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Mirrors > Home > MPE Home > Th. List > isnvi | Structured version Visualization version GIF version |
Description: Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
isnvi.5 | ⊢ 𝑋 = ran 𝐺 |
isnvi.6 | ⊢ 𝑍 = (GId‘𝐺) |
isnvi.7 | ⊢ 〈𝐺, 𝑆〉 ∈ CVecOLD |
isnvi.8 | ⊢ 𝑁:𝑋⟶ℝ |
isnvi.9 | ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑁‘𝑥) = 0) → 𝑥 = 𝑍) |
isnvi.10 | ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥))) |
isnvi.11 | ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) |
isnvi.12 | ⊢ 𝑈 = 〈〈𝐺, 𝑆〉, 𝑁〉 |
Ref | Expression |
---|---|
isnvi | ⊢ 𝑈 ∈ NrmCVec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnvi.12 | . 2 ⊢ 𝑈 = 〈〈𝐺, 𝑆〉, 𝑁〉 | |
2 | isnvi.7 | . . 3 ⊢ 〈𝐺, 𝑆〉 ∈ CVecOLD | |
3 | isnvi.8 | . . 3 ⊢ 𝑁:𝑋⟶ℝ | |
4 | isnvi.9 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑁‘𝑥) = 0) → 𝑥 = 𝑍) | |
5 | 4 | ex 449 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 → ((𝑁‘𝑥) = 0 → 𝑥 = 𝑍)) |
6 | isnvi.10 | . . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥))) | |
7 | 6 | ancoms 468 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ℂ) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥))) |
8 | 7 | ralrimiva 2949 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 → ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥))) |
9 | isnvi.11 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) | |
10 | 9 | ralrimiva 2949 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 → ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) |
11 | 5, 8, 10 | 3jca 1235 | . . . 4 ⊢ (𝑥 ∈ 𝑋 → (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))) |
12 | 11 | rgen 2906 | . . 3 ⊢ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) |
13 | isnvi.5 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
14 | isnvi.6 | . . . 4 ⊢ 𝑍 = (GId‘𝐺) | |
15 | 13, 14 | isnv 26851 | . . 3 ⊢ (〈〈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))))) |
16 | 2, 3, 12, 15 | mpbir3an 1237 | . 2 ⊢ 〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec |
17 | 1, 16 | eqeltri 2684 | 1 ⊢ 𝑈 ∈ NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 〈cop 4131 class class class wbr 4583 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 + caddc 9818 · cmul 9820 ≤ cle 9954 abscabs 13822 GIdcgi 26728 CVecOLDcvc 26797 NrmCVeccnv 26823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-vc 26798 df-nv 26831 |
This theorem is referenced by: cnnv 26916 hhnv 27406 hhssnv 27505 |
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