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Theorem islshp 34584
Description: The predicate "is a hyperplane" (of a left module or left vector space). (Contributed by NM, 29-Jun-2014.)
Hypotheses
Ref Expression
lshpset.v 𝑉 = (Base‘𝑊)
lshpset.n 𝑁 = (LSpan‘𝑊)
lshpset.s 𝑆 = (LSubSp‘𝑊)
lshpset.h 𝐻 = (LSHyp‘𝑊)
Assertion
Ref Expression
islshp (𝑊𝑋 → (𝑈𝐻 ↔ (𝑈𝑆𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
Distinct variable groups:   𝑣,𝑉   𝑣,𝑊   𝑣,𝑈
Allowed substitution hints:   𝑆(𝑣)   𝐻(𝑣)   𝑁(𝑣)   𝑋(𝑣)

Proof of Theorem islshp
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 lshpset.v . . . 4 𝑉 = (Base‘𝑊)
2 lshpset.n . . . 4 𝑁 = (LSpan‘𝑊)
3 lshpset.s . . . 4 𝑆 = (LSubSp‘𝑊)
4 lshpset.h . . . 4 𝐻 = (LSHyp‘𝑊)
51, 2, 3, 4lshpset 34583 . . 3 (𝑊𝑋𝐻 = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
65eleq2d 2716 . 2 (𝑊𝑋 → (𝑈𝐻𝑈 ∈ {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)}))
7 neeq1 2885 . . . . 5 (𝑠 = 𝑈 → (𝑠𝑉𝑈𝑉))
8 uneq1 3793 . . . . . . . 8 (𝑠 = 𝑈 → (𝑠 ∪ {𝑣}) = (𝑈 ∪ {𝑣}))
98fveq2d 6233 . . . . . . 7 (𝑠 = 𝑈 → (𝑁‘(𝑠 ∪ {𝑣})) = (𝑁‘(𝑈 ∪ {𝑣})))
109eqeq1d 2653 . . . . . 6 (𝑠 = 𝑈 → ((𝑁‘(𝑠 ∪ {𝑣})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))
1110rexbidv 3081 . . . . 5 (𝑠 = 𝑈 → (∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉 ↔ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))
127, 11anbi12d 747 . . . 4 (𝑠 = 𝑈 → ((𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉) ↔ (𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
1312elrab 3396 . . 3 (𝑈 ∈ {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ↔ (𝑈𝑆 ∧ (𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
14 3anass 1059 . . 3 ((𝑈𝑆𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉) ↔ (𝑈𝑆 ∧ (𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
1513, 14bitr4i 267 . 2 (𝑈 ∈ {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ↔ (𝑈𝑆𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))
166, 15syl6bb 276 1 (𝑊𝑋 → (𝑈𝐻 ↔ (𝑈𝑆𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wrex 2942  {crab 2945  cun 3605  {csn 4210  cfv 5926  Basecbs 15904  LSubSpclss 18980  LSpanclspn 19019  LSHypclsh 34580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-lshyp 34582
This theorem is referenced by:  islshpsm  34585  lshplss  34586  lshpne  34587  lshpnel2N  34590  lkrshp  34710  lshpset2N  34724  dochsatshp  37057
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