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Mirrors > Home > MPE Home > Th. List > nvvc | Structured version Visualization version GIF version |
Description: The vector space component of a normed complex vector space. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvvc.1 | ⊢ 𝑊 = (1st ‘𝑈) |
Ref | Expression |
---|---|
nvvc | ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvvc.1 | . . 3 ⊢ 𝑊 = (1st ‘𝑈) | |
2 | eqid 2818 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
3 | eqid 2818 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
4 | 1, 2, 3 | nvvop 28313 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = 〈( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)〉) |
5 | eqid 2818 | . . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
6 | eqid 2818 | . . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
7 | eqid 2818 | . . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
8 | 5, 2, 3, 6, 7 | nvi 28318 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (〈( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)〉 ∈ CVecOLD ∧ (normCV‘𝑈):(BaseSet‘𝑈)⟶ℝ ∧ ∀𝑥 ∈ (BaseSet‘𝑈)((((normCV‘𝑈)‘𝑥) = 0 → 𝑥 = (0vec‘𝑈)) ∧ ∀𝑦 ∈ ℂ ((normCV‘𝑈)‘(𝑦( ·𝑠OLD ‘𝑈)𝑥)) = ((abs‘𝑦) · ((normCV‘𝑈)‘𝑥)) ∧ ∀𝑦 ∈ (BaseSet‘𝑈)((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)𝑦)) ≤ (((normCV‘𝑈)‘𝑥) + ((normCV‘𝑈)‘𝑦))))) |
9 | 8 | simp1d 1134 | . 2 ⊢ (𝑈 ∈ NrmCVec → 〈( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)〉 ∈ CVecOLD) |
10 | 4, 9 | eqeltrd 2910 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 〈cop 4563 class class class wbr 5057 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 ℂcc 10523 ℝcr 10524 0cc0 10525 + caddc 10528 · cmul 10530 ≤ cle 10664 abscabs 14581 CVecOLDcvc 28262 NrmCVeccnv 28288 +𝑣 cpv 28289 BaseSetcba 28290 ·𝑠OLD cns 28291 0veccn0v 28292 normCVcnmcv 28294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 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 7148 df-oprab 7149 df-1st 7678 df-2nd 7679 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-nmcv 28304 |
This theorem is referenced by: nvablo 28320 nvsf 28323 nvscl 28330 nvsid 28331 nvsass 28332 nvdi 28334 nvdir 28335 nv2 28336 nv0 28341 nvsz 28342 nvinv 28343 phop 28522 ip0i 28529 ipdirilem 28533 hlvc 28597 |
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