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Theorem lcvbr 38493
Description: The covers relation for a left vector space (or a left module). (cvbr 32091 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
lcvbr (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
Distinct variable groups:   𝑆,𝑠   𝑊,𝑠   𝑇,𝑠   𝑈,𝑠
Allowed substitution hints:   𝜑(𝑠)   𝐶(𝑠)   𝑋(𝑠)

Proof of Theorem lcvbr
Dummy variables 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lcvfbr.t . . 3 (𝜑𝑇𝑆)
2 lcvfbr.u . . 3 (𝜑𝑈𝑆)
3 eleq1 2817 . . . . . 6 (𝑡 = 𝑇 → (𝑡𝑆𝑇𝑆))
43anbi1d 630 . . . . 5 (𝑡 = 𝑇 → ((𝑡𝑆𝑢𝑆) ↔ (𝑇𝑆𝑢𝑆)))
5 psseq1 4085 . . . . . 6 (𝑡 = 𝑇 → (𝑡𝑢𝑇𝑢))
6 psseq1 4085 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡𝑠𝑇𝑠))
76anbi1d 630 . . . . . . . 8 (𝑡 = 𝑇 → ((𝑡𝑠𝑠𝑢) ↔ (𝑇𝑠𝑠𝑢)))
87rexbidv 3175 . . . . . . 7 (𝑡 = 𝑇 → (∃𝑠𝑆 (𝑡𝑠𝑠𝑢) ↔ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)))
98notbid 318 . . . . . 6 (𝑡 = 𝑇 → (¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)))
105, 9anbi12d 631 . . . . 5 (𝑡 = 𝑇 → ((𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)) ↔ (𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢))))
114, 10anbi12d 631 . . . 4 (𝑡 = 𝑇 → (((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢))) ↔ ((𝑇𝑆𝑢𝑆) ∧ (𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)))))
12 eleq1 2817 . . . . . 6 (𝑢 = 𝑈 → (𝑢𝑆𝑈𝑆))
1312anbi2d 629 . . . . 5 (𝑢 = 𝑈 → ((𝑇𝑆𝑢𝑆) ↔ (𝑇𝑆𝑈𝑆)))
14 psseq2 4086 . . . . . 6 (𝑢 = 𝑈 → (𝑇𝑢𝑇𝑈))
15 psseq2 4086 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑠𝑢𝑠𝑈))
1615anbi2d 629 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑇𝑠𝑠𝑢) ↔ (𝑇𝑠𝑠𝑈)))
1716rexbidv 3175 . . . . . . 7 (𝑢 = 𝑈 → (∃𝑠𝑆 (𝑇𝑠𝑠𝑢) ↔ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
1817notbid 318 . . . . . 6 (𝑢 = 𝑈 → (¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
1914, 18anbi12d 631 . . . . 5 (𝑢 = 𝑈 → ((𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)) ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
2013, 19anbi12d 631 . . . 4 (𝑢 = 𝑈 → (((𝑇𝑆𝑢𝑆) ∧ (𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢))) ↔ ((𝑇𝑆𝑈𝑆) ∧ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))))
21 eqid 2728 . . . 4 {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))} = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))}
2211, 20, 21brabg 5541 . . 3 ((𝑇𝑆𝑈𝑆) → (𝑇{⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))}𝑈 ↔ ((𝑇𝑆𝑈𝑆) ∧ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))))
231, 2, 22syl2anc 583 . 2 (𝜑 → (𝑇{⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))}𝑈 ↔ ((𝑇𝑆𝑈𝑆) ∧ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))))
24 lcvfbr.s . . . 4 𝑆 = (LSubSp‘𝑊)
25 lcvfbr.c . . . 4 𝐶 = ( ⋖L𝑊)
26 lcvfbr.w . . . 4 (𝜑𝑊𝑋)
2724, 25, 26lcvfbr 38492 . . 3 (𝜑𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
2827breqd 5159 . 2 (𝜑 → (𝑇𝐶𝑈𝑇{⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))}𝑈))
291, 2jca 511 . . 3 (𝜑 → (𝑇𝑆𝑈𝑆))
3029biantrurd 532 . 2 (𝜑 → ((𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)) ↔ ((𝑇𝑆𝑈𝑆) ∧ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))))
3123, 28, 303bitr4d 311 1 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wrex 3067  wpss 3948   class class class wbr 5148  {copab 5210  cfv 6548  LSubSpclss 20814  L clcv 38490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-lcv 38491
This theorem is referenced by:  lcvbr2  38494  lcvbr3  38495  lcvpss  38496  lcvnbtwn  38497  lsatcv0  38503  mapdcv  41133
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