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Theorem lcvbr3 37514
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 37512 . 2 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
7 iman 403 . . . . . 6 (((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ¬ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
8 df-pss 3934 . . . . . . . . 9 (𝑇𝑠 ↔ (𝑇𝑠𝑇𝑠))
9 necom 2998 . . . . . . . . . 10 (𝑇𝑠𝑠𝑇)
109anbi2i 624 . . . . . . . . 9 ((𝑇𝑠𝑇𝑠) ↔ (𝑇𝑠𝑠𝑇))
118, 10bitri 275 . . . . . . . 8 (𝑇𝑠 ↔ (𝑇𝑠𝑠𝑇))
12 df-pss 3934 . . . . . . . 8 (𝑠𝑈 ↔ (𝑠𝑈𝑠𝑈))
1311, 12anbi12i 628 . . . . . . 7 ((𝑇𝑠𝑠𝑈) ↔ ((𝑇𝑠𝑠𝑇) ∧ (𝑠𝑈𝑠𝑈)))
14 an4 655 . . . . . . . 8 (((𝑇𝑠𝑠𝑇) ∧ (𝑠𝑈𝑠𝑈)) ↔ ((𝑇𝑠𝑠𝑈) ∧ (𝑠𝑇𝑠𝑈)))
15 neanior 3038 . . . . . . . . 9 ((𝑠𝑇𝑠𝑈) ↔ ¬ (𝑠 = 𝑇𝑠 = 𝑈))
1615anbi2i 624 . . . . . . . 8 (((𝑇𝑠𝑠𝑈) ∧ (𝑠𝑇𝑠𝑈)) ↔ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
1714, 16bitri 275 . . . . . . 7 (((𝑇𝑠𝑠𝑇) ∧ (𝑠𝑈𝑠𝑈)) ↔ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
1813, 17bitri 275 . . . . . 6 ((𝑇𝑠𝑠𝑈) ↔ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
197, 18xchbinxr 335 . . . . 5 (((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ¬ (𝑇𝑠𝑠𝑈))
2019ralbii 3097 . . . 4 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈))
21 ralnex 3076 . . . 4 (∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
2220, 21bitri 275 . . 3 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
2322anbi2i 624 . 2 ((𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈))) ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
246, 23bitr4di 289 1 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2944  wral 3065  wrex 3074  wss 3915  wpss 3916   class class class wbr 5110  cfv 6501  LSubSpclss 20408  L clcv 37509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-lcv 37510
This theorem is referenced by:  lcvexchlem4  37528  lcvexchlem5  37529
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