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Theorem nvs 27364
 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 2621 . . . . . . 7 ( +𝑣𝑈) = ( +𝑣𝑈)
3 nvs.4 . . . . . . 7 𝑆 = ( ·𝑠OLD𝑈)
4 eqid 2621 . . . . . . 7 (0vec𝑈) = (0vec𝑈)
5 nvs.6 . . . . . . 7 𝑁 = (normCV𝑈)
61, 2, 3, 4, 5nvi 27315 . . . . . 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 1073 . . . . 5 (𝑈 ∈ NrmCVec → ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = (0vec𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥( +𝑣𝑈)𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
8 simp2 1060 . . . . . 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 2947 . . . . 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 6612 . . . . . . 7 (𝑥 = 𝐵 → (𝑦𝑆𝑥) = (𝑦𝑆𝐵))
1211fveq2d 6152 . . . . . 6 (𝑥 = 𝐵 → (𝑁‘(𝑦𝑆𝑥)) = (𝑁‘(𝑦𝑆𝐵)))
13 fveq2 6148 . . . . . . 7 (𝑥 = 𝐵 → (𝑁𝑥) = (𝑁𝐵))
1413oveq2d 6620 . . . . . 6 (𝑥 = 𝐵 → ((abs‘𝑦) · (𝑁𝑥)) = ((abs‘𝑦) · (𝑁𝐵)))
1512, 14eqeq12d 2636 . . . . 5 (𝑥 = 𝐵 → ((𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ↔ (𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁𝐵))))
16 oveq1 6611 . . . . . . 7 (𝑦 = 𝐴 → (𝑦𝑆𝐵) = (𝐴𝑆𝐵))
1716fveq2d 6152 . . . . . 6 (𝑦 = 𝐴 → (𝑁‘(𝑦𝑆𝐵)) = (𝑁‘(𝐴𝑆𝐵)))
18 fveq2 6148 . . . . . . 7 (𝑦 = 𝐴 → (abs‘𝑦) = (abs‘𝐴))
1918oveq1d 6619 . . . . . 6 (𝑦 = 𝐴 → ((abs‘𝑦) · (𝑁𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))
2017, 19eqeq12d 2636 . . . . 5 (𝑦 = 𝐴 → ((𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁𝐵)) ↔ (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵))))
2115, 20rspc2v 3306 . . . 4 ((𝐵𝑋𝐴 ∈ ℂ) → (∀𝑥𝑋𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵))))
2210, 21syl5 34 . . 3 ((𝐵𝑋𝐴 ∈ ℂ) → (𝑈 ∈ NrmCVec → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵))))
23223impia 1258 . 2 ((𝐵𝑋𝐴 ∈ ℂ ∧ 𝑈 ∈ NrmCVec) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))
24233com13 1267 1 ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ⟨cop 4154   class class class wbr 4613  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  ℂcc 9878  ℝcr 9879  0cc0 9880   + caddc 9883   · cmul 9885   ≤ cle 10019  abscabs 13908  CVecOLDcvc 27259  NrmCVeccnv 27285   +𝑣 cpv 27286  BaseSetcba 27287   ·𝑠OLD cns 27288  0veccn0v 27289  normCVcnmcv 27291 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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 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 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-1st 7113  df-2nd 7114  df-vc 27260  df-nv 27293  df-va 27296  df-ba 27297  df-sm 27298  df-0v 27299  df-nmcv 27301 This theorem is referenced by:  nvsge0  27365  nvm1  27366  nvpi  27368  nvmtri  27372  smcnlem  27398  ipidsq  27411  minvecolem2  27577
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