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Theorem lcvfbr 36192
 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 3491 . . 3 (𝑊𝑋𝑊 ∈ V)
4 fveq2 6646 . . . . . . . . 9 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
5 lcvfbr.s . . . . . . . . 9 𝑆 = (LSubSp‘𝑊)
64, 5syl6eqr 2873 . . . . . . . 8 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
76eleq2d 2896 . . . . . . 7 (𝑤 = 𝑊 → (𝑡 ∈ (LSubSp‘𝑤) ↔ 𝑡𝑆))
86eleq2d 2896 . . . . . . 7 (𝑤 = 𝑊 → (𝑢 ∈ (LSubSp‘𝑤) ↔ 𝑢𝑆))
97, 8anbi12d 632 . . . . . 6 (𝑤 = 𝑊 → ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ↔ (𝑡𝑆𝑢𝑆)))
106rexeqdv 3399 . . . . . . . 8 (𝑤 = 𝑊 → (∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢) ↔ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))
1110notbid 320 . . . . . . 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 5108 . . . 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 36191 . . . 4 L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢)))})
165fvexi 6660 . . . . . 6 𝑆 ∈ V
1716, 16xpex 7454 . . . . 5 (𝑆 × 𝑆) ∈ V
18 opabssxp 5619 . . . . 5 {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))} ⊆ (𝑆 × 𝑆)
1917, 18ssexi 5202 . . . 4 {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))} ∈ V
2014, 15, 19fvmpt 6744 . . 3 (𝑊 ∈ V → ( ⋖L𝑊) = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
212, 3, 203syl 18 . 2 (𝜑 → ( ⋖L𝑊) = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
221, 21syl5eq 2867 1 (𝜑𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3126  Vcvv 3473   ⊊ wpss 3914  {copab 5104   × cxp 5529  ‘cfv 6331  LSubSpclss 19679   ⋖L clcv 36190 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-iota 6290  df-fun 6333  df-fv 6339  df-lcv 36191 This theorem is referenced by:  lcvbr  36193
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