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Theorem islfld 34168
Description: Properties that determine a linear functional. TODO: use this in place of islfl 34166 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 6182 . . . . . 6 (𝜑 → (Base‘𝐷) = (Base‘(Scalar‘𝑊)))
74, 6eqtrd 2654 . . . . 5 (𝜑𝐾 = (Base‘(Scalar‘𝑊)))
83, 7feq23d 6027 . . . 4 (𝜑 → (𝐺:𝑉𝐾𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊))))
92, 8mpbid 222 . . 3 (𝜑𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
10 islfld.l . . . . 5 ((𝜑 ∧ (𝑟𝐾𝑥𝑉𝑦𝑉)) → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
1110ralrimivvva 2969 . . . 4 (𝜑 → ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
12 islfld.a . . . . . . . . . 10 (𝜑+ = (+g𝑊))
13 islfld.s . . . . . . . . . . 11 (𝜑· = ( ·𝑠𝑊))
1413oveqd 6652 . . . . . . . . . 10 (𝜑 → (𝑟 · 𝑥) = (𝑟( ·𝑠𝑊)𝑥))
15 eqidd 2621 . . . . . . . . . 10 (𝜑𝑦 = 𝑦)
1612, 14, 15oveq123d 6656 . . . . . . . . 9 (𝜑 → ((𝑟 · 𝑥) + 𝑦) = ((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦))
1716fveq2d 6182 . . . . . . . 8 (𝜑 → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)))
18 islfld.p . . . . . . . . . 10 (𝜑 = (+g𝐷))
195fveq2d 6182 . . . . . . . . . 10 (𝜑 → (+g𝐷) = (+g‘(Scalar‘𝑊)))
2018, 19eqtrd 2654 . . . . . . . . 9 (𝜑 = (+g‘(Scalar‘𝑊)))
21 islfld.t . . . . . . . . . . 11 (𝜑× = (.r𝐷))
225fveq2d 6182 . . . . . . . . . . 11 (𝜑 → (.r𝐷) = (.r‘(Scalar‘𝑊)))
2321, 22eqtrd 2654 . . . . . . . . . 10 (𝜑× = (.r‘(Scalar‘𝑊)))
2423oveqd 6652 . . . . . . . . 9 (𝜑 → (𝑟 × (𝐺𝑥)) = (𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥)))
25 eqidd 2621 . . . . . . . . 9 (𝜑 → (𝐺𝑦) = (𝐺𝑦))
2620, 24, 25oveq123d 6656 . . . . . . . 8 (𝜑 → ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺𝑥))(+g‘(Scalar‘𝑊))(𝐺𝑦)))
2717, 26eqeq12d 2635 . . . . . . 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 3147 . . . . . 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 3147 . . . . 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 3147 . . . 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 222 . . 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 2620 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
33 eqid 2620 . . . . 5 (+g𝑊) = (+g𝑊)
34 eqid 2620 . . . . 5 (Scalar‘𝑊) = (Scalar‘𝑊)
35 eqid 2620 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
36 eqid 2620 . . . . 5 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
37 eqid 2620 . . . . 5 (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
38 eqid 2620 . . . . 5 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
39 eqid 2620 . . . . 5 (LFnl‘𝑊) = (LFnl‘𝑊)
4032, 33, 34, 35, 36, 37, 38, 39islfl 34166 . . . 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 502 . . 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 1322 . 2 (𝜑𝐺 ∈ (LFnl‘𝑊))
43 islfld.f . 2 (𝜑𝐹 = (LFnl‘𝑊))
4442, 43eleqtrrd 2702 1 (𝜑𝐺𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  wf 5872  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  .rcmulr 15923  Scalarcsca 15925   ·𝑠 cvsca 15926  LFnlclfn 34163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844  df-lfl 34164
This theorem is referenced by:  lflvscl  34183
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