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Theorem lshpset 34787
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 3364 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6333 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
4 lshpset.s . . . . . 6 𝑆 = (LSubSp‘𝑊)
53, 4syl6eqr 2823 . . . . 5 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
6 fveq2 6333 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7 lshpset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
86, 7syl6eqr 2823 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
98neeq2d 3003 . . . . . 6 (𝑤 = 𝑊 → (𝑠 ≠ (Base‘𝑤) ↔ 𝑠𝑉))
10 fveq2 6333 . . . . . . . . . 10 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
11 lshpset.n . . . . . . . . . 10 𝑁 = (LSpan‘𝑊)
1210, 11syl6eqr 2823 . . . . . . . . 9 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
1312fveq1d 6335 . . . . . . . 8 (𝑤 = 𝑊 → ((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (𝑁‘(𝑠 ∪ {𝑣})))
1413, 8eqeq12d 2786 . . . . . . 7 (𝑤 = 𝑊 → (((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤) ↔ (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉))
158, 14rexeqbidv 3302 . . . . . 6 (𝑤 = 𝑊 → (∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤) ↔ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉))
169, 15anbi12d 616 . . . . 5 (𝑤 = 𝑊 → ((𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤)) ↔ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)))
175, 16rabeqbidv 3345 . . . 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 34786 . . . 4 LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
194fvexi 6345 . . . . 5 𝑆 ∈ V
2019rabex 4947 . . . 4 {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ∈ V
2117, 18, 20fvmpt 6426 . . 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 2817 1 (𝑊𝑋𝐻 = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  wrex 3062  {crab 3065  Vcvv 3351  cun 3721  {csn 4317  cfv 6030  Basecbs 16064  LSubSpclss 19142  LSpanclspn 19184  LSHypclsh 34784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-lshyp 34786
This theorem is referenced by:  islshp  34788
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