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Theorem islfld 39044
Description: Properties that determine a linear functional. TODO: use this in place of islfl 39042 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 6911 . . . . . 6 (𝜑 → (Base‘𝐷) = (Base‘(Scalar‘𝑊)))
74, 6eqtrd 2775 . . . . 5 (𝜑𝐾 = (Base‘(Scalar‘𝑊)))
83, 7feq23d 6732 . . . 4 (𝜑 → (𝐺:𝑉𝐾𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊))))
92, 8mpbid 232 . . 3 (𝜑𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
10 islfld.l . . . . 5 ((𝜑 ∧ (𝑟𝐾𝑥𝑉𝑦𝑉)) → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
1110ralrimivvva 3203 . . . 4 (𝜑 → ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
12 islfld.a . . . . . . . . . 10 (𝜑+ = (+g𝑊))
13 islfld.s . . . . . . . . . . 11 (𝜑· = ( ·𝑠𝑊))
1413oveqd 7448 . . . . . . . . . 10 (𝜑 → (𝑟 · 𝑥) = (𝑟( ·𝑠𝑊)𝑥))
15 eqidd 2736 . . . . . . . . . 10 (𝜑𝑦 = 𝑦)
1612, 14, 15oveq123d 7452 . . . . . . . . 9 (𝜑 → ((𝑟 · 𝑥) + 𝑦) = ((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦))
1716fveq2d 6911 . . . . . . . 8 (𝜑 → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)))
18 islfld.p . . . . . . . . . 10 (𝜑 = (+g𝐷))
195fveq2d 6911 . . . . . . . . . 10 (𝜑 → (+g𝐷) = (+g‘(Scalar‘𝑊)))
2018, 19eqtrd 2775 . . . . . . . . 9 (𝜑 = (+g‘(Scalar‘𝑊)))
21 islfld.t . . . . . . . . . . 11 (𝜑× = (.r𝐷))
225fveq2d 6911 . . . . . . . . . . 11 (𝜑 → (.r𝐷) = (.r‘(Scalar‘𝑊)))
2321, 22eqtrd 2775 . . . . . . . . . 10 (𝜑× = (.r‘(Scalar‘𝑊)))
2423oveqd 7448 . . . . . . . . 9 (𝜑 → (𝑟 × (𝐺𝑥)) = (𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥)))
25 eqidd 2736 . . . . . . . . 9 (𝜑 → (𝐺𝑦) = (𝐺𝑦))
2620, 24, 25oveq123d 7452 . . . . . . . 8 (𝜑 → ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥))(+g‘(Scalar‘𝑊))(𝐺𝑦)))
2717, 26eqeq12d 2751 . . . . . . 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 3344 . . . . . 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 3344 . . . . 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 3344 . . . 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 232 . . 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 2735 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
33 eqid 2735 . . . . 5 (+g𝑊) = (+g𝑊)
34 eqid 2735 . . . . 5 (Scalar‘𝑊) = (Scalar‘𝑊)
35 eqid 2735 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
36 eqid 2735 . . . . 5 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
37 eqid 2735 . . . . 5 (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
38 eqid 2735 . . . . 5 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
39 eqid 2735 . . . . 5 (LFnl‘𝑊) = (LFnl‘𝑊)
4032, 33, 34, 35, 36, 37, 38, 39islfl 39042 . . . 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 477 . . 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 837 . 2 (𝜑𝐺 ∈ (LFnl‘𝑊))
43 islfld.f . 2 (𝜑𝐹 = (LFnl‘𝑊))
4442, 43eleqtrrd 2842 1 (𝜑𝐺𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  LFnlclfn 39039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-lfl 39040
This theorem is referenced by:  lflvscl  39059
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