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Theorem nvs 27852
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 2770 . . . . . . 7 ( +𝑣𝑈) = ( +𝑣𝑈)
3 nvs.4 . . . . . . 7 𝑆 = ( ·𝑠OLD𝑈)
4 eqid 2770 . . . . . . 7 (0vec𝑈) = (0vec𝑈)
5 nvs.6 . . . . . . 7 𝑁 = (normCV𝑈)
61, 2, 3, 4, 5nvi 27803 . . . . . 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 1137 . . . . 5 (𝑈 ∈ NrmCVec → ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = (0vec𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥( +𝑣𝑈)𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
8 simp2 1130 . . . . . 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 3100 . . . . 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 6800 . . . . . . 7 (𝑥 = 𝐵 → (𝑦𝑆𝑥) = (𝑦𝑆𝐵))
1211fveq2d 6336 . . . . . 6 (𝑥 = 𝐵 → (𝑁‘(𝑦𝑆𝑥)) = (𝑁‘(𝑦𝑆𝐵)))
13 fveq2 6332 . . . . . . 7 (𝑥 = 𝐵 → (𝑁𝑥) = (𝑁𝐵))
1413oveq2d 6808 . . . . . 6 (𝑥 = 𝐵 → ((abs‘𝑦) · (𝑁𝑥)) = ((abs‘𝑦) · (𝑁𝐵)))
1512, 14eqeq12d 2785 . . . . 5 (𝑥 = 𝐵 → ((𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ↔ (𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁𝐵))))
16 fvoveq1 6815 . . . . . 6 (𝑦 = 𝐴 → (𝑁‘(𝑦𝑆𝐵)) = (𝑁‘(𝐴𝑆𝐵)))
17 fveq2 6332 . . . . . . 7 (𝑦 = 𝐴 → (abs‘𝑦) = (abs‘𝐴))
1817oveq1d 6807 . . . . . 6 (𝑦 = 𝐴 → ((abs‘𝑦) · (𝑁𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))
1916, 18eqeq12d 2785 . . . . 5 (𝑦 = 𝐴 → ((𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁𝐵)) ↔ (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵))))
2015, 19rspc2v 3470 . . . 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 1108 . 2 ((𝐵𝑋𝐴 ∈ ℂ ∧ 𝑈 ∈ NrmCVec) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))
23223com13 1117 1 ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  cop 4320   class class class wbr 4784  wf 6027  cfv 6031  (class class class)co 6792  cc 10135  cr 10136  0cc0 10137   + caddc 10140   · cmul 10142  cle 10276  abscabs 14181  CVecOLDcvc 27747  NrmCVeccnv 27773   +𝑣 cpv 27774  BaseSetcba 27775   ·𝑠OLD cns 27776  0veccn0v 27777  normCVcnmcv 27779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-1st 7314  df-2nd 7315  df-vc 27748  df-nv 27781  df-va 27784  df-ba 27785  df-sm 27786  df-0v 27787  df-nmcv 27789
This theorem is referenced by:  nvsge0  27853  nvm1  27854  nvpi  27856  nvmtri  27860  smcnlem  27886  ipidsq  27899  minvecolem2  28065
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