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Theorem lshpset 33283
 Description: The set of all hyperplanes of a left module or left vector space. The vector 𝑣 is called a generating vector for the hyperplane. (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
lshpset (𝑊𝑋𝐻 = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
Distinct variable groups:   𝑆,𝑠   𝑣,𝑉   𝑣,𝑠,𝑊
Allowed substitution hints:   𝑆(𝑣)   𝐻(𝑣,𝑠)   𝑁(𝑣,𝑠)   𝑉(𝑠)   𝑋(𝑣,𝑠)

Proof of Theorem lshpset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 lshpset.h . 2 𝐻 = (LSHyp‘𝑊)
2 elex 3185 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
4 lshpset.s . . . . . 6 𝑆 = (LSubSp‘𝑊)
53, 4syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
6 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7 lshpset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
86, 7syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
98neeq2d 2842 . . . . . 6 (𝑤 = 𝑊 → (𝑠 ≠ (Base‘𝑤) ↔ 𝑠𝑉))
10 fveq2 6103 . . . . . . . . . 10 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
11 lshpset.n . . . . . . . . . 10 𝑁 = (LSpan‘𝑊)
1210, 11syl6eqr 2662 . . . . . . . . 9 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
1312fveq1d 6105 . . . . . . . 8 (𝑤 = 𝑊 → ((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (𝑁‘(𝑠 ∪ {𝑣})))
1413, 8eqeq12d 2625 . . . . . . 7 (𝑤 = 𝑊 → (((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤) ↔ (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉))
158, 14rexeqbidv 3130 . . . . . 6 (𝑤 = 𝑊 → (∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤) ↔ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉))
169, 15anbi12d 743 . . . . 5 (𝑤 = 𝑊 → ((𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤)) ↔ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)))
175, 16rabeqbidv 3168 . . . 4 (𝑤 = 𝑊 → {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))} = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
18 df-lshyp 33282 . . . 4 LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
19 fvex 6113 . . . . . 6 (LSubSp‘𝑊) ∈ V
204, 19eqeltri 2684 . . . . 5 𝑆 ∈ V
2120rabex 4740 . . . 4 {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ∈ V
2217, 18, 21fvmpt 6191 . . 3 (𝑊 ∈ V → (LSHyp‘𝑊) = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
232, 22syl 17 . 2 (𝑊𝑋 → (LSHyp‘𝑊) = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
241, 23syl5eq 2656 1 (𝑊𝑋𝐻 = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ∪ cun 3538  {csn 4125  ‘cfv 5804  Basecbs 15695  LSubSpclss 18753  LSpanclspn 18792  LSHypclsh 33280 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-lshyp 33282 This theorem is referenced by:  islshp  33284
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