MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvs Structured version   Visualization version   GIF version

Theorem nvs 30692
Description: Proportionality property of the norm of a scalar product in a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvs.1 𝑋 = (BaseSet‘𝑈)
nvs.4 𝑆 = ( ·𝑠OLD𝑈)
nvs.6 𝑁 = (normCV𝑈)
Assertion
Ref Expression
nvs ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))

Proof of Theorem nvs
Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvs.1 . . . . . . 7 𝑋 = (BaseSet‘𝑈)
2 eqid 2735 . . . . . . 7 ( +𝑣𝑈) = ( +𝑣𝑈)
3 nvs.4 . . . . . . 7 𝑆 = ( ·𝑠OLD𝑈)
4 eqid 2735 . . . . . . 7 (0vec𝑈) = (0vec𝑈)
5 nvs.6 . . . . . . 7 𝑁 = (normCV𝑈)
61, 2, 3, 4, 5nvi 30643 . . . . . 6 (𝑈 ∈ NrmCVec → (⟨( +𝑣𝑈), 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = (0vec𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥( +𝑣𝑈)𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
76simp3d 1143 . . . . 5 (𝑈 ∈ NrmCVec → ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = (0vec𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥( +𝑣𝑈)𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
8 simp2 1136 . . . . . 6 ((((𝑁𝑥) = 0 → 𝑥 = (0vec𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥( +𝑣𝑈)𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))
98ralimi 3081 . . . . 5 (∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = (0vec𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥( +𝑣𝑈)𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))
107, 9syl 17 . . . 4 (𝑈 ∈ NrmCVec → ∀𝑥𝑋𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))
11 oveq2 7439 . . . . . . 7 (𝑥 = 𝐵 → (𝑦𝑆𝑥) = (𝑦𝑆𝐵))
1211fveq2d 6911 . . . . . 6 (𝑥 = 𝐵 → (𝑁‘(𝑦𝑆𝑥)) = (𝑁‘(𝑦𝑆𝐵)))
13 fveq2 6907 . . . . . . 7 (𝑥 = 𝐵 → (𝑁𝑥) = (𝑁𝐵))
1413oveq2d 7447 . . . . . 6 (𝑥 = 𝐵 → ((abs‘𝑦) · (𝑁𝑥)) = ((abs‘𝑦) · (𝑁𝐵)))
1512, 14eqeq12d 2751 . . . . 5 (𝑥 = 𝐵 → ((𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ↔ (𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁𝐵))))
16 fvoveq1 7454 . . . . . 6 (𝑦 = 𝐴 → (𝑁‘(𝑦𝑆𝐵)) = (𝑁‘(𝐴𝑆𝐵)))
17 fveq2 6907 . . . . . . 7 (𝑦 = 𝐴 → (abs‘𝑦) = (abs‘𝐴))
1817oveq1d 7446 . . . . . 6 (𝑦 = 𝐴 → ((abs‘𝑦) · (𝑁𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))
1916, 18eqeq12d 2751 . . . . 5 (𝑦 = 𝐴 → ((𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁𝐵)) ↔ (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵))))
2015, 19rspc2v 3633 . . . 4 ((𝐵𝑋𝐴 ∈ ℂ) → (∀𝑥𝑋𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵))))
2110, 20syl5 34 . . 3 ((𝐵𝑋𝐴 ∈ ℂ) → (𝑈 ∈ NrmCVec → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵))))
22213impia 1116 . 2 ((𝐵𝑋𝐴 ∈ ℂ ∧ 𝑈 ∈ NrmCVec) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))
23223com13 1123 1 ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  cop 4637   class class class wbr 5148  wf 6559  cfv 6563  (class class class)co 7431  cc 11151  cr 11152  0cc0 11153   + caddc 11156   · cmul 11158  cle 11294  abscabs 15270  CVecOLDcvc 30587  NrmCVeccnv 30613   +𝑣 cpv 30614  BaseSetcba 30615   ·𝑠OLD cns 30616  0veccn0v 30617  normCVcnmcv 30619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-1st 8013  df-2nd 8014  df-vc 30588  df-nv 30621  df-va 30624  df-ba 30625  df-sm 30626  df-0v 30627  df-nmcv 30629
This theorem is referenced by:  nvsge0  30693  nvm1  30694  nvpi  30696  nvmtri  30700  smcnlem  30726  ipidsq  30739  minvecolem2  30904
  Copyright terms: Public domain W3C validator