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Theorem islfl 35081
Description: The predicate "is a linear functional". (Contributed by NM, 15-Apr-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
islfl (𝑊𝑋 → (𝐺𝐹 ↔ (𝐺:𝑉𝐾 ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))))
Distinct variable groups:   𝐾,𝑟   𝑥,𝑦,𝑉   𝑥,𝑟,𝑦,𝑊   𝐺,𝑟,𝑥,𝑦
Allowed substitution hints:   𝐷(𝑥,𝑦,𝑟)   + (𝑥,𝑦,𝑟)   (𝑥,𝑦,𝑟)   · (𝑥,𝑦,𝑟)   × (𝑥,𝑦,𝑟)   𝐹(𝑥,𝑦,𝑟)   𝐾(𝑥,𝑦)   𝑉(𝑟)   𝑋(𝑥,𝑦,𝑟)

Proof of Theorem islfl
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 lflset.v . . . 4 𝑉 = (Base‘𝑊)
2 lflset.a . . . 4 + = (+g𝑊)
3 lflset.d . . . 4 𝐷 = (Scalar‘𝑊)
4 lflset.s . . . 4 · = ( ·𝑠𝑊)
5 lflset.k . . . 4 𝐾 = (Base‘𝐷)
6 lflset.p . . . 4 = (+g𝐷)
7 lflset.t . . . 4 × = (.r𝐷)
8 lflset.f . . . 4 𝐹 = (LFnl‘𝑊)
91, 2, 3, 4, 5, 6, 7, 8lflset 35080 . . 3 (𝑊𝑋𝐹 = {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))})
109eleq2d 2864 . 2 (𝑊𝑋 → (𝐺𝐹𝐺 ∈ {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))}))
11 fveq1 6410 . . . . . . 7 (𝑓 = 𝐺 → (𝑓‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑟 · 𝑥) + 𝑦)))
12 fveq1 6410 . . . . . . . . 9 (𝑓 = 𝐺 → (𝑓𝑥) = (𝐺𝑥))
1312oveq2d 6894 . . . . . . . 8 (𝑓 = 𝐺 → (𝑟 × (𝑓𝑥)) = (𝑟 × (𝐺𝑥)))
14 fveq1 6410 . . . . . . . 8 (𝑓 = 𝐺 → (𝑓𝑦) = (𝐺𝑦))
1513, 14oveq12d 6896 . . . . . . 7 (𝑓 = 𝐺 → ((𝑟 × (𝑓𝑥)) (𝑓𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
1611, 15eqeq12d 2814 . . . . . 6 (𝑓 = 𝐺 → ((𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦)) ↔ (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
17162ralbidv 3170 . . . . 5 (𝑓 = 𝐺 → (∀𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦)) ↔ ∀𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
1817ralbidv 3167 . . . 4 (𝑓 = 𝐺 → (∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦)) ↔ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
1918elrab 3556 . . 3 (𝐺 ∈ {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))} ↔ (𝐺 ∈ (𝐾𝑚 𝑉) ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
205fvexi 6425 . . . . 5 𝐾 ∈ V
211fvexi 6425 . . . . 5 𝑉 ∈ V
2220, 21elmap 8124 . . . 4 (𝐺 ∈ (𝐾𝑚 𝑉) ↔ 𝐺:𝑉𝐾)
2322anbi1i 618 . . 3 ((𝐺 ∈ (𝐾𝑚 𝑉) ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))) ↔ (𝐺:𝑉𝐾 ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
2419, 23bitri 267 . 2 (𝐺 ∈ {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))} ↔ (𝐺:𝑉𝐾 ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
2510, 24syl6bb 279 1 (𝑊𝑋 → (𝐺𝐹 ↔ (𝐺:𝑉𝐾 ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  {crab 3093  wf 6097  cfv 6101  (class class class)co 6878  𝑚 cmap 8095  Basecbs 16184  +gcplusg 16267  .rcmulr 16268  Scalarcsca 16270   ·𝑠 cvsca 16271  LFnlclfn 35078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-lfl 35079
This theorem is referenced by:  lfli  35082  islfld  35083  lflf  35084  lfl0f  35090  lfladdcl  35092  lflnegcl  35096  lshpkrcl  35137
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