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Theorem nvi 27597
Description: The properties of a normed complex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvi.1 𝑋 = (BaseSet‘𝑈)
nvi.2 𝐺 = ( +𝑣𝑈)
nvi.4 𝑆 = ( ·𝑠OLD𝑈)
nvi.5 𝑍 = (0vec𝑈)
nvi.6 𝑁 = (normCV𝑈)
Assertion
Ref Expression
nvi (𝑈 ∈ NrmCVec → (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑈   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑈(𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem nvi
StepHypRef Expression
1 eqid 2651 . . . . . 6 (1st𝑈) = (1st𝑈)
2 nvi.6 . . . . . 6 𝑁 = (normCV𝑈)
31, 2nvop2 27591 . . . . 5 (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1st𝑈), 𝑁⟩)
4 nvi.2 . . . . . . 7 𝐺 = ( +𝑣𝑈)
5 nvi.4 . . . . . . 7 𝑆 = ( ·𝑠OLD𝑈)
61, 4, 5nvvop 27592 . . . . . 6 (𝑈 ∈ NrmCVec → (1st𝑈) = ⟨𝐺, 𝑆⟩)
76opeq1d 4439 . . . . 5 (𝑈 ∈ NrmCVec → ⟨(1st𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
83, 7eqtrd 2685 . . . 4 (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
9 id 22 . . . 4 (𝑈 ∈ NrmCVec → 𝑈 ∈ NrmCVec)
108, 9eqeltrrd 2731 . . 3 (𝑈 ∈ NrmCVec → ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec)
11 nvi.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
1211, 4bafval 27587 . . . 4 𝑋 = ran 𝐺
13 eqid 2651 . . . 4 (GId‘𝐺) = (GId‘𝐺)
1412, 13isnv 27595 . . 3 (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = (GId‘𝐺)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
1510, 14sylib 208 . 2 (𝑈 ∈ NrmCVec → (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = (GId‘𝐺)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
16 nvi.5 . . . . . . . 8 𝑍 = (0vec𝑈)
174, 160vfval 27589 . . . . . . 7 (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
1817eqeq2d 2661 . . . . . 6 (𝑈 ∈ NrmCVec → (𝑥 = 𝑍𝑥 = (GId‘𝐺)))
1918imbi2d 329 . . . . 5 (𝑈 ∈ NrmCVec → (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ↔ ((𝑁𝑥) = 0 → 𝑥 = (GId‘𝐺))))
20193anbi1d 1443 . . . 4 (𝑈 ∈ NrmCVec → ((((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ↔ (((𝑁𝑥) = 0 → 𝑥 = (GId‘𝐺)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
2120ralbidv 3015 . . 3 (𝑈 ∈ NrmCVec → (∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) ↔ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = (GId‘𝐺)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
22213anbi3d 1445 . 2 (𝑈 ∈ NrmCVec → ((⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))) ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = (GId‘𝐺)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))
2315, 22mpbird 247 1 (𝑈 ∈ NrmCVec → (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cop 4216   class class class wbr 4685  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  cc 9972  cr 9973  0cc0 9974   + caddc 9977   · cmul 9979  cle 10113  abscabs 14018  GIdcgi 27472  CVecOLDcvc 27541  NrmCVeccnv 27567   +𝑣 cpv 27568  BaseSetcba 27569   ·𝑠OLD cns 27570  0veccn0v 27571  normCVcnmcv 27573
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-pow 4873  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-1st 7210  df-2nd 7211  df-vc 27542  df-nv 27575  df-va 27578  df-ba 27579  df-sm 27580  df-0v 27581  df-nmcv 27583
This theorem is referenced by:  nvvc  27598  nvf  27643  nvs  27646  nvz  27652  nvtri  27653
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