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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 2825 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
3 | eqid 2825 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
4 | 1, 2, 3 | nvvop 28008 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = 〈( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)〉) |
5 | eqid 2825 | . . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
6 | eqid 2825 | . . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
7 | eqid 2825 | . . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
8 | 5, 2, 3, 6, 7 | nvi 28013 | . . 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 1176 | . 2 ⊢ (𝑈 ∈ NrmCVec → 〈( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)〉 ∈ CVecOLD) |
10 | 4, 9 | eqeltrd 2906 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ∀wral 3117 〈cop 4403 class class class wbr 4873 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 1st c1st 7426 ℂcc 10250 ℝcr 10251 0cc0 10252 + caddc 10255 · cmul 10257 ≤ cle 10392 abscabs 14351 CVecOLDcvc 27957 NrmCVeccnv 27983 +𝑣 cpv 27984 BaseSetcba 27985 ·𝑠OLD cns 27986 0veccn0v 27987 normCVcnmcv 27989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-1st 7428 df-2nd 7429 df-vc 27958 df-nv 27991 df-va 27994 df-ba 27995 df-sm 27996 df-0v 27997 df-nmcv 27999 |
This theorem is referenced by: nvablo 28015 nvsf 28018 nvscl 28025 nvsid 28026 nvsass 28027 nvdi 28029 nvdir 28030 nv2 28031 nv0 28036 nvsz 28037 nvinv 28038 phop 28217 ip0i 28224 ipdirilem 28228 hlvc 28293 |
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