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Theorem islfl 36236
 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 36235 . . 3 (𝑊𝑋𝐹 = {𝑓 ∈ (𝐾m 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))})
109eleq2d 2897 . 2 (𝑊𝑋 → (𝐺𝐹𝐺 ∈ {𝑓 ∈ (𝐾m 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))}))
11 fveq1 6642 . . . . . . 7 (𝑓 = 𝐺 → (𝑓‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑟 · 𝑥) + 𝑦)))
12 fveq1 6642 . . . . . . . . 9 (𝑓 = 𝐺 → (𝑓𝑥) = (𝐺𝑥))
1312oveq2d 7146 . . . . . . . 8 (𝑓 = 𝐺 → (𝑟 × (𝑓𝑥)) = (𝑟 × (𝐺𝑥)))
14 fveq1 6642 . . . . . . . 8 (𝑓 = 𝐺 → (𝑓𝑦) = (𝐺𝑦))
1513, 14oveq12d 7148 . . . . . . 7 (𝑓 = 𝐺 → ((𝑟 × (𝑓𝑥)) (𝑓𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
1611, 15eqeq12d 2837 . . . . . 6 (𝑓 = 𝐺 → ((𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦)) ↔ (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
17162ralbidv 3187 . . . . 5 (𝑓 = 𝐺 → (∀𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦)) ↔ ∀𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
1817ralbidv 3185 . . . 4 (𝑓 = 𝐺 → (∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦)) ↔ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
1918elrab 3657 . . 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 6657 . . . . 5 𝐾 ∈ V
211fvexi 6657 . . . . 5 𝑉 ∈ V
2220, 21elmap 8410 . . . 4 (𝐺 ∈ (𝐾m 𝑉) ↔ 𝐺:𝑉𝐾)
2322anbi1i 626 . . 3 ((𝐺 ∈ (𝐾m 𝑉) ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))) ↔ (𝐺:𝑉𝐾 ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
2419, 23bitri 278 . 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 290 1 (𝑊𝑋 → (𝐺𝐹 ↔ (𝐺:𝑉𝐾 ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3126  {crab 3130  ⟶wf 6324  ‘cfv 6328  (class class class)co 7130   ↑m cmap 8381  Basecbs 16461  +gcplusg 16543  .rcmulr 16544  Scalarcsca 16546   ·𝑠 cvsca 16547  LFnlclfn 36233 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-map 8383  df-lfl 36234 This theorem is referenced by:  lfli  36237  islfld  36238  lflf  36239  lfl0f  36245  lfladdcl  36247  lflnegcl  36251  lshpkrcl  36292
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