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Theorem lshpset 37296
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 3460 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6829 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
4 lshpset.s . . . . . 6 𝑆 = (LSubSp‘𝑊)
53, 4eqtr4di 2795 . . . . 5 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
6 fveq2 6829 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7 lshpset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
86, 7eqtr4di 2795 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
98neeq2d 3002 . . . . . 6 (𝑤 = 𝑊 → (𝑠 ≠ (Base‘𝑤) ↔ 𝑠𝑉))
10 fveq2 6829 . . . . . . . . . 10 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
11 lshpset.n . . . . . . . . . 10 𝑁 = (LSpan‘𝑊)
1210, 11eqtr4di 2795 . . . . . . . . 9 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
1312fveq1d 6831 . . . . . . . 8 (𝑤 = 𝑊 → ((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (𝑁‘(𝑠 ∪ {𝑣})))
1413, 8eqeq12d 2753 . . . . . . 7 (𝑤 = 𝑊 → (((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤) ↔ (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉))
158, 14rexeqbidv 3317 . . . . . 6 (𝑤 = 𝑊 → (∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤) ↔ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉))
169, 15anbi12d 632 . . . . 5 (𝑤 = 𝑊 → ((𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤)) ↔ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)))
175, 16rabeqbidv 3421 . . . 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 37295 . . . 4 LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
194fvexi 6843 . . . . 5 𝑆 ∈ V
2019rabex 5280 . . . 4 {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ∈ V
2117, 18, 20fvmpt 6935 . . 3 (𝑊 ∈ V → (LSHyp‘𝑊) = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
222, 21syl 17 . 2 (𝑊𝑋 → (LSHyp‘𝑊) = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
231, 22eqtrid 2789 1 (𝑊𝑋𝐻 = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1541  wcel 2106  wne 2941  wrex 3071  {crab 3404  Vcvv 3442  cun 3899  {csn 4577  cfv 6483  Basecbs 17009  LSubSpclss 20298  LSpanclspn 20338  LSHypclsh 37293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5247  ax-nul 5254  ax-pr 5376
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-br 5097  df-opab 5159  df-mpt 5180  df-id 5522  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6435  df-fun 6485  df-fv 6491  df-lshyp 37295
This theorem is referenced by:  islshp  37297
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