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Mirrors > Home > MPE Home > Th. List > imsdval | Structured version Visualization version GIF version |
Description: Value of the induced metric (distance function) of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
imsdval.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
imsdval.3 | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
imsdval.6 | ⊢ 𝑁 = (normCV‘𝑈) |
imsdval.8 | ⊢ 𝐷 = (IndMet‘𝑈) |
Ref | Expression |
---|---|
imsdval | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imsdval.3 | . . . . . 6 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
2 | imsdval.6 | . . . . . 6 ⊢ 𝑁 = (normCV‘𝑈) | |
3 | imsdval.8 | . . . . . 6 ⊢ 𝐷 = (IndMet‘𝑈) | |
4 | 1, 2, 3 | imsval 28468 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀)) |
5 | 4 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐷 = (𝑁 ∘ 𝑀)) |
6 | 5 | fveq1d 6647 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐷‘〈𝐴, 𝐵〉) = ((𝑁 ∘ 𝑀)‘〈𝐴, 𝐵〉)) |
7 | imsdval.1 | . . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈) | |
8 | 7, 1 | nvmf 28428 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋) |
9 | opelxpi 5556 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) | |
10 | fvco3 6737 | . . . . 5 ⊢ ((𝑀:(𝑋 × 𝑋)⟶𝑋 ∧ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) → ((𝑁 ∘ 𝑀)‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) | |
11 | 8, 9, 10 | syl2an 598 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑁 ∘ 𝑀)‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) |
12 | 11 | 3impb 1112 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁 ∘ 𝑀)‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) |
13 | 6, 12 | eqtrd 2833 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐷‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) |
14 | df-ov 7138 | . 2 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
15 | df-ov 7138 | . . 3 ⊢ (𝐴𝑀𝐵) = (𝑀‘〈𝐴, 𝐵〉) | |
16 | 15 | fveq2i 6648 | . 2 ⊢ (𝑁‘(𝐴𝑀𝐵)) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉)) |
17 | 13, 14, 16 | 3eqtr4g 2858 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 〈cop 4531 × cxp 5517 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 NrmCVeccnv 28367 BaseSetcba 28369 −𝑣 cnsb 28372 normCVcnmcv 28373 IndMetcims 28374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-neg 10862 df-grpo 28276 df-gid 28277 df-ginv 28278 df-gdiv 28279 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-vs 28382 df-nmcv 28383 df-ims 28384 |
This theorem is referenced by: imsdval2 28470 nvnd 28471 vacn 28477 smcnlem 28480 sspimsval 28521 blometi 28586 blocnilem 28587 ubthlem2 28654 minvecolem2 28658 minvecolem4 28663 minvecolem5 28664 minvecolem6 28665 h2hmetdval 28761 hhssmetdval 29060 |
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