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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 27540 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀)) |
5 | 4 | 3ad2ant1 1082 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐷 = (𝑁 ∘ 𝑀)) |
6 | 5 | fveq1d 6193 | . . 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 27500 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋) |
9 | opelxpi 5148 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) | |
10 | fvco3 6275 | . . . . 5 ⊢ ((𝑀:(𝑋 × 𝑋)⟶𝑋 ∧ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) → ((𝑁 ∘ 𝑀)‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) | |
11 | 8, 9, 10 | syl2an 494 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑁 ∘ 𝑀)‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) |
12 | 11 | 3impb 1260 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁 ∘ 𝑀)‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) |
13 | 6, 12 | eqtrd 2656 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐷‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) |
14 | df-ov 6653 | . 2 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
15 | df-ov 6653 | . . 3 ⊢ (𝐴𝑀𝐵) = (𝑀‘〈𝐴, 𝐵〉) | |
16 | 15 | fveq2i 6194 | . 2 ⊢ (𝑁‘(𝐴𝑀𝐵)) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉)) |
17 | 13, 14, 16 | 3eqtr4g 2681 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 〈cop 4183 × cxp 5112 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 NrmCVeccnv 27439 BaseSetcba 27441 −𝑣 cnsb 27444 normCVcnmcv 27445 IndMetcims 27446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 |
This theorem is referenced by: imsdval2 27542 nvnd 27543 vacn 27549 smcnlem 27552 sspimsval 27593 blometi 27658 blocnilem 27659 ubthlem2 27727 minvecolem2 27731 minvecolem4 27736 minvecolem5 27737 minvecolem6 27738 h2hmetdval 27835 hhssmetdval 28135 |
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