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Mirrors > Home > MPE Home > Th. List > nvnd | Structured version Visualization version GIF version |
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 28040 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
4 | 3 | adantr 474 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋) |
5 | eqid 2825 | . . . 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 28092 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝐴𝐷𝑍) = (𝑁‘(𝐴( −𝑣 ‘𝑈)𝑍))) |
9 | 4, 8 | mpd3an3 1590 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝑍) = (𝑁‘(𝐴( −𝑣 ‘𝑈)𝑍))) |
10 | eqid 2825 | . . . . . 6 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
11 | eqid 2825 | . . . . . 6 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
12 | 1, 10, 11, 5 | nvmval 28048 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝑍) = (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍))) |
13 | 4, 12 | mpd3an3 1590 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝑍) = (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍))) |
14 | neg1cn 11479 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
15 | 11, 2 | nvsz 28044 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ) → (-1( ·𝑠OLD ‘𝑈)𝑍) = 𝑍) |
16 | 14, 15 | mpan2 682 | . . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (-1( ·𝑠OLD ‘𝑈)𝑍) = 𝑍) |
17 | 16 | oveq2d 6926 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍)) = (𝐴( +𝑣 ‘𝑈)𝑍)) |
18 | 17 | adantr 474 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍)) = (𝐴( +𝑣 ‘𝑈)𝑍)) |
19 | 1, 10, 2 | nv0rid 28041 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)𝑍) = 𝐴) |
20 | 13, 18, 19 | 3eqtrd 2865 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝑍) = 𝐴) |
21 | 20 | fveq2d 6441 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( −𝑣 ‘𝑈)𝑍)) = (𝑁‘𝐴)) |
22 | 9, 21 | eqtr2d 2862 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐷𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 ℂcc 10257 1c1 10260 -cneg 10593 NrmCVeccnv 27990 +𝑣 cpv 27991 BaseSetcba 27992 ·𝑠OLD cns 27993 0veccn0v 27994 −𝑣 cnsb 27995 normCVcnmcv 27996 IndMetcims 27997 |
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 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 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-nel 3103 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 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-ltxr 10403 df-sub 10594 df-neg 10595 df-grpo 27899 df-gid 27900 df-ginv 27901 df-gdiv 27902 df-ablo 27951 df-vc 27965 df-nv 27998 df-va 28001 df-ba 28002 df-sm 28003 df-0v 28004 df-vs 28005 df-nmcv 28006 df-ims 28007 |
This theorem is referenced by: ubthlem1 28277 |
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