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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 2797 | . . . . . . 7 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
3 | nvs.4 | . . . . . . 7 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
4 | eqid 2797 | . . . . . . 7 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
5 | nvs.6 | . . . . . . 7 ⊢ 𝑁 = (normCV‘𝑈) | |
6 | 1, 2, 3, 4, 5 | nvi 27986 | . . . . . 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 1175 | . . . . 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 1168 | . . . . . 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 3131 | . . . . 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 6884 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑦𝑆𝑥) = (𝑦𝑆𝐵)) | |
12 | 11 | fveq2d 6413 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑁‘(𝑦𝑆𝑥)) = (𝑁‘(𝑦𝑆𝐵))) |
13 | fveq2 6409 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑁‘𝑥) = (𝑁‘𝐵)) | |
14 | 13 | oveq2d 6892 | . . . . . 6 ⊢ (𝑥 = 𝐵 → ((abs‘𝑦) · (𝑁‘𝑥)) = ((abs‘𝑦) · (𝑁‘𝐵))) |
15 | 12, 14 | eqeq12d 2812 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ↔ (𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁‘𝐵)))) |
16 | fvoveq1 6899 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑁‘(𝑦𝑆𝐵)) = (𝑁‘(𝐴𝑆𝐵))) | |
17 | fveq2 6409 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (abs‘𝑦) = (abs‘𝐴)) | |
18 | 17 | oveq1d 6891 | . . . . . 6 ⊢ (𝑦 = 𝐴 → ((abs‘𝑦) · (𝑁‘𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) |
19 | 16, 18 | eqeq12d 2812 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁‘𝐵)) ↔ (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵)))) |
20 | 15, 19 | rspc2v 3508 | . . . 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 1146 | . 2 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝑈 ∈ NrmCVec) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) |
23 | 22 | 3com13 1155 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) |
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 ℂcc 10220 ℝcr 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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