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Theorem lcvfbr 33787
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 3198 . . 3 (𝑊𝑋𝑊 ∈ V)
4 fveq2 6148 . . . . . . . . 9 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
5 lcvfbr.s . . . . . . . . 9 𝑆 = (LSubSp‘𝑊)
64, 5syl6eqr 2673 . . . . . . . 8 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
76eleq2d 2684 . . . . . . 7 (𝑤 = 𝑊 → (𝑡 ∈ (LSubSp‘𝑤) ↔ 𝑡𝑆))
86eleq2d 2684 . . . . . . 7 (𝑤 = 𝑊 → (𝑢 ∈ (LSubSp‘𝑤) ↔ 𝑢𝑆))
97, 8anbi12d 746 . . . . . 6 (𝑤 = 𝑊 → ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ↔ (𝑡𝑆𝑢𝑆)))
106rexeqdv 3134 . . . . . . . 8 (𝑤 = 𝑊 → (∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢) ↔ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))
1110notbid 308 . . . . . . 7 (𝑤 = 𝑊 → (¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢) ↔ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))
1211anbi2d 739 . . . . . 6 (𝑤 = 𝑊 → ((𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢)) ↔ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢))))
139, 12anbi12d 746 . . . . 5 (𝑤 = 𝑊 → (((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢))) ↔ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))))
1413opabbidv 4678 . . . 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 33786 . . . 4 L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢)))})
16 fvex 6158 . . . . . . 7 (LSubSp‘𝑊) ∈ V
175, 16eqeltri 2694 . . . . . 6 𝑆 ∈ V
1817, 17xpex 6915 . . . . 5 (𝑆 × 𝑆) ∈ V
19 opabssxp 5154 . . . . 5 {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))} ⊆ (𝑆 × 𝑆)
2018, 19ssexi 4763 . . . 4 {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))} ∈ V
2114, 15, 20fvmpt 6239 . . 3 (𝑊 ∈ V → ( ⋖L𝑊) = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
222, 3, 213syl 18 . 2 (𝜑 → ( ⋖L𝑊) = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
231, 22syl5eq 2667 1 (𝜑𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  wpss 3556  {copab 4672   × cxp 5072  cfv 5847  LSubSpclss 18851  L clcv 33785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-lcv 33786
This theorem is referenced by:  lcvbr  33788
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