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Theorem nvi 27439
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 2620 . . . . . 6 (1st𝑈) = (1st𝑈)
2 nvi.6 . . . . . 6 𝑁 = (normCV𝑈)
31, 2nvop2 27433 . . . . 5 (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1st𝑈), 𝑁⟩)
4 nvi.2 . . . . . . 7 𝐺 = ( +𝑣𝑈)
5 nvi.4 . . . . . . 7 𝑆 = ( ·𝑠OLD𝑈)
61, 4, 5nvvop 27434 . . . . . 6 (𝑈 ∈ NrmCVec → (1st𝑈) = ⟨𝐺, 𝑆⟩)
76opeq1d 4399 . . . . 5 (𝑈 ∈ NrmCVec → ⟨(1st𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
83, 7eqtrd 2654 . . . 4 (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
9 id 22 . . . 4 (𝑈 ∈ NrmCVec → 𝑈 ∈ NrmCVec)
108, 9eqeltrrd 2700 . . 3 (𝑈 ∈ NrmCVec → ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec)
11 nvi.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
1211, 4bafval 27429 . . . 4 𝑋 = ran 𝐺
13 eqid 2620 . . . 4 (GId‘𝐺) = (GId‘𝐺)
1412, 13isnv 27437 . . 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 27431 . . . . . . 7 (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
1817eqeq2d 2630 . . . . . 6 (𝑈 ∈ NrmCVec → (𝑥 = 𝑍𝑥 = (GId‘𝐺)))
1918imbi2d 330 . . . . 5 (𝑈 ∈ NrmCVec → (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ↔ ((𝑁𝑥) = 0 → 𝑥 = (GId‘𝐺))))
20193anbi1d 1401 . . . 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 2983 . . 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 1403 . 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 1036   = wceq 1481  wcel 1988  wral 2909  cop 4174   class class class wbr 4644  wf 5872  cfv 5876  (class class class)co 6635  1st c1st 7151  cc 9919  cr 9920  0cc0 9921   + caddc 9924   · cmul 9926  cle 10060  abscabs 13955  GIdcgi 27314  CVecOLDcvc 27383  NrmCVeccnv 27409   +𝑣 cpv 27410  BaseSetcba 27411   ·𝑠OLD cns 27412  0veccn0v 27413  normCVcnmcv 27415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-1st 7153  df-2nd 7154  df-vc 27384  df-nv 27417  df-va 27420  df-ba 27421  df-sm 27422  df-0v 27423  df-nmcv 27425
This theorem is referenced by:  nvvc  27440  nvf  27485  nvs  27488  nvz  27494  nvtri  27495
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