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| Description: The norm of a normed complex vector space expressed in terms of the distance function of its induced metric. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| nvnd.1 | ⊢ 𝑋 = (BaseSet‘𝑈) | 
| nvnd.5 | ⊢ 𝑍 = (0vec‘𝑈) | 
| nvnd.6 | ⊢ 𝑁 = (normCV‘𝑈) | 
| nvnd.8 | ⊢ 𝐷 = (IndMet‘𝑈) | 
| Ref | Expression | 
|---|---|
| nvnd | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐷𝑍)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nvnd.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 2 | nvnd.5 | . . . . 5 ⊢ 𝑍 = (0vec‘𝑈) | |
| 3 | 1, 2 | nvzcl 30653 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) | 
| 4 | 3 | adantr 480 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋) | 
| 5 | eqid 2737 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈) | |
| 6 | nvnd.6 | . . . 4 ⊢ 𝑁 = (normCV‘𝑈) | |
| 7 | nvnd.8 | . . . 4 ⊢ 𝐷 = (IndMet‘𝑈) | |
| 8 | 1, 5, 6, 7 | imsdval 30705 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝐴𝐷𝑍) = (𝑁‘(𝐴( −𝑣 ‘𝑈)𝑍))) | 
| 9 | 4, 8 | mpd3an3 1464 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝑍) = (𝑁‘(𝐴( −𝑣 ‘𝑈)𝑍))) | 
| 10 | eqid 2737 | . . . . . 6 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
| 11 | eqid 2737 | . . . . . 6 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
| 12 | 1, 10, 11, 5 | nvmval 30661 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝑍) = (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍))) | 
| 13 | 4, 12 | mpd3an3 1464 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝑍) = (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍))) | 
| 14 | neg1cn 12380 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
| 15 | 11, 2 | nvsz 30657 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ) → (-1( ·𝑠OLD ‘𝑈)𝑍) = 𝑍) | 
| 16 | 14, 15 | mpan2 691 | . . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (-1( ·𝑠OLD ‘𝑈)𝑍) = 𝑍) | 
| 17 | 16 | oveq2d 7447 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍)) = (𝐴( +𝑣 ‘𝑈)𝑍)) | 
| 18 | 17 | adantr 480 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍)) = (𝐴( +𝑣 ‘𝑈)𝑍)) | 
| 19 | 1, 10, 2 | nv0rid 30654 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)𝑍) = 𝐴) | 
| 20 | 13, 18, 19 | 3eqtrd 2781 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝑍) = 𝐴) | 
| 21 | 20 | fveq2d 6910 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( −𝑣 ‘𝑈)𝑍)) = (𝑁‘𝐴)) | 
| 22 | 9, 21 | eqtr2d 2778 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐷𝑍)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 1c1 11156 -cneg 11493 NrmCVeccnv 30603 +𝑣 cpv 30604 BaseSetcba 30605 ·𝑠OLD cns 30606 0veccn0v 30607 −𝑣 cnsb 30608 normCVcnmcv 30609 IndMetcims 30610 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 df-grpo 30512 df-gid 30513 df-ginv 30514 df-gdiv 30515 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-vs 30618 df-nmcv 30619 df-ims 30620 | 
| This theorem is referenced by: ubthlem1 30889 | 
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