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Theorem nvf 28535
 Description: Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvf.1 𝑋 = (BaseSet‘𝑈)
nvf.6 𝑁 = (normCV𝑈)
Assertion
Ref Expression
nvf (𝑈 ∈ NrmCVec → 𝑁:𝑋⟶ℝ)

Proof of Theorem nvf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvf.1 . . 3 𝑋 = (BaseSet‘𝑈)
2 eqid 2759 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
3 eqid 2759 . . 3 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
4 eqid 2759 . . 3 (0vec𝑈) = (0vec𝑈)
5 nvf.6 . . 3 𝑁 = (normCV𝑈)
61, 2, 3, 4, 5nvi 28489 . 2 (𝑈 ∈ NrmCVec → (⟨( +𝑣𝑈), ( ·𝑠OLD𝑈)⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = (0vec𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦( ·𝑠OLD𝑈)𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥( +𝑣𝑈)𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
76simp2d 1141 1 (𝑈 ∈ NrmCVec → 𝑁:𝑋⟶ℝ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ⟨cop 4529   class class class wbr 5033  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151  ℂcc 10566  ℝcr 10567  0cc0 10568   + caddc 10571   · cmul 10573   ≤ cle 10707  abscabs 14634  CVecOLDcvc 28433  NrmCVeccnv 28459   +𝑣 cpv 28460  BaseSetcba 28461   ·𝑠OLD cns 28462  0veccn0v 28463  normCVcnmcv 28465 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  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 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-1st 7694  df-2nd 7695  df-vc 28434  df-nv 28467  df-va 28470  df-ba 28471  df-sm 28472  df-0v 28473  df-nmcv 28475 This theorem is referenced by:  nvcl  28536  imsdf  28564  nmcvcn  28570  sspn  28611  hilnormi  29038
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