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Theorem lshpset 36271
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 3459 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6645 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
4 lshpset.s . . . . . 6 𝑆 = (LSubSp‘𝑊)
53, 4eqtr4di 2851 . . . . 5 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
6 fveq2 6645 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7 lshpset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
86, 7eqtr4di 2851 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
98neeq2d 3047 . . . . . 6 (𝑤 = 𝑊 → (𝑠 ≠ (Base‘𝑤) ↔ 𝑠𝑉))
10 fveq2 6645 . . . . . . . . . 10 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
11 lshpset.n . . . . . . . . . 10 𝑁 = (LSpan‘𝑊)
1210, 11eqtr4di 2851 . . . . . . . . 9 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
1312fveq1d 6647 . . . . . . . 8 (𝑤 = 𝑊 → ((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (𝑁‘(𝑠 ∪ {𝑣})))
1413, 8eqeq12d 2814 . . . . . . 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 633 . . . . 5 (𝑤 = 𝑊 → ((𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤)) ↔ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)))
175, 16rabeqbidv 3433 . . . 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 36270 . . . 4 LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
194fvexi 6659 . . . . 5 𝑆 ∈ V
2019rabex 5199 . . . 4 {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ∈ V
2117, 18, 20fvmpt 6745 . . 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, 22syl5eq 2845 1 (𝑊𝑋𝐻 = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  wne 2987  wrex 3107  {crab 3110  Vcvv 3441  cun 3879  {csn 4525  cfv 6324  Basecbs 16475  LSubSpclss 19696  LSpanclspn 19736  LSHypclsh 36268
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-lshyp 36270
This theorem is referenced by:  islshp  36272
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