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Theorem islshp 37849
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 37848 . . 3 (π‘Š ∈ 𝑋 β†’ 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)})
65eleq2d 2820 . 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 3004 . . . . 5 (𝑠 = π‘ˆ β†’ (𝑠 β‰  𝑉 ↔ π‘ˆ β‰  𝑉))
8 uneq1 4157 . . . . . . 7 (𝑠 = π‘ˆ β†’ (𝑠 βˆͺ {𝑣}) = (π‘ˆ βˆͺ {𝑣}))
98fveqeq2d 6900 . . . . . 6 (𝑠 = π‘ˆ β†’ ((π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉 ↔ (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉))
109rexbidv 3179 . . . . 5 (𝑠 = π‘ˆ β†’ (βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉 ↔ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉))
117, 10anbi12d 632 . . . 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 3684 . . 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 1096 . . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆƒwrex 3071  {crab 3433   βˆͺ cun 3947  {csn 4629  β€˜cfv 6544  Basecbs 17144  LSubSpclss 20542  LSpanclspn 20582  LSHypclsh 37845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-lshyp 37847
This theorem is referenced by:  islshpsm  37850  lshplss  37851  lshpne  37852  lshpnel2N  37855  lkrshp  37975  lshpset2N  37989  dochsatshp  40322
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