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Theorem lcvbr 33827
Description: The covers relation for a left vector space (or a left module). (cvbr 29029 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 2686 . . . . . 6 (𝑡 = 𝑇 → (𝑡𝑆𝑇𝑆))
43anbi1d 740 . . . . 5 (𝑡 = 𝑇 → ((𝑡𝑆𝑢𝑆) ↔ (𝑇𝑆𝑢𝑆)))
5 psseq1 3678 . . . . . 6 (𝑡 = 𝑇 → (𝑡𝑢𝑇𝑢))
6 psseq1 3678 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡𝑠𝑇𝑠))
76anbi1d 740 . . . . . . . 8 (𝑡 = 𝑇 → ((𝑡𝑠𝑠𝑢) ↔ (𝑇𝑠𝑠𝑢)))
87rexbidv 3047 . . . . . . 7 (𝑡 = 𝑇 → (∃𝑠𝑆 (𝑡𝑠𝑠𝑢) ↔ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)))
98notbid 308 . . . . . 6 (𝑡 = 𝑇 → (¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)))
105, 9anbi12d 746 . . . . 5 (𝑡 = 𝑇 → ((𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)) ↔ (𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢))))
114, 10anbi12d 746 . . . 4 (𝑡 = 𝑇 → (((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢))) ↔ ((𝑇𝑆𝑢𝑆) ∧ (𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)))))
12 eleq1 2686 . . . . . 6 (𝑢 = 𝑈 → (𝑢𝑆𝑈𝑆))
1312anbi2d 739 . . . . 5 (𝑢 = 𝑈 → ((𝑇𝑆𝑢𝑆) ↔ (𝑇𝑆𝑈𝑆)))
14 psseq2 3679 . . . . . 6 (𝑢 = 𝑈 → (𝑇𝑢𝑇𝑈))
15 psseq2 3679 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑠𝑢𝑠𝑈))
1615anbi2d 739 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑇𝑠𝑠𝑢) ↔ (𝑇𝑠𝑠𝑈)))
1716rexbidv 3047 . . . . . . 7 (𝑢 = 𝑈 → (∃𝑠𝑆 (𝑇𝑠𝑠𝑢) ↔ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
1817notbid 308 . . . . . 6 (𝑢 = 𝑈 → (¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
1914, 18anbi12d 746 . . . . 5 (𝑢 = 𝑈 → ((𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)) ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
2013, 19anbi12d 746 . . . 4 (𝑢 = 𝑈 → (((𝑇𝑆𝑢𝑆) ∧ (𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢))) ↔ ((𝑇𝑆𝑈𝑆) ∧ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))))
21 eqid 2621 . . . 4 {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))} = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))}
2211, 20, 21brabg 4964 . . 3 ((𝑇𝑆𝑈𝑆) → (𝑇{⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))}𝑈 ↔ ((𝑇𝑆𝑈𝑆) ∧ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))))
231, 2, 22syl2anc 692 . 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 33826 . . 3 (𝜑𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
2827breqd 4634 . 2 (𝜑 → (𝑇𝐶𝑈𝑇{⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))}𝑈))
291, 2jca 554 . . 3 (𝜑 → (𝑇𝑆𝑈𝑆))
3029biantrurd 529 . 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 384   = wceq 1480  wcel 1987  wrex 2909  wpss 3561   class class class wbr 4623  {copab 4682  cfv 5857  LSubSpclss 18872  L clcv 33824
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 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
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-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-lcv 33825
This theorem is referenced by:  lcvbr2  33828  lcvbr3  33829  lcvpss  33830  lcvnbtwn  33831  lsatcv0  33837  mapdcv  36468
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