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Theorem lcvbr 36597
 Description: The covers relation for a left vector space (or a left module). (cvbr 30164 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 2839 . . . . . 6 (𝑡 = 𝑇 → (𝑡𝑆𝑇𝑆))
43anbi1d 632 . . . . 5 (𝑡 = 𝑇 → ((𝑡𝑆𝑢𝑆) ↔ (𝑇𝑆𝑢𝑆)))
5 psseq1 3993 . . . . . 6 (𝑡 = 𝑇 → (𝑡𝑢𝑇𝑢))
6 psseq1 3993 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡𝑠𝑇𝑠))
76anbi1d 632 . . . . . . . 8 (𝑡 = 𝑇 → ((𝑡𝑠𝑠𝑢) ↔ (𝑇𝑠𝑠𝑢)))
87rexbidv 3221 . . . . . . 7 (𝑡 = 𝑇 → (∃𝑠𝑆 (𝑡𝑠𝑠𝑢) ↔ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)))
98notbid 321 . . . . . 6 (𝑡 = 𝑇 → (¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)))
105, 9anbi12d 633 . . . . 5 (𝑡 = 𝑇 → ((𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)) ↔ (𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢))))
114, 10anbi12d 633 . . . 4 (𝑡 = 𝑇 → (((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢))) ↔ ((𝑇𝑆𝑢𝑆) ∧ (𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)))))
12 eleq1 2839 . . . . . 6 (𝑢 = 𝑈 → (𝑢𝑆𝑈𝑆))
1312anbi2d 631 . . . . 5 (𝑢 = 𝑈 → ((𝑇𝑆𝑢𝑆) ↔ (𝑇𝑆𝑈𝑆)))
14 psseq2 3994 . . . . . 6 (𝑢 = 𝑈 → (𝑇𝑢𝑇𝑈))
15 psseq2 3994 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑠𝑢𝑠𝑈))
1615anbi2d 631 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑇𝑠𝑠𝑢) ↔ (𝑇𝑠𝑠𝑈)))
1716rexbidv 3221 . . . . . . 7 (𝑢 = 𝑈 → (∃𝑠𝑆 (𝑇𝑠𝑠𝑢) ↔ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
1817notbid 321 . . . . . 6 (𝑢 = 𝑈 → (¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢) ↔ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
1914, 18anbi12d 633 . . . . 5 (𝑢 = 𝑈 → ((𝑇𝑢 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑢)) ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
2013, 19anbi12d 633 . . . 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 5396 . . 3 ((𝑇𝑆𝑈𝑆) → (𝑇{⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))}𝑈 ↔ ((𝑇𝑆𝑈𝑆) ∧ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))))
231, 2, 22syl2anc 587 . 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 36596 . . 3 (𝜑𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})
2827breqd 5043 . 2 (𝜑 → (𝑇𝐶𝑈𝑇{⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))}𝑈))
291, 2jca 515 . . 3 (𝜑 → (𝑇𝑆𝑈𝑆))
3029biantrurd 536 . 2 (𝜑 → ((𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)) ↔ ((𝑇𝑆𝑈𝑆) ∧ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))))
3123, 28, 303bitr4d 314 1 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3071   ⊊ wpss 3859   class class class wbr 5032  {copab 5094  ‘cfv 6335  LSubSpclss 19771   ⋖L clcv 36594 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-lcv 36595 This theorem is referenced by:  lcvbr2  36598  lcvbr3  36599  lcvpss  36600  lcvnbtwn  36601  lsatcv0  36607  mapdcv  39236
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