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Theorem lcvbr3 36599
 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 36597 . 2 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
7 iman 405 . . . . . 6 (((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ¬ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
8 df-pss 3877 . . . . . . . . 9 (𝑇𝑠 ↔ (𝑇𝑠𝑇𝑠))
9 necom 3004 . . . . . . . . . 10 (𝑇𝑠𝑠𝑇)
109anbi2i 625 . . . . . . . . 9 ((𝑇𝑠𝑇𝑠) ↔ (𝑇𝑠𝑠𝑇))
118, 10bitri 278 . . . . . . . 8 (𝑇𝑠 ↔ (𝑇𝑠𝑠𝑇))
12 df-pss 3877 . . . . . . . 8 (𝑠𝑈 ↔ (𝑠𝑈𝑠𝑈))
1311, 12anbi12i 629 . . . . . . 7 ((𝑇𝑠𝑠𝑈) ↔ ((𝑇𝑠𝑠𝑇) ∧ (𝑠𝑈𝑠𝑈)))
14 an4 655 . . . . . . . 8 (((𝑇𝑠𝑠𝑇) ∧ (𝑠𝑈𝑠𝑈)) ↔ ((𝑇𝑠𝑠𝑈) ∧ (𝑠𝑇𝑠𝑈)))
15 neanior 3043 . . . . . . . . 9 ((𝑠𝑇𝑠𝑈) ↔ ¬ (𝑠 = 𝑇𝑠 = 𝑈))
1615anbi2i 625 . . . . . . . 8 (((𝑇𝑠𝑠𝑈) ∧ (𝑠𝑇𝑠𝑈)) ↔ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
1714, 16bitri 278 . . . . . . 7 (((𝑇𝑠𝑠𝑇) ∧ (𝑠𝑈𝑠𝑈)) ↔ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
1813, 17bitri 278 . . . . . 6 ((𝑇𝑠𝑠𝑈) ↔ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
197, 18xchbinxr 338 . . . . 5 (((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ¬ (𝑇𝑠𝑠𝑈))
2019ralbii 3097 . . . 4 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈))
21 ralnex 3163 . . . 4 (∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
2220, 21bitri 278 . . 3 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
2322anbi2i 625 . 2 ((𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈))) ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
246, 23bitr4di 292 1 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071   ⊆ wss 3858   ⊊ wpss 3859   class class class wbr 5032  ‘cfv 6335  LSubSpclss 19771   ⋖L clcv 36594 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-lcv 36595 This theorem is referenced by:  lcvexchlem4  36613  lcvexchlem5  36614
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