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Theorem isnvi 27808
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 397 . . . . 5 (𝑥𝑋 → ((𝑁𝑥) = 0 → 𝑥 = 𝑍))
6 isnvi.10 . . . . . . 7 ((𝑦 ∈ ℂ ∧ 𝑥𝑋) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))
76ancoms 455 . . . . . 6 ((𝑥𝑋𝑦 ∈ ℂ) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))
87ralrimiva 3115 . . . . 5 (𝑥𝑋 → ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))
9 isnvi.11 . . . . . 6 ((𝑥𝑋𝑦𝑋) → (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
109ralrimiva 3115 . . . . 5 (𝑥𝑋 → ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
115, 8, 103jca 1122 . . . 4 (𝑥𝑋 → (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
1211rgen 3071 . . 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 27807 . . 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 1426 . 2 ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec
171, 16eqeltri 2846 1 𝑈 ∈ NrmCVec
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  cop 4322   class class class wbr 4786  ran crn 5250  wf 6027  cfv 6031  (class class class)co 6793  cc 10136  cr 10137  0cc0 10138   + caddc 10141   · cmul 10143  cle 10277  abscabs 14182  GIdcgi 27684  CVecOLDcvc 27753  NrmCVeccnv 27779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-vc 27754  df-nv 27787
This theorem is referenced by:  cnnv  27872  hhnv  28362  hhssnv  28461
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