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Mirrors > Home > MPE Home > Th. List > nvmtri | Structured version Visualization version GIF version |
Description: Triangle inequality for the norm of a vector difference. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvmtri.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvmtri.3 | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
nvmtri.6 | ⊢ 𝑁 = (normCV‘𝑈) |
Ref | Expression |
---|---|
nvmtri | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑀𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1cn 12359 | . . . . 5 ⊢ -1 ∈ ℂ | |
2 | nvmtri.1 | . . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | eqid 2725 | . . . . . 6 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
4 | 2, 3 | nvscl 30508 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐵) ∈ 𝑋) |
5 | 1, 4 | mp3an2 1445 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐵) ∈ 𝑋) |
6 | 5 | 3adant2 1128 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐵) ∈ 𝑋) |
7 | eqid 2725 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
8 | nvmtri.6 | . . . 4 ⊢ 𝑁 = (normCV‘𝑈) | |
9 | 2, 7, 8 | nvtri 30552 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ (-1( ·𝑠OLD ‘𝑈)𝐵) ∈ 𝑋) → (𝑁‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵))) ≤ ((𝑁‘𝐴) + (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵)))) |
10 | 6, 9 | syld3an3 1406 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵))) ≤ ((𝑁‘𝐴) + (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵)))) |
11 | nvmtri.3 | . . . 4 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
12 | 2, 7, 3, 11 | nvmval 30524 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵))) |
13 | 12 | fveq2d 6900 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑀𝐵)) = (𝑁‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))) |
14 | 2, 3, 8 | nvs 30545 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵)) = ((abs‘-1) · (𝑁‘𝐵))) |
15 | 1, 14 | mp3an2 1445 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵)) = ((abs‘-1) · (𝑁‘𝐵))) |
16 | ax-1cn 11198 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
17 | 16 | absnegi 15383 | . . . . . . . 8 ⊢ (abs‘-1) = (abs‘1) |
18 | abs1 15280 | . . . . . . . 8 ⊢ (abs‘1) = 1 | |
19 | 17, 18 | eqtri 2753 | . . . . . . 7 ⊢ (abs‘-1) = 1 |
20 | 19 | oveq1i 7429 | . . . . . 6 ⊢ ((abs‘-1) · (𝑁‘𝐵)) = (1 · (𝑁‘𝐵)) |
21 | 2, 8 | nvcl 30543 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ ℝ) |
22 | 21 | recnd 11274 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ ℂ) |
23 | 22 | mullidd 11264 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (1 · (𝑁‘𝐵)) = (𝑁‘𝐵)) |
24 | 20, 23 | eqtrid 2777 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → ((abs‘-1) · (𝑁‘𝐵)) = (𝑁‘𝐵)) |
25 | 15, 24 | eqtr2d 2766 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) = (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵))) |
26 | 25 | 3adant2 1128 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) = (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵))) |
27 | 26 | oveq2d 7435 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴) + (𝑁‘𝐵)) = ((𝑁‘𝐴) + (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵)))) |
28 | 10, 13, 27 | 3brtr4d 5181 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑀𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 1c1 11141 + caddc 11143 · cmul 11145 ≤ cle 11281 -cneg 11477 abscabs 15217 NrmCVeccnv 30466 +𝑣 cpv 30467 BaseSetcba 30468 ·𝑠OLD cns 30469 −𝑣 cnsb 30471 normCVcnmcv 30472 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
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-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-grpo 30375 df-gid 30376 df-ginv 30377 df-gdiv 30378 df-ablo 30427 df-vc 30441 df-nv 30474 df-va 30477 df-ba 30478 df-sm 30479 df-0v 30480 df-vs 30481 df-nmcv 30482 |
This theorem is referenced by: ubthlem2 30753 |
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