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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 11316 | . . . . 5 ⊢ -1 ∈ ℂ | |
2 | nvmtri.1 | . . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | eqid 2760 | . . . . . 6 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
4 | 2, 3 | nvscl 27790 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐵) ∈ 𝑋) |
5 | 1, 4 | mp3an2 1561 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐵) ∈ 𝑋) |
6 | 5 | 3adant2 1126 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐵) ∈ 𝑋) |
7 | eqid 2760 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
8 | nvmtri.6 | . . . 4 ⊢ 𝑁 = (normCV‘𝑈) | |
9 | 2, 7, 8 | nvtri 27834 | . . 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 1516 | . 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 27806 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵))) |
13 | 12 | fveq2d 6356 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑀𝐵)) = (𝑁‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))) |
14 | 2, 3, 8 | nvs 27827 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵)) = ((abs‘-1) · (𝑁‘𝐵))) |
15 | 1, 14 | mp3an2 1561 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵)) = ((abs‘-1) · (𝑁‘𝐵))) |
16 | ax-1cn 10186 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
17 | 16 | absnegi 14338 | . . . . . . . 8 ⊢ (abs‘-1) = (abs‘1) |
18 | abs1 14236 | . . . . . . . 8 ⊢ (abs‘1) = 1 | |
19 | 17, 18 | eqtri 2782 | . . . . . . 7 ⊢ (abs‘-1) = 1 |
20 | 19 | oveq1i 6823 | . . . . . 6 ⊢ ((abs‘-1) · (𝑁‘𝐵)) = (1 · (𝑁‘𝐵)) |
21 | 2, 8 | nvcl 27825 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ ℝ) |
22 | 21 | recnd 10260 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ ℂ) |
23 | 22 | mulid2d 10250 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (1 · (𝑁‘𝐵)) = (𝑁‘𝐵)) |
24 | 20, 23 | syl5eq 2806 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → ((abs‘-1) · (𝑁‘𝐵)) = (𝑁‘𝐵)) |
25 | 15, 24 | eqtr2d 2795 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) = (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵))) |
26 | 25 | 3adant2 1126 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) = (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵))) |
27 | 26 | oveq2d 6829 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴) + (𝑁‘𝐵)) = ((𝑁‘𝐴) + (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵)))) |
28 | 10, 13, 27 | 3brtr4d 4836 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑀𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 class class class wbr 4804 ‘cfv 6049 (class class class)co 6813 ℂcc 10126 1c1 10129 + caddc 10131 · cmul 10133 ≤ cle 10267 -cneg 10459 abscabs 14173 NrmCVeccnv 27748 +𝑣 cpv 27749 BaseSetcba 27750 ·𝑠OLD cns 27751 −𝑣 cnsb 27753 normCVcnmcv 27754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-sup 8513 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-n0 11485 df-z 11570 df-uz 11880 df-rp 12026 df-seq 12996 df-exp 13055 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-grpo 27656 df-gid 27657 df-ginv 27658 df-gdiv 27659 df-ablo 27708 df-vc 27723 df-nv 27756 df-va 27759 df-ba 27760 df-sm 27761 df-0v 27762 df-vs 27763 df-nmcv 27764 |
This theorem is referenced by: ubthlem2 28036 |
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