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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 28409 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋) |
5 | eqid 2820 | . . . 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 28461 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝐴𝐷𝑍) = (𝑁‘(𝐴( −𝑣 ‘𝑈)𝑍))) |
9 | 4, 8 | mpd3an3 1457 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝑍) = (𝑁‘(𝐴( −𝑣 ‘𝑈)𝑍))) |
10 | eqid 2820 | . . . . . 6 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
11 | eqid 2820 | . . . . . 6 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
12 | 1, 10, 11, 5 | nvmval 28417 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝑍) = (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍))) |
13 | 4, 12 | mpd3an3 1457 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝑍) = (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍))) |
14 | neg1cn 11745 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
15 | 11, 2 | nvsz 28413 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ) → (-1( ·𝑠OLD ‘𝑈)𝑍) = 𝑍) |
16 | 14, 15 | mpan2 689 | . . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (-1( ·𝑠OLD ‘𝑈)𝑍) = 𝑍) |
17 | 16 | oveq2d 7165 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍)) = (𝐴( +𝑣 ‘𝑈)𝑍)) |
18 | 17 | adantr 483 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍)) = (𝐴( +𝑣 ‘𝑈)𝑍)) |
19 | 1, 10, 2 | nv0rid 28410 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)𝑍) = 𝐴) |
20 | 13, 18, 19 | 3eqtrd 2859 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝑍) = 𝐴) |
21 | 20 | fveq2d 6667 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( −𝑣 ‘𝑈)𝑍)) = (𝑁‘𝐴)) |
22 | 9, 21 | eqtr2d 2856 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐷𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 ℂcc 10528 1c1 10531 -cneg 10864 NrmCVeccnv 28359 +𝑣 cpv 28360 BaseSetcba 28361 ·𝑠OLD cns 28362 0veccn0v 28363 −𝑣 cnsb 28364 normCVcnmcv 28365 IndMetcims 28366 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-1st 7682 df-2nd 7683 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-ltxr 10673 df-sub 10865 df-neg 10866 df-grpo 28268 df-gid 28269 df-ginv 28270 df-gdiv 28271 df-ablo 28320 df-vc 28334 df-nv 28367 df-va 28370 df-ba 28371 df-sm 28372 df-0v 28373 df-vs 28374 df-nmcv 28375 df-ims 28376 |
This theorem is referenced by: ubthlem1 28645 |
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