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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 2737 | . . . . . . 7 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
| 3 | nvs.4 | . . . . . . 7 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
| 4 | eqid 2737 | . . . . . . 7 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
| 5 | nvs.6 | . . . . . . 7 ⊢ 𝑁 = (normCV‘𝑈) | |
| 6 | 1, 2, 3, 4, 5 | nvi 30633 | . . . . . 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 1145 | . . . . 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 1138 | . . . . . 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 3083 | . . . . 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 7439 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑦𝑆𝑥) = (𝑦𝑆𝐵)) | |
| 12 | 11 | fveq2d 6910 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑁‘(𝑦𝑆𝑥)) = (𝑁‘(𝑦𝑆𝐵))) |
| 13 | fveq2 6906 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑁‘𝑥) = (𝑁‘𝐵)) | |
| 14 | 13 | oveq2d 7447 | . . . . . 6 ⊢ (𝑥 = 𝐵 → ((abs‘𝑦) · (𝑁‘𝑥)) = ((abs‘𝑦) · (𝑁‘𝐵))) |
| 15 | 12, 14 | eqeq12d 2753 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ↔ (𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁‘𝐵)))) |
| 16 | fvoveq1 7454 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑁‘(𝑦𝑆𝐵)) = (𝑁‘(𝐴𝑆𝐵))) | |
| 17 | fveq2 6906 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (abs‘𝑦) = (abs‘𝐴)) | |
| 18 | 17 | oveq1d 7446 | . . . . . 6 ⊢ (𝑦 = 𝐴 → ((abs‘𝑦) · (𝑁‘𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) |
| 19 | 16, 18 | eqeq12d 2753 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁‘𝐵)) ↔ (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵)))) |
| 20 | 15, 19 | rspc2v 3633 | . . . 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 1118 | . 2 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝑈 ∈ NrmCVec) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) |
| 23 | 22 | 3com13 1125 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 〈cop 4632 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 + caddc 11158 · cmul 11160 ≤ cle 11296 abscabs 15273 CVecOLDcvc 30577 NrmCVeccnv 30603 +𝑣 cpv 30604 BaseSetcba 30605 ·𝑠OLD cns 30606 0veccn0v 30607 normCVcnmcv 30609 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-1st 8014 df-2nd 8015 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-nmcv 30619 |
| This theorem is referenced by: nvsge0 30683 nvm1 30684 nvpi 30686 nvmtri 30690 smcnlem 30716 ipidsq 30729 minvecolem2 30894 |
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