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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 2820 | . . . . . . 7 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
3 | nvs.4 | . . . . . . 7 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
4 | eqid 2820 | . . . . . . 7 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
5 | nvs.6 | . . . . . . 7 ⊢ 𝑁 = (normCV‘𝑈) | |
6 | 1, 2, 3, 4, 5 | nvi 28389 | . . . . . 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 1139 | . . . . 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 1132 | . . . . . 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 3159 | . . . . 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 7157 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑦𝑆𝑥) = (𝑦𝑆𝐵)) | |
12 | 11 | fveq2d 6667 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑁‘(𝑦𝑆𝑥)) = (𝑁‘(𝑦𝑆𝐵))) |
13 | fveq2 6663 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑁‘𝑥) = (𝑁‘𝐵)) | |
14 | 13 | oveq2d 7165 | . . . . . 6 ⊢ (𝑥 = 𝐵 → ((abs‘𝑦) · (𝑁‘𝑥)) = ((abs‘𝑦) · (𝑁‘𝐵))) |
15 | 12, 14 | eqeq12d 2836 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ↔ (𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁‘𝐵)))) |
16 | fvoveq1 7172 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑁‘(𝑦𝑆𝐵)) = (𝑁‘(𝐴𝑆𝐵))) | |
17 | fveq2 6663 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (abs‘𝑦) = (abs‘𝐴)) | |
18 | 17 | oveq1d 7164 | . . . . . 6 ⊢ (𝑦 = 𝐴 → ((abs‘𝑦) · (𝑁‘𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) |
19 | 16, 18 | eqeq12d 2836 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁‘𝐵)) ↔ (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵)))) |
20 | 15, 19 | rspc2v 3630 | . . . 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 1112 | . 2 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝑈 ∈ NrmCVec) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) |
23 | 22 | 3com13 1119 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ∀wral 3137 〈cop 4566 class class class wbr 5059 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 ℂcc 10528 ℝcr 10529 0cc0 10530 + caddc 10533 · cmul 10535 ≤ cle 10669 abscabs 14588 CVecOLDcvc 28333 NrmCVeccnv 28359 +𝑣 cpv 28360 BaseSetcba 28361 ·𝑠OLD cns 28362 0veccn0v 28363 normCVcnmcv 28365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-1st 7682 df-2nd 7683 df-vc 28334 df-nv 28367 df-va 28370 df-ba 28371 df-sm 28372 df-0v 28373 df-nmcv 28375 |
This theorem is referenced by: nvsge0 28439 nvm1 28440 nvpi 28442 nvmtri 28446 smcnlem 28472 ipidsq 28485 minvecolem2 28650 |
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