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Theorem islfld 33367
 Description: Properties that determine a linear functional. TODO: use this in place of islfl 33365 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)
Hypotheses
Ref Expression
islfld.v (𝜑𝑉 = (Base‘𝑊))
islfld.a (𝜑+ = (+g𝑊))
islfld.d (𝜑𝐷 = (Scalar‘𝑊))
islfld.s (𝜑· = ( ·𝑠𝑊))
islfld.k (𝜑𝐾 = (Base‘𝐷))
islfld.p (𝜑 = (+g𝐷))
islfld.t (𝜑× = (.r𝐷))
islfld.f (𝜑𝐹 = (LFnl‘𝑊))
islfld.u (𝜑𝐺:𝑉𝐾)
islfld.l ((𝜑 ∧ (𝑟𝐾𝑥𝑉𝑦𝑉)) → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
islfld.w (𝜑𝑊𝑋)
Assertion
Ref Expression
islfld (𝜑𝐺𝐹)
Distinct variable groups:   𝑥,𝑟,𝑦,𝐺   𝐾,𝑟,𝑥,𝑦   𝑥,𝑉,𝑦   𝑊,𝑟,𝑥,𝑦   𝜑,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)   𝑉(𝑟)   𝑋(𝑥,𝑦,𝑟)

Proof of Theorem islfld
StepHypRef Expression
1 islfld.w . . 3 (𝜑𝑊𝑋)
2 islfld.u . . . 4 (𝜑𝐺:𝑉𝐾)
3 islfld.v . . . . 5 (𝜑𝑉 = (Base‘𝑊))
4 islfld.k . . . . . 6 (𝜑𝐾 = (Base‘𝐷))
5 islfld.d . . . . . . 7 (𝜑𝐷 = (Scalar‘𝑊))
65fveq2d 6107 . . . . . 6 (𝜑 → (Base‘𝐷) = (Base‘(Scalar‘𝑊)))
74, 6eqtrd 2644 . . . . 5 (𝜑𝐾 = (Base‘(Scalar‘𝑊)))
83, 7feq23d 5953 . . . 4 (𝜑 → (𝐺:𝑉𝐾𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊))))
92, 8mpbid 221 . . 3 (𝜑𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
10 islfld.l . . . . 5 ((𝜑 ∧ (𝑟𝐾𝑥𝑉𝑦𝑉)) → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
1110ralrimivvva 2955 . . . 4 (𝜑 → ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
12 islfld.a . . . . . . . . . 10 (𝜑+ = (+g𝑊))
13 islfld.s . . . . . . . . . . 11 (𝜑· = ( ·𝑠𝑊))
1413oveqd 6566 . . . . . . . . . 10 (𝜑 → (𝑟 · 𝑥) = (𝑟( ·𝑠𝑊)𝑥))
15 eqidd 2611 . . . . . . . . . 10 (𝜑𝑦 = 𝑦)
1612, 14, 15oveq123d 6570 . . . . . . . . 9 (𝜑 → ((𝑟 · 𝑥) + 𝑦) = ((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦))
1716fveq2d 6107 . . . . . . . 8 (𝜑 → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)))
18 islfld.p . . . . . . . . . 10 (𝜑 = (+g𝐷))
195fveq2d 6107 . . . . . . . . . 10 (𝜑 → (+g𝐷) = (+g‘(Scalar‘𝑊)))
2018, 19eqtrd 2644 . . . . . . . . 9 (𝜑 = (+g‘(Scalar‘𝑊)))
21 islfld.t . . . . . . . . . . 11 (𝜑× = (.r𝐷))
225fveq2d 6107 . . . . . . . . . . 11 (𝜑 → (.r𝐷) = (.r‘(Scalar‘𝑊)))
2321, 22eqtrd 2644 . . . . . . . . . 10 (𝜑× = (.r‘(Scalar‘𝑊)))
2423oveqd 6566 . . . . . . . . 9 (𝜑 → (𝑟 × (𝐺𝑥)) = (𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥)))
25 eqidd 2611 . . . . . . . . 9 (𝜑 → (𝐺𝑦) = (𝐺𝑦))
2620, 24, 25oveq123d 6570 . . . . . . . 8 (𝜑 → ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥))(+g‘(Scalar‘𝑊))(𝐺𝑦)))
2717, 26eqeq12d 2625 . . . . . . 7 (𝜑 → ((𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) ↔ (𝐺‘((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥))(+g‘(Scalar‘𝑊))(𝐺𝑦))))
283, 27raleqbidv 3129 . . . . . 6 (𝜑 → (∀𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) ↔ ∀𝑦 ∈ (Base‘𝑊)(𝐺‘((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥))(+g‘(Scalar‘𝑊))(𝐺𝑦))))
293, 28raleqbidv 3129 . . . . 5 (𝜑 → (∀𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) ↔ ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(𝐺‘((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥))(+g‘(Scalar‘𝑊))(𝐺𝑦))))
307, 29raleqbidv 3129 . . . 4 (𝜑 → (∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) ↔ ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(𝐺‘((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥))(+g‘(Scalar‘𝑊))(𝐺𝑦))))
3111, 30mpbid 221 . . 3 (𝜑 → ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(𝐺‘((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥))(+g‘(Scalar‘𝑊))(𝐺𝑦)))
32 eqid 2610 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
33 eqid 2610 . . . . 5 (+g𝑊) = (+g𝑊)
34 eqid 2610 . . . . 5 (Scalar‘𝑊) = (Scalar‘𝑊)
35 eqid 2610 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
36 eqid 2610 . . . . 5 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
37 eqid 2610 . . . . 5 (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
38 eqid 2610 . . . . 5 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
39 eqid 2610 . . . . 5 (LFnl‘𝑊) = (LFnl‘𝑊)
4032, 33, 34, 35, 36, 37, 38, 39islfl 33365 . . . 4 (𝑊𝑋 → (𝐺 ∈ (LFnl‘𝑊) ↔ (𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)) ∧ ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(𝐺‘((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥))(+g‘(Scalar‘𝑊))(𝐺𝑦)))))
4140biimpar 501 . . 3 ((𝑊𝑋 ∧ (𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)) ∧ ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(𝐺‘((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥))(+g‘(Scalar‘𝑊))(𝐺𝑦)))) → 𝐺 ∈ (LFnl‘𝑊))
421, 9, 31, 41syl12anc 1316 . 2 (𝜑𝐺 ∈ (LFnl‘𝑊))
43 islfld.f . 2 (𝜑𝐹 = (LFnl‘𝑊))
4442, 43eleqtrrd 2691 1 (𝜑𝐺𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  LFnlclfn 33362 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-lfl 33363 This theorem is referenced by:  lflvscl  33382
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