![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 30714 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀)) |
5 | 4 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐷 = (𝑁 ∘ 𝑀)) |
6 | 5 | fveq1d 6909 | . . 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 30674 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋) |
9 | opelxpi 5726 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) | |
10 | fvco3 7008 | . . . . 5 ⊢ ((𝑀:(𝑋 × 𝑋)⟶𝑋 ∧ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) → ((𝑁 ∘ 𝑀)‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) | |
11 | 8, 9, 10 | syl2an 596 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑁 ∘ 𝑀)‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) |
12 | 11 | 3impb 1114 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁 ∘ 𝑀)‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) |
13 | 6, 12 | eqtrd 2775 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐷‘〈𝐴, 𝐵〉) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉))) |
14 | df-ov 7434 | . 2 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
15 | df-ov 7434 | . . 3 ⊢ (𝐴𝑀𝐵) = (𝑀‘〈𝐴, 𝐵〉) | |
16 | 15 | fveq2i 6910 | . 2 ⊢ (𝑁‘(𝐴𝑀𝐵)) = (𝑁‘(𝑀‘〈𝐴, 𝐵〉)) |
17 | 13, 14, 16 | 3eqtr4g 2800 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 〈cop 4637 × cxp 5687 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 NrmCVeccnv 30613 BaseSetcba 30615 −𝑣 cnsb 30618 normCVcnmcv 30619 IndMetcims 30620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-neg 11493 df-grpo 30522 df-gid 30523 df-ginv 30524 df-gdiv 30525 df-ablo 30574 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-vs 30628 df-nmcv 30629 df-ims 30630 |
This theorem is referenced by: imsdval2 30716 nvnd 30717 vacn 30723 smcnlem 30726 sspimsval 30767 blometi 30832 blocnilem 30833 ubthlem2 30900 minvecolem2 30904 minvecolem4 30909 minvecolem5 30910 minvecolem6 30911 h2hmetdval 31007 hhssmetdval 31306 |
Copyright terms: Public domain | W3C validator |