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Theorem nvvc 28308
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 2825 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
3 eqid 2825 . . 3 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
41, 2, 3nvvop 28302 . 2 (𝑈 ∈ NrmCVec → 𝑊 = ⟨( +𝑣𝑈), ( ·𝑠OLD𝑈)⟩)
5 eqid 2825 . . . 4 (BaseSet‘𝑈) = (BaseSet‘𝑈)
6 eqid 2825 . . . 4 (0vec𝑈) = (0vec𝑈)
7 eqid 2825 . . . 4 (normCV𝑈) = (normCV𝑈)
85, 2, 3, 6, 7nvi 28307 . . 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 1136 . 2 (𝑈 ∈ NrmCVec → ⟨( +𝑣𝑈), ( ·𝑠OLD𝑈)⟩ ∈ CVecOLD)
104, 9eqeltrd 2917 1 (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081   = wceq 1530  wcel 2107  wral 3142  cop 4569   class class class wbr 5062  wf 6347  cfv 6351  (class class class)co 7151  1st c1st 7681  cc 10527  cr 10528  0cc0 10529   + caddc 10532   · cmul 10534  cle 10668  abscabs 14586  CVecOLDcvc 28251  NrmCVeccnv 28277   +𝑣 cpv 28278  BaseSetcba 28279   ·𝑠OLD cns 28280  0veccn0v 28281  normCVcnmcv 28283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-1st 7683  df-2nd 7684  df-vc 28252  df-nv 28285  df-va 28288  df-ba 28289  df-sm 28290  df-0v 28291  df-nmcv 28293
This theorem is referenced by:  nvablo  28309  nvsf  28312  nvscl  28319  nvsid  28320  nvsass  28321  nvdi  28323  nvdir  28324  nv2  28325  nv0  28330  nvsz  28331  nvinv  28332  phop  28511  ip0i  28518  ipdirilem  28522  hlvc  28586
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