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| Description: Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| nvf.1 | ⊢ 𝑋 = (BaseSet‘𝑈) | 
| nvf.6 | ⊢ 𝑁 = (normCV‘𝑈) | 
| Ref | Expression | 
|---|---|
| nvf | ⊢ (𝑈 ∈ NrmCVec → 𝑁:𝑋⟶ℝ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nvf.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 2 | eqid 2736 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
| 3 | eqid 2736 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
| 4 | eqid 2736 | . . 3 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
| 5 | nvf.6 | . . 3 ⊢ 𝑁 = (normCV‘𝑈) | |
| 6 | 1, 2, 3, 4, 5 | nvi 30634 | . 2 ⊢ (𝑈 ∈ NrmCVec → (〈( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)〉 ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = (0vec‘𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦( ·𝑠OLD ‘𝑈)𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥( +𝑣 ‘𝑈)𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) | 
| 7 | 6 | simp2d 1143 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑁:𝑋⟶ℝ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 〈cop 4631 class class class wbr 5142 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ℝcr 11155 0cc0 11156 + caddc 11159 · cmul 11161 ≤ cle 11297 abscabs 15274 CVecOLDcvc 30578 NrmCVeccnv 30604 +𝑣 cpv 30605 BaseSetcba 30606 ·𝑠OLD cns 30607 0veccn0v 30608 normCVcnmcv 30610 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-1st 8015 df-2nd 8016 df-vc 30579 df-nv 30612 df-va 30615 df-ba 30616 df-sm 30617 df-0v 30618 df-nmcv 30620 | 
| This theorem is referenced by: nvcl 30681 imsdf 30709 nmcvcn 30715 sspn 30756 hilnormi 31183 | 
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