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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 2798 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
3 | eqid 2798 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
4 | 1, 2, 3 | nvvop 28392 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = 〈( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)〉) |
5 | eqid 2798 | . . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
6 | eqid 2798 | . . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
7 | eqid 2798 | . . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
8 | 5, 2, 3, 6, 7 | nvi 28397 | . . 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 1139 | . 2 ⊢ (𝑈 ∈ NrmCVec → 〈( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)〉 ∈ CVecOLD) |
10 | 4, 9 | eqeltrd 2890 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 〈cop 4531 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 ℂcc 10524 ℝcr 10525 0cc0 10526 + caddc 10529 · cmul 10531 ≤ cle 10665 abscabs 14585 CVecOLDcvc 28341 NrmCVeccnv 28367 +𝑣 cpv 28368 BaseSetcba 28369 ·𝑠OLD cns 28370 0veccn0v 28371 normCVcnmcv 28373 |
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-pow 5231 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-1st 7671 df-2nd 7672 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-nmcv 28383 |
This theorem is referenced by: nvablo 28399 nvsf 28402 nvscl 28409 nvsid 28410 nvsass 28411 nvdi 28413 nvdir 28414 nv2 28415 nv0 28420 nvsz 28421 nvinv 28422 phop 28601 ip0i 28608 ipdirilem 28612 hlvc 28676 |
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