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Theorem lflset 34927
 Description: The set of linear functionals in a left module or left vector space. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
lflset.v
lflset.a
lflset.d Scalar
lflset.s
lflset.k
lflset.p
lflset.t
lflset.f LFnl
Assertion
Ref Expression
lflset
Distinct variable groups:   ,,   ,,,   ,   ,,,
Allowed substitution hints:   (,,,)   (,,,)   (,,,)   (,,,)   (,,,)   (,,,)   (,)   ()   (,,,)

Proof of Theorem lflset
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3118 . 2
2 lflset.f . . 3 LFnl
3 fveq2 5872 . . . . . . . . 9 Scalar Scalar
4 lflset.d . . . . . . . . 9 Scalar
53, 4syl6eqr 2516 . . . . . . . 8 Scalar
65fveq2d 5876 . . . . . . 7 Scalar
7 lflset.k . . . . . . 7
86, 7syl6eqr 2516 . . . . . 6 Scalar
9 fveq2 5872 . . . . . . 7
10 lflset.v . . . . . . 7
119, 10syl6eqr 2516 . . . . . 6
128, 11oveq12d 6314 . . . . 5 Scalar
13 fveq2 5872 . . . . . . . . . . . 12
14 lflset.a . . . . . . . . . . . 12
1513, 14syl6eqr 2516 . . . . . . . . . . 11
16 fveq2 5872 . . . . . . . . . . . . 13
17 lflset.s . . . . . . . . . . . . 13
1816, 17syl6eqr 2516 . . . . . . . . . . . 12
1918oveqd 6313 . . . . . . . . . . 11
20 eqidd 2458 . . . . . . . . . . 11
2115, 19, 20oveq123d 6317 . . . . . . . . . 10
2221fveq2d 5876 . . . . . . . . 9
235fveq2d 5876 . . . . . . . . . . 11 Scalar
24 lflset.p . . . . . . . . . . 11
2523, 24syl6eqr 2516 . . . . . . . . . 10 Scalar
265fveq2d 5876 . . . . . . . . . . . 12 Scalar
27 lflset.t . . . . . . . . . . . 12
2826, 27syl6eqr 2516 . . . . . . . . . . 11 Scalar
2928oveqd 6313 . . . . . . . . . 10 Scalar
30 eqidd 2458 . . . . . . . . . 10
3125, 29, 30oveq123d 6317 . . . . . . . . 9 Scalar Scalar
3222, 31eqeq12d 2479 . . . . . . . 8 Scalar Scalar
3311, 32raleqbidv 3068 . . . . . . 7 Scalar Scalar
3411, 33raleqbidv 3068 . . . . . 6 Scalar Scalar
358, 34raleqbidv 3068 . . . . 5 Scalar Scalar Scalar
3612, 35rabeqbidv 3104 . . . 4 Scalar Scalar Scalar Scalar
37 df-lfl 34926 . . . 4 LFnl Scalar Scalar Scalar Scalar
38 ovex 6324 . . . . 5
3938rabex 4607 . . . 4
4036, 37, 39fvmpt 5956 . . 3 LFnl
412, 40syl5eq 2510 . 2
421, 41syl 16 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1395   wcel 1819  wral 2807  crab 2811  cvv 3109  cfv 5594  (class class class)co 6296   cmap 7438  cbs 14644   cplusg 14712  cmulr 14713  Scalarcsca 14715  cvsca 14716  LFnlclfn 34925 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-lfl 34926 This theorem is referenced by:  islfl  34928
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