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Theorem lshpset 37469
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 3466 . . 3 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
3 fveq2 6847 . . . . . 6 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = (LSubSpβ€˜π‘Š))
4 lshpset.s . . . . . 6 𝑆 = (LSubSpβ€˜π‘Š)
53, 4eqtr4di 2795 . . . . 5 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = 𝑆)
6 fveq2 6847 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
7 lshpset.v . . . . . . . 8 𝑉 = (Baseβ€˜π‘Š)
86, 7eqtr4di 2795 . . . . . . 7 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝑉)
98neeq2d 3005 . . . . . 6 (𝑀 = π‘Š β†’ (𝑠 β‰  (Baseβ€˜π‘€) ↔ 𝑠 β‰  𝑉))
10 fveq2 6847 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (LSpanβ€˜π‘€) = (LSpanβ€˜π‘Š))
11 lshpset.n . . . . . . . . . 10 𝑁 = (LSpanβ€˜π‘Š)
1210, 11eqtr4di 2795 . . . . . . . . 9 (𝑀 = π‘Š β†’ (LSpanβ€˜π‘€) = 𝑁)
1312fveq1d 6849 . . . . . . . 8 (𝑀 = π‘Š β†’ ((LSpanβ€˜π‘€)β€˜(𝑠 βˆͺ {𝑣})) = (π‘β€˜(𝑠 βˆͺ {𝑣})))
1413, 8eqeq12d 2753 . . . . . . 7 (𝑀 = π‘Š β†’ (((LSpanβ€˜π‘€)β€˜(𝑠 βˆͺ {𝑣})) = (Baseβ€˜π‘€) ↔ (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉))
158, 14rexeqbidv 3323 . . . . . 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 632 . . . . 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 3427 . . . 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 37468 . . . 4 LSHyp = (𝑀 ∈ V ↦ {𝑠 ∈ (LSubSpβ€˜π‘€) ∣ (𝑠 β‰  (Baseβ€˜π‘€) ∧ βˆƒπ‘£ ∈ (Baseβ€˜π‘€)((LSpanβ€˜π‘€)β€˜(𝑠 βˆͺ {𝑣})) = (Baseβ€˜π‘€))})
194fvexi 6861 . . . . 5 𝑆 ∈ V
2019rabex 5294 . . . 4 {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)} ∈ V
2117, 18, 20fvmpt 6953 . . 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 2789 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074  {crab 3410  Vcvv 3448   βˆͺ cun 3913  {csn 4591  β€˜cfv 6501  Basecbs 17090  LSubSpclss 20408  LSpanclspn 20448  LSHypclsh 37466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-lshyp 37468
This theorem is referenced by:  islshp  37470
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