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Mirrors > Home > MPE Home > Th. List > isnv | Structured version Visualization version GIF version |
Description: The predicate "is a normed complex vector space." (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
isnv.1 | ⊢ 𝑋 = ran 𝐺 |
isnv.2 | ⊢ 𝑍 = (GId‘𝐺) |
Ref | Expression |
---|---|
isnv | ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec ↔ (〈𝐺, 𝑆〉 ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvex 28394 | . 2 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V)) | |
2 | vcex 28361 | . . . . 5 ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V)) | |
3 | 2 | adantr 484 | . . . 4 ⊢ ((〈𝐺, 𝑆〉 ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ) → (𝐺 ∈ V ∧ 𝑆 ∈ V)) |
4 | isnv.1 | . . . . . . 7 ⊢ 𝑋 = ran 𝐺 | |
5 | 2 | simpld 498 | . . . . . . . 8 ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD → 𝐺 ∈ V) |
6 | rnexg 7595 | . . . . . . . 8 ⊢ (𝐺 ∈ V → ran 𝐺 ∈ V) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD → ran 𝐺 ∈ V) |
8 | 4, 7 | eqeltrid 2894 | . . . . . 6 ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD → 𝑋 ∈ V) |
9 | fex 6966 | . . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V) → 𝑁 ∈ V) | |
10 | 8, 9 | sylan2 595 | . . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ 〈𝐺, 𝑆〉 ∈ CVecOLD) → 𝑁 ∈ V) |
11 | 10 | ancoms 462 | . . . 4 ⊢ ((〈𝐺, 𝑆〉 ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ) → 𝑁 ∈ V) |
12 | df-3an 1086 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ↔ ((𝐺 ∈ V ∧ 𝑆 ∈ V) ∧ 𝑁 ∈ V)) | |
13 | 3, 11, 12 | sylanbrc 586 | . . 3 ⊢ ((〈𝐺, 𝑆〉 ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ) → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V)) |
14 | 13 | 3adant3 1129 | . 2 ⊢ ((〈𝐺, 𝑆〉 ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))) → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V)) |
15 | isnv.2 | . . 3 ⊢ 𝑍 = (GId‘𝐺) | |
16 | 4, 15 | isnvlem 28393 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) → (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec ↔ (〈𝐺, 𝑆〉 ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))) |
17 | 1, 14, 16 | pm5.21nii 383 | 1 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec ↔ (〈𝐺, 𝑆〉 ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 〈cop 4531 class class class wbr 5030 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 + caddc 10529 · cmul 10531 ≤ cle 10665 abscabs 14585 GIdcgi 28273 CVecOLDcvc 28341 NrmCVeccnv 28367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-vc 28342 df-nv 28375 |
This theorem is referenced by: isnvi 28396 nvi 28397 |
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