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Theorem lcvfbr 39021
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 3501 . . 3 (𝑊𝑋𝑊 ∈ V)
4 fveq2 6906 . . . . . . . . 9 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
5 lcvfbr.s . . . . . . . . 9 𝑆 = (LSubSp‘𝑊)
64, 5eqtr4di 2795 . . . . . . . 8 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
76eleq2d 2827 . . . . . . 7 (𝑤 = 𝑊 → (𝑡 ∈ (LSubSp‘𝑤) ↔ 𝑡𝑆))
86eleq2d 2827 . . . . . . 7 (𝑤 = 𝑊 → (𝑢 ∈ (LSubSp‘𝑤) ↔ 𝑢𝑆))
97, 8anbi12d 632 . . . . . 6 (𝑤 = 𝑊 → ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ↔ (𝑡𝑆𝑢𝑆)))
106rexeqdv 3327 . . . . . . . 8 (𝑤 = 𝑊 → (∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢) ↔ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))
1110notbid 318 . . . . . . 7 (𝑤 = 𝑊 → (¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢) ↔ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))
1211anbi2d 630 . . . . . 6 (𝑤 = 𝑊 → ((𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢)) ↔ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢))))
139, 12anbi12d 632 . . . . 5 (𝑤 = 𝑊 → (((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢))) ↔ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))))
1413opabbidv 5209 . . . 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 39020 . . . 4 L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢)))})
165fvexi 6920 . . . . . 6 𝑆 ∈ V
1716, 16xpex 7773 . . . . 5 (𝑆 × 𝑆) ∈ V
18 opabssxp 5778 . . . . 5 {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))} ⊆ (𝑆 × 𝑆)
1917, 18ssexi 5322 . . . 4 {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))} ∈ V
2014, 15, 19fvmpt 7016 . . 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 2789 1 (𝜑𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  wpss 3952  {copab 5205   × cxp 5683  cfv 6561  LSubSpclss 20929  L clcv 39019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-lcv 39020
This theorem is referenced by:  lcvbr  39022
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