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Theorem lflset 36354
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 3462 . 2 (𝑊𝑋𝑊 ∈ V)
2 lflset.f . . 3 𝐹 = (LFnl‘𝑊)
3 fveq2 6649 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
4 lflset.d . . . . . . . . 9 𝐷 = (Scalar‘𝑊)
53, 4eqtr4di 2854 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷)
65fveq2d 6653 . . . . . . 7 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐷))
7 lflset.k . . . . . . 7 𝐾 = (Base‘𝐷)
86, 7eqtr4di 2854 . . . . . 6 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
9 fveq2 6649 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
10 lflset.v . . . . . . 7 𝑉 = (Base‘𝑊)
119, 10eqtr4di 2854 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
128, 11oveq12d 7157 . . . . 5 (𝑤 = 𝑊 → ((Base‘(Scalar‘𝑤)) ↑m (Base‘𝑤)) = (𝐾m 𝑉))
13 fveq2 6649 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (+g𝑤) = (+g𝑊))
14 lflset.a . . . . . . . . . . . 12 + = (+g𝑊)
1513, 14eqtr4di 2854 . . . . . . . . . . 11 (𝑤 = 𝑊 → (+g𝑤) = + )
16 fveq2 6649 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
17 lflset.s . . . . . . . . . . . . 13 · = ( ·𝑠𝑊)
1816, 17eqtr4di 2854 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
1918oveqd 7156 . . . . . . . . . . 11 (𝑤 = 𝑊 → (𝑟( ·𝑠𝑤)𝑥) = (𝑟 · 𝑥))
20 eqidd 2802 . . . . . . . . . . 11 (𝑤 = 𝑊𝑦 = 𝑦)
2115, 19, 20oveq123d 7160 . . . . . . . . . 10 (𝑤 = 𝑊 → ((𝑟( ·𝑠𝑤)𝑥)(+g𝑤)𝑦) = ((𝑟 · 𝑥) + 𝑦))
2221fveq2d 6653 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑓‘((𝑟( ·𝑠𝑤)𝑥)(+g𝑤)𝑦)) = (𝑓‘((𝑟 · 𝑥) + 𝑦)))
235fveq2d 6653 . . . . . . . . . . 11 (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = (+g𝐷))
24 lflset.p . . . . . . . . . . 11 = (+g𝐷)
2523, 24eqtr4di 2854 . . . . . . . . . 10 (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = )
265fveq2d 6653 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = (.r𝐷))
27 lflset.t . . . . . . . . . . . 12 × = (.r𝐷)
2826, 27eqtr4di 2854 . . . . . . . . . . 11 (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = × )
2928oveqd 7156 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑟(.r‘(Scalar‘𝑤))(𝑓𝑥)) = (𝑟 × (𝑓𝑥)))
30 eqidd 2802 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑓𝑦) = (𝑓𝑦))
3125, 29, 30oveq123d 7160 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑟(.r‘(Scalar‘𝑤))(𝑓𝑥))(+g‘(Scalar‘𝑤))(𝑓𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦)))
3222, 31eqeq12d 2817 . . . . . . . 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 3357 . . . . . . 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 3357 . . . . . 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 3357 . . . . 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 3436 . . . 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 36353 . . . 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 7172 . . . . 5 (𝐾m 𝑉) ∈ V
3938rabex 5202 . . . 4 {𝑓 ∈ (𝐾m 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))} ∈ V
4036, 37, 39fvmpt 6749 . . 3 (𝑊 ∈ V → (LFnl‘𝑊) = {𝑓 ∈ (𝐾m 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))})
412, 40syl5eq 2848 . 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 1538  wcel 2112  wral 3109  {crab 3113  Vcvv 3444  cfv 6328  (class class class)co 7139  m cmap 8393  Basecbs 16479  +gcplusg 16561  .rcmulr 16562  Scalarcsca 16564   ·𝑠 cvsca 16565  LFnlclfn 36352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-lfl 36353
This theorem is referenced by:  islfl  36355
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