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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 28971 | . 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 28976 | . . 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 1141 | . 2 ⊢ (𝑈 ∈ NrmCVec → 〈( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)〉 ∈ CVecOLD) |
10 | 4, 9 | eqeltrd 2839 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 〈cop 4567 class class class wbr 5074 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 1st c1st 7829 ℂcc 10869 ℝcr 10870 0cc0 10871 + caddc 10874 · cmul 10876 ≤ cle 11010 abscabs 14945 CVecOLDcvc 28920 NrmCVeccnv 28946 +𝑣 cpv 28947 BaseSetcba 28948 ·𝑠OLD cns 28949 0veccn0v 28950 normCVcnmcv 28952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-1st 7831 df-2nd 7832 df-vc 28921 df-nv 28954 df-va 28957 df-ba 28958 df-sm 28959 df-0v 28960 df-nmcv 28962 |
This theorem is referenced by: nvablo 28978 nvsf 28981 nvscl 28988 nvsid 28989 nvsass 28990 nvdi 28992 nvdir 28993 nv2 28994 nv0 28999 nvsz 29000 nvinv 29001 phop 29180 ip0i 29187 ipdirilem 29191 hlvc 29255 |
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