Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  islfl Structured version   Visualization version   GIF version

Theorem islfl 33862
 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 33861 . . 3 (𝑊𝑋𝐹 = {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))})
109eleq2d 2684 . 2 (𝑊𝑋 → (𝐺𝐹𝐺 ∈ {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))}))
11 fveq1 6152 . . . . . . 7 (𝑓 = 𝐺 → (𝑓‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑟 · 𝑥) + 𝑦)))
12 fveq1 6152 . . . . . . . . 9 (𝑓 = 𝐺 → (𝑓𝑥) = (𝐺𝑥))
1312oveq2d 6626 . . . . . . . 8 (𝑓 = 𝐺 → (𝑟 × (𝑓𝑥)) = (𝑟 × (𝐺𝑥)))
14 fveq1 6152 . . . . . . . 8 (𝑓 = 𝐺 → (𝑓𝑦) = (𝐺𝑦))
1513, 14oveq12d 6628 . . . . . . 7 (𝑓 = 𝐺 → ((𝑟 × (𝑓𝑥)) (𝑓𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
1611, 15eqeq12d 2636 . . . . . 6 (𝑓 = 𝐺 → ((𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦)) ↔ (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
17162ralbidv 2984 . . . . 5 (𝑓 = 𝐺 → (∀𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦)) ↔ ∀𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
1817ralbidv 2981 . . . 4 (𝑓 = 𝐺 → (∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦)) ↔ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
1918elrab 3350 . . 3 (𝐺 ∈ {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))} ↔ (𝐺 ∈ (𝐾𝑚 𝑉) ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
20 fvex 6163 . . . . . 6 (Base‘𝐷) ∈ V
215, 20eqeltri 2694 . . . . 5 𝐾 ∈ V
22 fvex 6163 . . . . . 6 (Base‘𝑊) ∈ V
231, 22eqeltri 2694 . . . . 5 𝑉 ∈ V
2421, 23elmap 7838 . . . 4 (𝐺 ∈ (𝐾𝑚 𝑉) ↔ 𝐺:𝑉𝐾)
2524anbi1i 730 . . 3 ((𝐺 ∈ (𝐾𝑚 𝑉) ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))) ↔ (𝐺:𝑉𝐾 ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
2619, 25bitri 264 . 2 (𝐺 ∈ {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))} ↔ (𝐺:𝑉𝐾 ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦))))
2710, 26syl6bb 276 1 (𝑊𝑋 → (𝐺𝐹 ↔ (𝐺:𝑉𝐾 ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  {crab 2911  Vcvv 3189  ⟶wf 5848  ‘cfv 5852  (class class class)co 6610   ↑𝑚 cmap 7809  Basecbs 15792  +gcplusg 15873  .rcmulr 15874  Scalarcsca 15876   ·𝑠 cvsca 15877  LFnlclfn 33859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-lfl 33860 This theorem is referenced by:  lfli  33863  islfld  33864  lflf  33865  lfl0f  33871  lfladdcl  33873  lflnegcl  33877  lshpkrcl  33918
 Copyright terms: Public domain W3C validator