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Theorem lcvbr3 36029
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 36027 . 2 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
7 iman 402 . . . . . 6 (((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ¬ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
8 df-pss 3958 . . . . . . . . 9 (𝑇𝑠 ↔ (𝑇𝑠𝑇𝑠))
9 necom 3074 . . . . . . . . . 10 (𝑇𝑠𝑠𝑇)
109anbi2i 622 . . . . . . . . 9 ((𝑇𝑠𝑇𝑠) ↔ (𝑇𝑠𝑠𝑇))
118, 10bitri 276 . . . . . . . 8 (𝑇𝑠 ↔ (𝑇𝑠𝑠𝑇))
12 df-pss 3958 . . . . . . . 8 (𝑠𝑈 ↔ (𝑠𝑈𝑠𝑈))
1311, 12anbi12i 626 . . . . . . 7 ((𝑇𝑠𝑠𝑈) ↔ ((𝑇𝑠𝑠𝑇) ∧ (𝑠𝑈𝑠𝑈)))
14 an4 652 . . . . . . . 8 (((𝑇𝑠𝑠𝑇) ∧ (𝑠𝑈𝑠𝑈)) ↔ ((𝑇𝑠𝑠𝑈) ∧ (𝑠𝑇𝑠𝑈)))
15 neanior 3114 . . . . . . . . 9 ((𝑠𝑇𝑠𝑈) ↔ ¬ (𝑠 = 𝑇𝑠 = 𝑈))
1615anbi2i 622 . . . . . . . 8 (((𝑇𝑠𝑠𝑈) ∧ (𝑠𝑇𝑠𝑈)) ↔ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
1714, 16bitri 276 . . . . . . 7 (((𝑇𝑠𝑠𝑇) ∧ (𝑠𝑈𝑠𝑈)) ↔ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
1813, 17bitri 276 . . . . . 6 ((𝑇𝑠𝑠𝑈) ↔ ((𝑇𝑠𝑠𝑈) ∧ ¬ (𝑠 = 𝑇𝑠 = 𝑈)))
197, 18xchbinxr 336 . . . . 5 (((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ¬ (𝑇𝑠𝑠𝑈))
2019ralbii 3170 . . . 4 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈))
21 ralnex 3241 . . . 4 (∀𝑠𝑆 ¬ (𝑇𝑠𝑠𝑈) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
2220, 21bitri 276 . . 3 (∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))
2322anbi2i 622 . 2 ((𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈))) ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
246, 23syl6bbr 290 1 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  wss 3940  wpss 3941   class class class wbr 5063  cfv 6352  LSubSpclss 19623  L clcv 36024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6312  df-fun 6354  df-fv 6360  df-lcv 36025
This theorem is referenced by:  lcvexchlem4  36043  lcvexchlem5  36044
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