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Theorem nvvc 30540
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 2725 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
3 eqid 2725 . . 3 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
41, 2, 3nvvop 30534 . 2 (𝑈 ∈ NrmCVec → 𝑊 = ⟨( +𝑣𝑈), ( ·𝑠OLD𝑈)⟩)
5 eqid 2725 . . . 4 (BaseSet‘𝑈) = (BaseSet‘𝑈)
6 eqid 2725 . . . 4 (0vec𝑈) = (0vec𝑈)
7 eqid 2725 . . . 4 (normCV𝑈) = (normCV𝑈)
85, 2, 3, 6, 7nvi 30539 . . 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 1139 . 2 (𝑈 ∈ NrmCVec → ⟨( +𝑣𝑈), ( ·𝑠OLD𝑈)⟩ ∈ CVecOLD)
104, 9eqeltrd 2825 1 (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  wral 3050  cop 4638   class class class wbr 5152  wf 6549  cfv 6553  (class class class)co 7423  1st c1st 8000  cc 11152  cr 11153  0cc0 11154   + caddc 11157   · cmul 11159  cle 11295  abscabs 15234  CVecOLDcvc 30483  NrmCVeccnv 30509   +𝑣 cpv 30510  BaseSetcba 30511   ·𝑠OLD cns 30512  0veccn0v 30513  normCVcnmcv 30515
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7426  df-oprab 7427  df-1st 8002  df-2nd 8003  df-vc 30484  df-nv 30517  df-va 30520  df-ba 30521  df-sm 30522  df-0v 30523  df-nmcv 30525
This theorem is referenced by:  nvablo  30541  nvsf  30544  nvscl  30551  nvsid  30552  nvsass  30553  nvdi  30555  nvdir  30556  nv2  30557  nv0  30562  nvsz  30563  nvinv  30564  phop  30743  ip0i  30750  ipdirilem  30754  hlvc  30818
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