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Theorem lflset 34861
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 𝑉 = (Base‘𝑊)
lflset.a + = (+g𝑊)
lflset.d 𝐷 = (Scalar‘𝑊)
lflset.s · = ( ·𝑠𝑊)
lflset.k 𝐾 = (Base‘𝐷)
lflset.p = (+g𝐷)
lflset.t × = (.r𝐷)
lflset.f 𝐹 = (LFnl‘𝑊)
Assertion
Ref Expression
lflset (𝑊𝑋𝐹 = {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))})
Distinct variable groups:   𝑓,𝑟,𝐾   𝑥,𝑓,𝑦,𝑉   𝑓,𝑊   𝑥,𝑟,𝑦,𝑊
Allowed substitution hints:   𝐷(𝑥,𝑦,𝑓,𝑟)   + (𝑥,𝑦,𝑓,𝑟)   (𝑥,𝑦,𝑓,𝑟)   · (𝑥,𝑦,𝑓,𝑟)   × (𝑥,𝑦,𝑓,𝑟)   𝐹(𝑥,𝑦,𝑓,𝑟)   𝐾(𝑥,𝑦)   𝑉(𝑟)   𝑋(𝑥,𝑦,𝑓,𝑟)

Proof of Theorem lflset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elex 3361 . 2 (𝑊𝑋𝑊 ∈ V)
2 lflset.f . . 3 𝐹 = (LFnl‘𝑊)
3 fveq2 6332 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
4 lflset.d . . . . . . . . 9 𝐷 = (Scalar‘𝑊)
53, 4syl6eqr 2822 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷)
65fveq2d 6336 . . . . . . 7 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐷))
7 lflset.k . . . . . . 7 𝐾 = (Base‘𝐷)
86, 7syl6eqr 2822 . . . . . 6 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
9 fveq2 6332 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
10 lflset.v . . . . . . 7 𝑉 = (Base‘𝑊)
119, 10syl6eqr 2822 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
128, 11oveq12d 6810 . . . . 5 (𝑤 = 𝑊 → ((Base‘(Scalar‘𝑤)) ↑𝑚 (Base‘𝑤)) = (𝐾𝑚 𝑉))
13 fveq2 6332 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (+g𝑤) = (+g𝑊))
14 lflset.a . . . . . . . . . . . 12 + = (+g𝑊)
1513, 14syl6eqr 2822 . . . . . . . . . . 11 (𝑤 = 𝑊 → (+g𝑤) = + )
16 fveq2 6332 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
17 lflset.s . . . . . . . . . . . . 13 · = ( ·𝑠𝑊)
1816, 17syl6eqr 2822 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
1918oveqd 6809 . . . . . . . . . . 11 (𝑤 = 𝑊 → (𝑟( ·𝑠𝑤)𝑥) = (𝑟 · 𝑥))
20 eqidd 2771 . . . . . . . . . . 11 (𝑤 = 𝑊𝑦 = 𝑦)
2115, 19, 20oveq123d 6813 . . . . . . . . . 10 (𝑤 = 𝑊 → ((𝑟( ·𝑠𝑤)𝑥)(+g𝑤)𝑦) = ((𝑟 · 𝑥) + 𝑦))
2221fveq2d 6336 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑓‘((𝑟( ·𝑠𝑤)𝑥)(+g𝑤)𝑦)) = (𝑓‘((𝑟 · 𝑥) + 𝑦)))
235fveq2d 6336 . . . . . . . . . . 11 (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = (+g𝐷))
24 lflset.p . . . . . . . . . . 11 = (+g𝐷)
2523, 24syl6eqr 2822 . . . . . . . . . 10 (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = )
265fveq2d 6336 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = (.r𝐷))
27 lflset.t . . . . . . . . . . . 12 × = (.r𝐷)
2826, 27syl6eqr 2822 . . . . . . . . . . 11 (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = × )
2928oveqd 6809 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑟(.r‘(Scalar‘𝑤))(𝑓𝑥)) = (𝑟 × (𝑓𝑥)))
30 eqidd 2771 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑓𝑦) = (𝑓𝑦))
3125, 29, 30oveq123d 6813 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑟(.r‘(Scalar‘𝑤))(𝑓𝑥))(+g‘(Scalar‘𝑤))(𝑓𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦)))
3222, 31eqeq12d 2785 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑓‘((𝑟( ·𝑠𝑤)𝑥)(+g𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓𝑥))(+g‘(Scalar‘𝑤))(𝑓𝑦)) ↔ (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))))
3311, 32raleqbidv 3300 . . . . . . 7 (𝑤 = 𝑊 → (∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·𝑠𝑤)𝑥)(+g𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓𝑥))(+g‘(Scalar‘𝑤))(𝑓𝑦)) ↔ ∀𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))))
3411, 33raleqbidv 3300 . . . . . 6 (𝑤 = 𝑊 → (∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·𝑠𝑤)𝑥)(+g𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓𝑥))(+g‘(Scalar‘𝑤))(𝑓𝑦)) ↔ ∀𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))))
358, 34raleqbidv 3300 . . . . 5 (𝑤 = 𝑊 → (∀𝑟 ∈ (Base‘(Scalar‘𝑤))∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·𝑠𝑤)𝑥)(+g𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓𝑥))(+g‘(Scalar‘𝑤))(𝑓𝑦)) ↔ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))))
3612, 35rabeqbidv 3344 . . . 4 (𝑤 = 𝑊 → {𝑓 ∈ ((Base‘(Scalar‘𝑤)) ↑𝑚 (Base‘𝑤)) ∣ ∀𝑟 ∈ (Base‘(Scalar‘𝑤))∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·𝑠𝑤)𝑥)(+g𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓𝑥))(+g‘(Scalar‘𝑤))(𝑓𝑦))} = {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))})
37 df-lfl 34860 . . . 4 LFnl = (𝑤 ∈ V ↦ {𝑓 ∈ ((Base‘(Scalar‘𝑤)) ↑𝑚 (Base‘𝑤)) ∣ ∀𝑟 ∈ (Base‘(Scalar‘𝑤))∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·𝑠𝑤)𝑥)(+g𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓𝑥))(+g‘(Scalar‘𝑤))(𝑓𝑦))})
38 ovex 6822 . . . . 5 (𝐾𝑚 𝑉) ∈ V
3938rabex 4943 . . . 4 {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))} ∈ V
4036, 37, 39fvmpt 6424 . . 3 (𝑊 ∈ V → (LFnl‘𝑊) = {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))})
412, 40syl5eq 2816 . 2 (𝑊 ∈ V → 𝐹 = {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))})
421, 41syl 17 1 (𝑊𝑋𝐹 = {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  wral 3060  {crab 3064  Vcvv 3349  cfv 6031  (class class class)co 6792  𝑚 cmap 8008  Basecbs 16063  +gcplusg 16148  .rcmulr 16149  Scalarcsca 16151   ·𝑠 cvsca 16152  LFnlclfn 34859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-lfl 34860
This theorem is referenced by:  islfl  34862
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