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Theorem islshp 38360
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 = (Baseβ€˜π‘Š)
lshpset.n 𝑁 = (LSpanβ€˜π‘Š)
lshpset.s 𝑆 = (LSubSpβ€˜π‘Š)
lshpset.h 𝐻 = (LSHypβ€˜π‘Š)
Assertion
Ref Expression
islshp (π‘Š ∈ 𝑋 β†’ (π‘ˆ ∈ 𝐻 ↔ (π‘ˆ ∈ 𝑆 ∧ π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉)))
Distinct variable groups:   𝑣,𝑉   𝑣,π‘Š   𝑣,π‘ˆ
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 = (Baseβ€˜π‘Š)
2 lshpset.n . . . 4 𝑁 = (LSpanβ€˜π‘Š)
3 lshpset.s . . . 4 𝑆 = (LSubSpβ€˜π‘Š)
4 lshpset.h . . . 4 𝐻 = (LSHypβ€˜π‘Š)
51, 2, 3, 4lshpset 38359 . . 3 (π‘Š ∈ 𝑋 β†’ 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)})
65eleq2d 2813 . 2 (π‘Š ∈ 𝑋 β†’ (π‘ˆ ∈ 𝐻 ↔ π‘ˆ ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)}))
7 neeq1 2997 . . . . 5 (𝑠 = π‘ˆ β†’ (𝑠 β‰  𝑉 ↔ π‘ˆ β‰  𝑉))
8 uneq1 4151 . . . . . . 7 (𝑠 = π‘ˆ β†’ (𝑠 βˆͺ {𝑣}) = (π‘ˆ βˆͺ {𝑣}))
98fveqeq2d 6892 . . . . . 6 (𝑠 = π‘ˆ β†’ ((π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉 ↔ (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉))
109rexbidv 3172 . . . . 5 (𝑠 = π‘ˆ β†’ (βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉 ↔ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉))
117, 10anbi12d 630 . . . 4 (𝑠 = π‘ˆ β†’ ((𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉) ↔ (π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉)))
1211elrab 3678 . . 3 (π‘ˆ ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)} ↔ (π‘ˆ ∈ 𝑆 ∧ (π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉)))
13 3anass 1092 . . 3 ((π‘ˆ ∈ 𝑆 ∧ π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉) ↔ (π‘ˆ ∈ 𝑆 ∧ (π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉)))
1412, 13bitr4i 278 . 2 (π‘ˆ ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)} ↔ (π‘ˆ ∈ 𝑆 ∧ π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉))
156, 14bitrdi 287 1 (π‘Š ∈ 𝑋 β†’ (π‘ˆ ∈ 𝐻 ↔ (π‘ˆ ∈ 𝑆 ∧ π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆƒwrex 3064  {crab 3426   βˆͺ cun 3941  {csn 4623  β€˜cfv 6536  Basecbs 17151  LSubSpclss 20776  LSpanclspn 20816  LSHypclsh 38356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-lshyp 38358
This theorem is referenced by:  islshpsm  38361  lshplss  38362  lshpne  38363  lshpnel2N  38366  lkrshp  38486  lshpset2N  38500  dochsatshp  40833
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