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Theorem lcvfbr 37885
Description: The covers relation for a left vector space (or a left module). (Contributed by NM, 7-Jan-2015.)
Hypotheses
Ref Expression
lcvfbr.s 𝑆 = (LSubSp‘𝑊)
lcvfbr.c 𝐶 = ( ⋖L𝑊)
lcvfbr.w (𝜑𝑊𝑋)
Assertion
Ref Expression
lcvfbr (𝜑𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
Distinct variable groups:   𝑡,𝑠,𝑢,𝑆   𝑊,𝑠,𝑡,𝑢
Allowed substitution hints:   𝜑(𝑢,𝑡,𝑠)   𝐶(𝑢,𝑡,𝑠)   𝑋(𝑢,𝑡,𝑠)

Proof of Theorem lcvfbr
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 lcvfbr.c . 2 𝐶 = ( ⋖L𝑊)
2 lcvfbr.w . . 3 (𝜑𝑊𝑋)
3 elex 3492 . . 3 (𝑊𝑋𝑊 ∈ V)
4 fveq2 6891 . . . . . . . . 9 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
5 lcvfbr.s . . . . . . . . 9 𝑆 = (LSubSp‘𝑊)
64, 5eqtr4di 2790 . . . . . . . 8 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
76eleq2d 2819 . . . . . . 7 (𝑤 = 𝑊 → (𝑡 ∈ (LSubSp‘𝑤) ↔ 𝑡𝑆))
86eleq2d 2819 . . . . . . 7 (𝑤 = 𝑊 → (𝑢 ∈ (LSubSp‘𝑤) ↔ 𝑢𝑆))
97, 8anbi12d 631 . . . . . 6 (𝑤 = 𝑊 → ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ↔ (𝑡𝑆𝑢𝑆)))
106rexeqdv 3326 . . . . . . . 8 (𝑤 = 𝑊 → (∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢) ↔ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))
1110notbid 317 . . . . . . 7 (𝑤 = 𝑊 → (¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢) ↔ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))
1211anbi2d 629 . . . . . 6 (𝑤 = 𝑊 → ((𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢)) ↔ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢))))
139, 12anbi12d 631 . . . . 5 (𝑤 = 𝑊 → (((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢))) ↔ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))))
1413opabbidv 5214 . . . 4 (𝑤 = 𝑊 → {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢)))} = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
15 df-lcv 37884 . . . 4 L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢)))})
165fvexi 6905 . . . . . 6 𝑆 ∈ V
1716, 16xpex 7739 . . . . 5 (𝑆 × 𝑆) ∈ V
18 opabssxp 5768 . . . . 5 {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))} ⊆ (𝑆 × 𝑆)
1917, 18ssexi 5322 . . . 4 {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))} ∈ V
2014, 15, 19fvmpt 6998 . . 3 (𝑊 ∈ V → ( ⋖L𝑊) = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
212, 3, 203syl 18 . 2 (𝜑 → ( ⋖L𝑊) = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
221, 21eqtrid 2784 1 (𝜑𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wrex 3070  Vcvv 3474  wpss 3949  {copab 5210   × cxp 5674  cfv 6543  LSubSpclss 20541  L clcv 37883
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-lcv 37884
This theorem is referenced by:  lcvbr  37886
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