MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnvi Structured version   Visualization version   GIF version

Theorem isnvi 30632
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 412 . . . . 5 (𝑥𝑋 → ((𝑁𝑥) = 0 → 𝑥 = 𝑍))
6 isnvi.10 . . . . . . 7 ((𝑦 ∈ ℂ ∧ 𝑥𝑋) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))
76ancoms 458 . . . . . 6 ((𝑥𝑋𝑦 ∈ ℂ) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))
87ralrimiva 3146 . . . . 5 (𝑥𝑋 → ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))
9 isnvi.11 . . . . . 6 ((𝑥𝑋𝑦𝑋) → (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
109ralrimiva 3146 . . . . 5 (𝑥𝑋 → ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
115, 8, 103jca 1129 . . . 4 (𝑥𝑋 → (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
1211rgen 3063 . . 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 30631 . . 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 1342 . 2 ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec
171, 16eqeltri 2837 1 𝑈 ∈ NrmCVec
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cop 4632   class class class wbr 5143  ran crn 5686  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  cr 11154  0cc0 11155   + caddc 11158   · cmul 11160  cle 11296  abscabs 15273  GIdcgi 30509  CVecOLDcvc 30577  NrmCVeccnv 30603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-vc 30578  df-nv 30611
This theorem is referenced by:  cnnv  30696  hhnv  31184  hhssnv  31283
  Copyright terms: Public domain W3C validator