555  wpss 3556   class class class wbr 4613  cfv 5847  LSubSpclss 18851  L clcv 33785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-lcv 33786
This theorem is referenced by:  lcvexchlem4  33804  lcvexchlem5  33805
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