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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 30559 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
4 | 3 | adantr 479 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋) |
5 | eqid 2725 | . . . 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 30611 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝐴𝐷𝑍) = (𝑁‘(𝐴( −𝑣 ‘𝑈)𝑍))) |
9 | 4, 8 | mpd3an3 1458 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝑍) = (𝑁‘(𝐴( −𝑣 ‘𝑈)𝑍))) |
10 | eqid 2725 | . . . . . 6 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
11 | eqid 2725 | . . . . . 6 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
12 | 1, 10, 11, 5 | nvmval 30567 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝑍) = (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍))) |
13 | 4, 12 | mpd3an3 1458 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝑍) = (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍))) |
14 | neg1cn 12373 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
15 | 11, 2 | nvsz 30563 | . . . . . . 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 7439 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍)) = (𝐴( +𝑣 ‘𝑈)𝑍)) |
18 | 17 | adantr 479 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑍)) = (𝐴( +𝑣 ‘𝑈)𝑍)) |
19 | 1, 10, 2 | nv0rid 30560 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)𝑍) = 𝐴) |
20 | 13, 18, 19 | 3eqtrd 2769 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝑍) = 𝐴) |
21 | 20 | fveq2d 6904 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( −𝑣 ‘𝑈)𝑍)) = (𝑁‘𝐴)) |
22 | 9, 21 | eqtr2d 2766 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐷𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7423 ℂcc 11152 1c1 11155 -cneg 11491 NrmCVeccnv 30509 +𝑣 cpv 30510 BaseSetcba 30511 ·𝑠OLD cns 30512 0veccn0v 30513 −𝑣 cnsb 30514 normCVcnmcv 30515 IndMetcims 30516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-po 5593 df-so 5594 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-1st 8002 df-2nd 8003 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-ltxr 11299 df-sub 11492 df-neg 11493 df-grpo 30418 df-gid 30419 df-ginv 30420 df-gdiv 30421 df-ablo 30470 df-vc 30484 df-nv 30517 df-va 30520 df-ba 30521 df-sm 30522 df-0v 30523 df-vs 30524 df-nmcv 30525 df-ims 30526 |
This theorem is referenced by: ubthlem1 30795 |
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