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Theorem lshpset 38936
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 3509 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6922 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
4 lshpset.s . . . . . 6 𝑆 = (LSubSp‘𝑊)
53, 4eqtr4di 2798 . . . . 5 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
6 fveq2 6922 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7 lshpset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
86, 7eqtr4di 2798 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
98neeq2d 3007 . . . . . 6 (𝑤 = 𝑊 → (𝑠 ≠ (Base‘𝑤) ↔ 𝑠𝑉))
10 fveq2 6922 . . . . . . . . . 10 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
11 lshpset.n . . . . . . . . . 10 𝑁 = (LSpan‘𝑊)
1210, 11eqtr4di 2798 . . . . . . . . 9 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
1312fveq1d 6924 . . . . . . . 8 (𝑤 = 𝑊 → ((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (𝑁‘(𝑠 ∪ {𝑣})))
1413, 8eqeq12d 2756 . . . . . . 7 (𝑤 = 𝑊 → (((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤) ↔ (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉))
158, 14rexeqbidv 3355 . . . . . 6 (𝑤 = 𝑊 → (∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤) ↔ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉))
169, 15anbi12d 631 . . . . 5 (𝑤 = 𝑊 → ((𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤)) ↔ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)))
175, 16rabeqbidv 3462 . . . 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 38935 . . . 4 LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
194fvexi 6936 . . . . 5 𝑆 ∈ V
2019rabex 5357 . . . 4 {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ∈ V
2117, 18, 20fvmpt 7031 . . 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 2792 1 (𝑊𝑋𝐻 = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  wne 2946  wrex 3076  {crab 3443  Vcvv 3488  cun 3974  {csn 4648  cfv 6575  Basecbs 17260  LSubSpclss 20954  LSpanclspn 20994  LSHypclsh 38933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6527  df-fun 6577  df-fv 6583  df-lshyp 38935
This theorem is referenced by:  islshp  38937
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