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Theorem nvvc 28550
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 2738 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
3 eqid 2738 . . 3 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
41, 2, 3nvvop 28544 . 2 (𝑈 ∈ NrmCVec → 𝑊 = ⟨( +𝑣𝑈), ( ·𝑠OLD𝑈)⟩)
5 eqid 2738 . . . 4 (BaseSet‘𝑈) = (BaseSet‘𝑈)
6 eqid 2738 . . . 4 (0vec𝑈) = (0vec𝑈)
7 eqid 2738 . . . 4 (normCV𝑈) = (normCV𝑈)
85, 2, 3, 6, 7nvi 28549 . . 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 1143 . 2 (𝑈 ∈ NrmCVec → ⟨( +𝑣𝑈), ( ·𝑠OLD𝑈)⟩ ∈ CVecOLD)
104, 9eqeltrd 2833 1 (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2114  wral 3053  cop 4522   class class class wbr 5030  wf 6335  cfv 6339  (class class class)co 7170  1st c1st 7712  cc 10613  cr 10614  0cc0 10615   + caddc 10618   · cmul 10620  cle 10754  abscabs 14683  CVecOLDcvc 28493  NrmCVeccnv 28519   +𝑣 cpv 28520  BaseSetcba 28521   ·𝑠OLD cns 28522  0veccn0v 28523  normCVcnmcv 28525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-1st 7714  df-2nd 7715  df-vc 28494  df-nv 28527  df-va 28530  df-ba 28531  df-sm 28532  df-0v 28533  df-nmcv 28535
This theorem is referenced by:  nvablo  28551  nvsf  28554  nvscl  28561  nvsid  28562  nvsass  28563  nvdi  28565  nvdir  28566  nv2  28567  nv0  28572  nvsz  28573  nvinv  28574  phop  28753  ip0i  28760  ipdirilem  28764  hlvc  28828
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