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Theorem lshpset 38482
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 = (Baseβ€˜π‘Š)
lshpset.n 𝑁 = (LSpanβ€˜π‘Š)
lshpset.s 𝑆 = (LSubSpβ€˜π‘Š)
lshpset.h 𝐻 = (LSHypβ€˜π‘Š)
Assertion
Ref Expression
lshpset (π‘Š ∈ 𝑋 β†’ 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)})
Distinct variable groups:   𝑆,𝑠   𝑣,𝑉   𝑣,𝑠,π‘Š
Allowed substitution hints:   𝑆(𝑣)   𝐻(𝑣,𝑠)   𝑁(𝑣,𝑠)   𝑉(𝑠)   𝑋(𝑣,𝑠)

Proof of Theorem lshpset
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 lshpset.h . 2 𝐻 = (LSHypβ€˜π‘Š)
2 elex 3492 . . 3 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
3 fveq2 6902 . . . . . 6 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = (LSubSpβ€˜π‘Š))
4 lshpset.s . . . . . 6 𝑆 = (LSubSpβ€˜π‘Š)
53, 4eqtr4di 2786 . . . . 5 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = 𝑆)
6 fveq2 6902 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
7 lshpset.v . . . . . . . 8 𝑉 = (Baseβ€˜π‘Š)
86, 7eqtr4di 2786 . . . . . . 7 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝑉)
98neeq2d 2998 . . . . . 6 (𝑀 = π‘Š β†’ (𝑠 β‰  (Baseβ€˜π‘€) ↔ 𝑠 β‰  𝑉))
10 fveq2 6902 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (LSpanβ€˜π‘€) = (LSpanβ€˜π‘Š))
11 lshpset.n . . . . . . . . . 10 𝑁 = (LSpanβ€˜π‘Š)
1210, 11eqtr4di 2786 . . . . . . . . 9 (𝑀 = π‘Š β†’ (LSpanβ€˜π‘€) = 𝑁)
1312fveq1d 6904 . . . . . . . 8 (𝑀 = π‘Š β†’ ((LSpanβ€˜π‘€)β€˜(𝑠 βˆͺ {𝑣})) = (π‘β€˜(𝑠 βˆͺ {𝑣})))
1413, 8eqeq12d 2744 . . . . . . 7 (𝑀 = π‘Š β†’ (((LSpanβ€˜π‘€)β€˜(𝑠 βˆͺ {𝑣})) = (Baseβ€˜π‘€) ↔ (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉))
158, 14rexeqbidv 3341 . . . . . 6 (𝑀 = π‘Š β†’ (βˆƒπ‘£ ∈ (Baseβ€˜π‘€)((LSpanβ€˜π‘€)β€˜(𝑠 βˆͺ {𝑣})) = (Baseβ€˜π‘€) ↔ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉))
169, 15anbi12d 630 . . . . 5 (𝑀 = π‘Š β†’ ((𝑠 β‰  (Baseβ€˜π‘€) ∧ βˆƒπ‘£ ∈ (Baseβ€˜π‘€)((LSpanβ€˜π‘€)β€˜(𝑠 βˆͺ {𝑣})) = (Baseβ€˜π‘€)) ↔ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)))
175, 16rabeqbidv 3448 . . . 4 (𝑀 = π‘Š β†’ {𝑠 ∈ (LSubSpβ€˜π‘€) ∣ (𝑠 β‰  (Baseβ€˜π‘€) ∧ βˆƒπ‘£ ∈ (Baseβ€˜π‘€)((LSpanβ€˜π‘€)β€˜(𝑠 βˆͺ {𝑣})) = (Baseβ€˜π‘€))} = {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)})
18 df-lshyp 38481 . . . 4 LSHyp = (𝑀 ∈ V ↦ {𝑠 ∈ (LSubSpβ€˜π‘€) ∣ (𝑠 β‰  (Baseβ€˜π‘€) ∧ βˆƒπ‘£ ∈ (Baseβ€˜π‘€)((LSpanβ€˜π‘€)β€˜(𝑠 βˆͺ {𝑣})) = (Baseβ€˜π‘€))})
194fvexi 6916 . . . . 5 𝑆 ∈ V
2019rabex 5338 . . . 4 {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)} ∈ V
2117, 18, 20fvmpt 7010 . . 3 (π‘Š ∈ V β†’ (LSHypβ€˜π‘Š) = {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)})
222, 21syl 17 . 2 (π‘Š ∈ 𝑋 β†’ (LSHypβ€˜π‘Š) = {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)})
231, 22eqtrid 2780 1 (π‘Š ∈ 𝑋 β†’ 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  βˆƒwrex 3067  {crab 3430  Vcvv 3473   βˆͺ cun 3947  {csn 4632  β€˜cfv 6553  Basecbs 17187  LSubSpclss 20822  LSpanclspn 20862  LSHypclsh 38479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-lshyp 38481
This theorem is referenced by:  islshp  38483
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