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Theorem isnvi 27596
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 2995 . . . . 5 (𝑥𝑋 → ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))
9 isnvi.11 . . . . . 6 ((𝑥𝑋𝑦𝑋) → (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
109ralrimiva 2995 . . . . 5 (𝑥𝑋 → ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
115, 8, 103jca 1261 . . . 4 (𝑥𝑋 → (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
1211rgen 2951 . . 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 27595 . . 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 1263 . 2 ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec
171, 16eqeltri 2726 1 𝑈 ∈ NrmCVec
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cop 4216   class class class wbr 4685  ran crn 5144  wf 5922  cfv 5926  (class class class)co 6690  cc 9972  cr 9973  0cc0 9974   + caddc 9977   · cmul 9979  cle 10113  abscabs 14018  GIdcgi 27472  CVecOLDcvc 27541  NrmCVeccnv 27567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-vc 27542  df-nv 27575
This theorem is referenced by:  cnnv  27660  hhnv  28150  hhssnv  28249
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