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Theorem islshp 38980
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 38979 . . 3 (𝑊𝑋𝐻 = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
65eleq2d 2827 . 2 (𝑊𝑋 → (𝑈𝐻𝑈 ∈ {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)}))
7 neeq1 3003 . . . . 5 (𝑠 = 𝑈 → (𝑠𝑉𝑈𝑉))
8 uneq1 4161 . . . . . . 7 (𝑠 = 𝑈 → (𝑠 ∪ {𝑣}) = (𝑈 ∪ {𝑣}))
98fveqeq2d 6914 . . . . . 6 (𝑠 = 𝑈 → ((𝑁‘(𝑠 ∪ {𝑣})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))
109rexbidv 3179 . . . . 5 (𝑠 = 𝑈 → (∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉 ↔ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))
117, 10anbi12d 632 . . . 4 (𝑠 = 𝑈 → ((𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉) ↔ (𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
1211elrab 3692 . . 3 (𝑈 ∈ {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ↔ (𝑈𝑆 ∧ (𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
13 3anass 1095 . . 3 ((𝑈𝑆𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉) ↔ (𝑈𝑆 ∧ (𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
1412, 13bitr4i 278 . 2 (𝑈 ∈ {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ↔ (𝑈𝑆𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))
156, 14bitrdi 287 1 (𝑊𝑋 → (𝑈𝐻 ↔ (𝑈𝑆𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070  {crab 3436  cun 3949  {csn 4626  cfv 6561  Basecbs 17247  LSubSpclss 20929  LSpanclspn 20969  LSHypclsh 38976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-lshyp 38978
This theorem is referenced by:  islshpsm  38981  lshplss  38982  lshpne  38983  lshpnel2N  38986  lkrshp  39106  lshpset2N  39120  dochsatshp  41453
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