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Mirrors > Home > MPE Home > Th. List > nvs | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
nvs.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvs.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
nvs.6 | ⊢ 𝑁 = (normCV‘𝑈) |
Ref | Expression |
---|---|
nvs | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvs.1 | . . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | eqid 2725 | . . . . . . 7 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
3 | nvs.4 | . . . . . . 7 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
4 | eqid 2725 | . . . . . . 7 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
5 | nvs.6 | . . . . . . 7 ⊢ 𝑁 = (normCV‘𝑈) | |
6 | 1, 2, 3, 4, 5 | nvi 30516 | . . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (〈( +𝑣 ‘𝑈), 𝑆〉 ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = (0vec‘𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥( +𝑣 ‘𝑈)𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) |
7 | 6 | simp3d 1141 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = (0vec‘𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥( +𝑣 ‘𝑈)𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))) |
8 | simp2 1134 | . . . . . 6 ⊢ ((((𝑁‘𝑥) = 0 → 𝑥 = (0vec‘𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥( +𝑣 ‘𝑈)𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥))) | |
9 | 8 | ralimi 3072 | . . . . 5 ⊢ (∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = (0vec‘𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥( +𝑣 ‘𝑈)𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥))) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥))) |
11 | oveq2 7427 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑦𝑆𝑥) = (𝑦𝑆𝐵)) | |
12 | 11 | fveq2d 6900 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑁‘(𝑦𝑆𝑥)) = (𝑁‘(𝑦𝑆𝐵))) |
13 | fveq2 6896 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑁‘𝑥) = (𝑁‘𝐵)) | |
14 | 13 | oveq2d 7435 | . . . . . 6 ⊢ (𝑥 = 𝐵 → ((abs‘𝑦) · (𝑁‘𝑥)) = ((abs‘𝑦) · (𝑁‘𝐵))) |
15 | 12, 14 | eqeq12d 2741 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ↔ (𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁‘𝐵)))) |
16 | fvoveq1 7442 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑁‘(𝑦𝑆𝐵)) = (𝑁‘(𝐴𝑆𝐵))) | |
17 | fveq2 6896 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (abs‘𝑦) = (abs‘𝐴)) | |
18 | 17 | oveq1d 7434 | . . . . . 6 ⊢ (𝑦 = 𝐴 → ((abs‘𝑦) · (𝑁‘𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) |
19 | 16, 18 | eqeq12d 2741 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁‘𝐵)) ↔ (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵)))) |
20 | 15, 19 | rspc2v 3617 | . . . 4 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵)))) |
21 | 10, 20 | syl5 34 | . . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ) → (𝑈 ∈ NrmCVec → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵)))) |
22 | 21 | 3impia 1114 | . 2 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝑈 ∈ NrmCVec) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) |
23 | 22 | 3com13 1121 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3050 〈cop 4636 class class class wbr 5149 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ℂcc 11143 ℝcr 11144 0cc0 11145 + caddc 11148 · cmul 11150 ≤ cle 11286 abscabs 15225 CVecOLDcvc 30460 NrmCVeccnv 30486 +𝑣 cpv 30487 BaseSetcba 30488 ·𝑠OLD cns 30489 0veccn0v 30490 normCVcnmcv 30492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-1st 7994 df-2nd 7995 df-vc 30461 df-nv 30494 df-va 30497 df-ba 30498 df-sm 30499 df-0v 30500 df-nmcv 30502 |
This theorem is referenced by: nvsge0 30566 nvm1 30567 nvpi 30569 nvmtri 30573 smcnlem 30599 ipidsq 30612 minvecolem2 30777 |
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