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Theorem isnvi 26852
Description: Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
isnvi.5 𝑋 = ran 𝐺
isnvi.6 𝑍 = (GId‘𝐺)
isnvi.7 𝐺, 𝑆⟩ ∈ CVecOLD
isnvi.8 𝑁:𝑋⟶ℝ
isnvi.9 ((𝑥𝑋 ∧ (𝑁𝑥) = 0) → 𝑥 = 𝑍)
isnvi.10 ((𝑦 ∈ ℂ ∧ 𝑥𝑋) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))
isnvi.11 ((𝑥𝑋𝑦𝑋) → (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
isnvi.12 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁
Assertion
Ref Expression
isnvi 𝑈 ∈ NrmCVec
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem isnvi
StepHypRef Expression
1 isnvi.12 . 2 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁
2 isnvi.7 . . 3 𝐺, 𝑆⟩ ∈ CVecOLD
3 isnvi.8 . . 3 𝑁:𝑋⟶ℝ
4 isnvi.9 . . . . . 6 ((𝑥𝑋 ∧ (𝑁𝑥) = 0) → 𝑥 = 𝑍)
54ex 449 . . . . 5 (𝑥𝑋 → ((𝑁𝑥) = 0 → 𝑥 = 𝑍))
6 isnvi.10 . . . . . . 7 ((𝑦 ∈ ℂ ∧ 𝑥𝑋) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))
76ancoms 468 . . . . . 6 ((𝑥𝑋𝑦 ∈ ℂ) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))
87ralrimiva 2949 . . . . 5 (𝑥𝑋 → ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))
9 isnvi.11 . . . . . 6 ((𝑥𝑋𝑦𝑋) → (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
109ralrimiva 2949 . . . . 5 (𝑥𝑋 → ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
115, 8, 103jca 1235 . . . 4 (𝑥𝑋 → (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
1211rgen 2906 . . 3 𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
13 isnvi.5 . . . 4 𝑋 = ran 𝐺
14 isnvi.6 . . . 4 𝑍 = (GId‘𝐺)
1513, 14isnv 26851 . . 3 (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
162, 3, 12, 15mpbir3an 1237 . 2 ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec
171, 16eqeltri 2684 1 𝑈 ∈ NrmCVec
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  cop 4131   class class class wbr 4583  ran crn 5039  wf 5800  cfv 5804  (class class class)co 6549  cc 9813  cr 9814  0cc0 9815   + caddc 9818   · cmul 9820  cle 9954  abscabs 13822  GIdcgi 26728  CVecOLDcvc 26797  NrmCVeccnv 26823
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-rep 4699  ax-sep 4709  ax-nul 4717  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-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  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-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-vc 26798  df-nv 26831
This theorem is referenced by:  cnnv  26916  hhnv  27406  hhssnv  27505
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