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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 28464 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀)) |
5 | 4 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐷 = (𝑁 ∘ 𝑀)) |
6 | 5 | fveq1d 6674 | . . 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 28424 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋) |
9 | opelxpi 5594 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) | |
10 | fvco3 6762 | . . . . 5 ⊢ ((𝑀:(𝑋 × 𝑋)⟶𝑋 ∧ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) → ((𝑁 ∘ 𝑀)‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) | |
11 | 8, 9, 10 | syl2an 597 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑁 ∘ 𝑀)‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) |
12 | 11 | 3impb 1111 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁 ∘ 𝑀)‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) |
13 | 6, 12 | eqtrd 2858 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐷‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) |
14 | df-ov 7161 | . 2 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
15 | df-ov 7161 | . . 3 ⊢ (𝐴𝑀𝐵) = (𝑀‘〈𝐴, 𝐵〉) | |
16 | 15 | fveq2i 6675 | . 2 ⊢ (𝑁‘(𝐴𝑀𝐵)) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉)) |
17 | 13, 14, 16 | 3eqtr4g 2883 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 〈cop 4575 × cxp 5555 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 NrmCVeccnv 28363 BaseSetcba 28365 −𝑣 cnsb 28368 normCVcnmcv 28369 IndMetcims 28370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-neg 10875 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-vs 28378 df-nmcv 28379 df-ims 28380 |
This theorem is referenced by: imsdval2 28466 nvnd 28467 vacn 28473 smcnlem 28476 sspimsval 28517 blometi 28582 blocnilem 28583 ubthlem2 28650 minvecolem2 28654 minvecolem4 28659 minvecolem5 28660 minvecolem6 28661 h2hmetdval 28757 hhssmetdval 29056 |
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