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Theorem nvvc 27331
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 2621 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
3 eqid 2621 . . 3 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
41, 2, 3nvvop 27325 . 2 (𝑈 ∈ NrmCVec → 𝑊 = ⟨( +𝑣𝑈), ( ·𝑠OLD𝑈)⟩)
5 eqid 2621 . . . 4 (BaseSet‘𝑈) = (BaseSet‘𝑈)
6 eqid 2621 . . . 4 (0vec𝑈) = (0vec𝑈)
7 eqid 2621 . . . 4 (normCV𝑈) = (normCV𝑈)
85, 2, 3, 6, 7nvi 27330 . . 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 1071 . 2 (𝑈 ∈ NrmCVec → ⟨( +𝑣𝑈), ( ·𝑠OLD𝑈)⟩ ∈ CVecOLD)
104, 9eqeltrd 2698 1 (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  wral 2907  cop 4156   class class class wbr 4615  wf 5845  cfv 5849  (class class class)co 6607  1st c1st 7114  cc 9881  cr 9882  0cc0 9883   + caddc 9886   · cmul 9888  cle 10022  abscabs 13911  CVecOLDcvc 27274  NrmCVeccnv 27300   +𝑣 cpv 27301  BaseSetcba 27302   ·𝑠OLD cns 27303  0veccn0v 27304  normCVcnmcv 27306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-1st 7116  df-2nd 7117  df-vc 27275  df-nv 27308  df-va 27311  df-ba 27312  df-sm 27313  df-0v 27314  df-nmcv 27316
This theorem is referenced by:  nvablo  27332  nvsf  27335  nvscl  27342  nvsid  27343  nvsass  27344  nvdi  27346  nvdir  27347  nv2  27348  nv0  27353  nvsz  27354  nvinv  27355  phop  27534  ip0i  27541  ipdirilem  27545  hlvc  27610
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