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Theorem nvvc 28319
Description: The vector space component of a normed complex vector space. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
nvvc.1 𝑊 = (1st𝑈)
Assertion
Ref Expression
nvvc (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)

Proof of Theorem nvvc
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvvc.1 . . 3 𝑊 = (1st𝑈)
2 eqid 2818 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
3 eqid 2818 . . 3 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
41, 2, 3nvvop 28313 . 2 (𝑈 ∈ NrmCVec → 𝑊 = ⟨( +𝑣𝑈), ( ·𝑠OLD𝑈)⟩)
5 eqid 2818 . . . 4 (BaseSet‘𝑈) = (BaseSet‘𝑈)
6 eqid 2818 . . . 4 (0vec𝑈) = (0vec𝑈)
7 eqid 2818 . . . 4 (normCV𝑈) = (normCV𝑈)
85, 2, 3, 6, 7nvi 28318 . . 3 (𝑈 ∈ NrmCVec → (⟨( +𝑣𝑈), ( ·𝑠OLD𝑈)⟩ ∈ CVecOLD ∧ (normCV𝑈):(BaseSet‘𝑈)⟶ℝ ∧ ∀𝑥 ∈ (BaseSet‘𝑈)((((normCV𝑈)‘𝑥) = 0 → 𝑥 = (0vec𝑈)) ∧ ∀𝑦 ∈ ℂ ((normCV𝑈)‘(𝑦( ·𝑠OLD𝑈)𝑥)) = ((abs‘𝑦) · ((normCV𝑈)‘𝑥)) ∧ ∀𝑦 ∈ (BaseSet‘𝑈)((normCV𝑈)‘(𝑥( +𝑣𝑈)𝑦)) ≤ (((normCV𝑈)‘𝑥) + ((normCV𝑈)‘𝑦)))))
98simp1d 1134 . 2 (𝑈 ∈ NrmCVec → ⟨( +𝑣𝑈), ( ·𝑠OLD𝑈)⟩ ∈ CVecOLD)
104, 9eqeltrd 2910 1 (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079   = wceq 1528  wcel 2105  wral 3135  cop 4563   class class class wbr 5057  wf 6344  cfv 6348  (class class class)co 7145  1st c1st 7676  cc 10523  cr 10524  0cc0 10525   + caddc 10528   · cmul 10530  cle 10664  abscabs 14581  CVecOLDcvc 28262  NrmCVeccnv 28288   +𝑣 cpv 28289  BaseSetcba 28290   ·𝑠OLD cns 28291  0veccn0v 28292  normCVcnmcv 28294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-1st 7678  df-2nd 7679  df-vc 28263  df-nv 28296  df-va 28299  df-ba 28300  df-sm 28301  df-0v 28302  df-nmcv 28304
This theorem is referenced by:  nvablo  28320  nvsf  28323  nvscl  28330  nvsid  28331  nvsass  28332  nvdi  28334  nvdir  28335  nv2  28336  nv0  28341  nvsz  28342  nvinv  28343  phop  28522  ip0i  28529  ipdirilem  28533  hlvc  28597
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