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Theorem nvs 28035
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 2797 . . . . . . 7 ( +𝑣𝑈) = ( +𝑣𝑈)
3 nvs.4 . . . . . . 7 𝑆 = ( ·𝑠OLD𝑈)
4 eqid 2797 . . . . . . 7 (0vec𝑈) = (0vec𝑈)
5 nvs.6 . . . . . . 7 𝑁 = (normCV𝑈)
61, 2, 3, 4, 5nvi 27986 . . . . . 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 1175 . . . . 5 (𝑈 ∈ NrmCVec → ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = (0vec𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥( +𝑣𝑈)𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
8 simp2 1168 . . . . . 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 3131 . . . . 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 6884 . . . . . . 7 (𝑥 = 𝐵 → (𝑦𝑆𝑥) = (𝑦𝑆𝐵))
1211fveq2d 6413 . . . . . 6 (𝑥 = 𝐵 → (𝑁‘(𝑦𝑆𝑥)) = (𝑁‘(𝑦𝑆𝐵)))
13 fveq2 6409 . . . . . . 7 (𝑥 = 𝐵 → (𝑁𝑥) = (𝑁𝐵))
1413oveq2d 6892 . . . . . 6 (𝑥 = 𝐵 → ((abs‘𝑦) · (𝑁𝑥)) = ((abs‘𝑦) · (𝑁𝐵)))
1512, 14eqeq12d 2812 . . . . 5 (𝑥 = 𝐵 → ((𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ↔ (𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁𝐵))))
16 fvoveq1 6899 . . . . . 6 (𝑦 = 𝐴 → (𝑁‘(𝑦𝑆𝐵)) = (𝑁‘(𝐴𝑆𝐵)))
17 fveq2 6409 . . . . . . 7 (𝑦 = 𝐴 → (abs‘𝑦) = (abs‘𝐴))
1817oveq1d 6891 . . . . . 6 (𝑦 = 𝐴 → ((abs‘𝑦) · (𝑁𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))
1916, 18eqeq12d 2812 . . . . 5 (𝑦 = 𝐴 → ((𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁𝐵)) ↔ (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵))))
2015, 19rspc2v 3508 . . . 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 1146 . 2 ((𝐵𝑋𝐴 ∈ ℂ ∧ 𝑈 ∈ NrmCVec) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))
23223com13 1155 1 ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3087  cop 4372   class class class wbr 4841  wf 6095  cfv 6099  (class class class)co 6876  cc 10220  cr 10221  0cc0 10222   + caddc 10225   · cmul 10227  cle 10362  abscabs 14312  CVecOLDcvc 27930  NrmCVeccnv 27956   +𝑣 cpv 27957  BaseSetcba 27958   ·𝑠OLD cns 27959  0veccn0v 27960  normCVcnmcv 27962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-1st 7399  df-2nd 7400  df-vc 27931  df-nv 27964  df-va 27967  df-ba 27968  df-sm 27969  df-0v 27970  df-nmcv 27972
This theorem is referenced by:  nvsge0  28036  nvm1  28037  nvpi  28039  nvmtri  28043  smcnlem  28069  ipidsq  28082  minvecolem2  28248
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