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Theorem lcvbr 34809
Description: The covers relation for a left vector space (or a left module). (cvbr 29448 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 2825 . . . . . 6 (𝑡 = 𝑇 → (𝑡𝑆𝑇𝑆))
43anbi1d 743 . . . . 5 (𝑡 = 𝑇 → ((𝑡𝑆𝑢𝑆) ↔ (𝑇𝑆𝑢𝑆)))
5 psseq1 3834 . . . . . 6 (𝑡 = 𝑇 → (𝑡𝑢𝑇𝑢))
6 psseq1 3834 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡𝑠𝑇𝑠))
76anbi1d 743 . . . . . . . 8 (𝑡 = 𝑇 → ((𝑡𝑠𝑠𝑢) ↔ (𝑇𝑠𝑠𝑢)))
87rexbidv 3188 . . . . . . 7 (𝑡 = 𝑇 → (∃𝑠𝑆 (𝑡𝑠𝑠𝑢) ↔ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)))
98notbid 307 . . . . . 6 (𝑡 = 𝑇 → (¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)))
105, 9anbi12d 749 . . . . 5 (𝑡 = 𝑇 → ((𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)) ↔ (𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢))))
114, 10anbi12d 749 . . . 4 (𝑡 = 𝑇 → (((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢))) ↔ ((𝑇𝑆𝑢𝑆) ∧ (𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)))))
12 eleq1 2825 . . . . . 6 (𝑢 = 𝑈 → (𝑢𝑆𝑈𝑆))
1312anbi2d 742 . . . . 5 (𝑢 = 𝑈 → ((𝑇𝑆𝑢𝑆) ↔ (𝑇𝑆𝑈𝑆)))
14 psseq2 3835 . . . . . 6 (𝑢 = 𝑈 → (𝑇𝑢𝑇𝑈))
15 psseq2 3835 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑠𝑢𝑠𝑈))
1615anbi2d 742 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑇𝑠𝑠𝑢) ↔ (𝑇𝑠𝑠𝑈)))
1716rexbidv 3188 . . . . . . 7 (𝑢 = 𝑈 → (∃𝑠𝑆 (𝑇𝑠𝑠𝑢) ↔ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
1817notbid 307 . . . . . 6 (𝑢 = 𝑈 → (¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
1914, 18anbi12d 749 . . . . 5 (𝑢 = 𝑈 → ((𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)) ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
2013, 19anbi12d 749 . . . 4 (𝑢 = 𝑈 → (((𝑇𝑆𝑢𝑆) ∧ (𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢))) ↔ ((𝑇𝑆𝑈𝑆) ∧ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))))
21 eqid 2758 . . . 4 {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))} = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))}
2211, 20, 21brabg 5142 . . 3 ((𝑇𝑆𝑈𝑆) → (𝑇{⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))}𝑈 ↔ ((𝑇𝑆𝑈𝑆) ∧ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))))
231, 2, 22syl2anc 696 . 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 34808 . . 3 (𝜑𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
2827breqd 4813 . 2 (𝜑 → (𝑇𝐶𝑈𝑇{⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))}𝑈))
291, 2jca 555 . . 3 (𝜑 → (𝑇𝑆𝑈𝑆))
3029biantrurd 530 . 2 (𝜑 → ((𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)) ↔ ((𝑇𝑆𝑈𝑆) ∧ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))))
3123, 28, 303bitr4d 300 1 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1630  wcel 2137  wrex 3049  wpss 3714   class class class wbr 4802  {copab 4862  cfv 6047  LSubSpclss 19132  L clcv 34806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-iota 6010  df-fun 6049  df-fv 6055  df-lcv 34807
This theorem is referenced by:  lcvbr2  34810  lcvbr3  34811  lcvpss  34812  lcvnbtwn  34813  lsatcv0  34819  mapdcv  37449
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