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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 2738 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
3 | eqid 2738 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
4 | 1, 2, 3 | nvvop 28544 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = 〈( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)〉) |
5 | eqid 2738 | . . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
6 | eqid 2738 | . . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
7 | eqid 2738 | . . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
8 | 5, 2, 3, 6, 7 | nvi 28549 | . . 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 1143 | . 2 ⊢ (𝑈 ∈ NrmCVec → 〈( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)〉 ∈ CVecOLD) |
10 | 4, 9 | eqeltrd 2833 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3053 〈cop 4522 class class class wbr 5030 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 1st c1st 7712 ℂcc 10613 ℝcr 10614 0cc0 10615 + caddc 10618 · cmul 10620 ≤ cle 10754 abscabs 14683 CVecOLDcvc 28493 NrmCVeccnv 28519 +𝑣 cpv 28520 BaseSetcba 28521 ·𝑠OLD cns 28522 0veccn0v 28523 normCVcnmcv 28525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-oprab 7174 df-1st 7714 df-2nd 7715 df-vc 28494 df-nv 28527 df-va 28530 df-ba 28531 df-sm 28532 df-0v 28533 df-nmcv 28535 |
This theorem is referenced by: nvablo 28551 nvsf 28554 nvscl 28561 nvsid 28562 nvsass 28563 nvdi 28565 nvdir 28566 nv2 28567 nv0 28572 nvsz 28573 nvinv 28574 phop 28753 ip0i 28760 ipdirilem 28764 hlvc 28828 |
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