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Theorem islfld 36631
 Description: Properties that determine a linear functional. TODO: use this in place of islfl 36629 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 6663 . . . . . 6 (𝜑 → (Base‘𝐷) = (Base‘(Scalar‘𝑊)))
74, 6eqtrd 2794 . . . . 5 (𝜑𝐾 = (Base‘(Scalar‘𝑊)))
83, 7feq23d 6494 . . . 4 (𝜑 → (𝐺:𝑉𝐾𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊))))
92, 8mpbid 235 . . 3 (𝜑𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
10 islfld.l . . . . 5 ((𝜑 ∧ (𝑟𝐾𝑥𝑉𝑦𝑉)) → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
1110ralrimivvva 3122 . . . 4 (𝜑 → ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
12 islfld.a . . . . . . . . . 10 (𝜑+ = (+g𝑊))
13 islfld.s . . . . . . . . . . 11 (𝜑· = ( ·𝑠𝑊))
1413oveqd 7168 . . . . . . . . . 10 (𝜑 → (𝑟 · 𝑥) = (𝑟( ·𝑠𝑊)𝑥))
15 eqidd 2760 . . . . . . . . . 10 (𝜑𝑦 = 𝑦)
1612, 14, 15oveq123d 7172 . . . . . . . . 9 (𝜑 → ((𝑟 · 𝑥) + 𝑦) = ((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦))
1716fveq2d 6663 . . . . . . . 8 (𝜑 → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)))
18 islfld.p . . . . . . . . . 10 (𝜑 = (+g𝐷))
195fveq2d 6663 . . . . . . . . . 10 (𝜑 → (+g𝐷) = (+g‘(Scalar‘𝑊)))
2018, 19eqtrd 2794 . . . . . . . . 9 (𝜑 = (+g‘(Scalar‘𝑊)))
21 islfld.t . . . . . . . . . . 11 (𝜑× = (.r𝐷))
225fveq2d 6663 . . . . . . . . . . 11 (𝜑 → (.r𝐷) = (.r‘(Scalar‘𝑊)))
2321, 22eqtrd 2794 . . . . . . . . . 10 (𝜑× = (.r‘(Scalar‘𝑊)))
2423oveqd 7168 . . . . . . . . 9 (𝜑 → (𝑟 × (𝐺𝑥)) = (𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥)))
25 eqidd 2760 . . . . . . . . 9 (𝜑 → (𝐺𝑦) = (𝐺𝑦))
2620, 24, 25oveq123d 7172 . . . . . . . 8 (𝜑 → ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥))(+g‘(Scalar‘𝑊))(𝐺𝑦)))
2717, 26eqeq12d 2775 . . . . . . 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 3320 . . . . . 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 3320 . . . . 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 3320 . . . 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 235 . . 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 2759 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
33 eqid 2759 . . . . 5 (+g𝑊) = (+g𝑊)
34 eqid 2759 . . . . 5 (Scalar‘𝑊) = (Scalar‘𝑊)
35 eqid 2759 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
36 eqid 2759 . . . . 5 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
37 eqid 2759 . . . . 5 (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
38 eqid 2759 . . . . 5 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
39 eqid 2759 . . . . 5 (LFnl‘𝑊) = (LFnl‘𝑊)
4032, 33, 34, 35, 36, 37, 38, 39islfl 36629 . . . 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 482 . . 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 836 . 2 (𝜑𝐺 ∈ (LFnl‘𝑊))
43 islfld.f . 2 (𝜑𝐹 = (LFnl‘𝑊))
4442, 43eleqtrrd 2856 1 (𝜑𝐺𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151  Basecbs 16534  +gcplusg 16616  .rcmulr 16617  Scalarcsca 16619   ·𝑠 cvsca 16620  LFnlclfn 36626 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8419  df-lfl 36627 This theorem is referenced by:  lflvscl  36646
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