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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 12380 | . . . . 5 ⊢ -1 ∈ ℂ | |
| 2 | nvmtri.1 | . . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 3 | eqid 2737 | . . . . . 6 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
| 4 | 2, 3 | nvscl 30645 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐵) ∈ 𝑋) | 
| 5 | 1, 4 | mp3an2 1451 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐵) ∈ 𝑋) | 
| 6 | 5 | 3adant2 1132 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐵) ∈ 𝑋) | 
| 7 | eqid 2737 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
| 8 | nvmtri.6 | . . . 4 ⊢ 𝑁 = (normCV‘𝑈) | |
| 9 | 2, 7, 8 | nvtri 30689 | . . 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 1411 | . 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 30661 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵))) | 
| 13 | 12 | fveq2d 6910 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑀𝐵)) = (𝑁‘(𝐴( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝐵)))) | 
| 14 | 2, 3, 8 | nvs 30682 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵)) = ((abs‘-1) · (𝑁‘𝐵))) | 
| 15 | 1, 14 | mp3an2 1451 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵)) = ((abs‘-1) · (𝑁‘𝐵))) | 
| 16 | ax-1cn 11213 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
| 17 | 16 | absnegi 15439 | . . . . . . . 8 ⊢ (abs‘-1) = (abs‘1) | 
| 18 | abs1 15336 | . . . . . . . 8 ⊢ (abs‘1) = 1 | |
| 19 | 17, 18 | eqtri 2765 | . . . . . . 7 ⊢ (abs‘-1) = 1 | 
| 20 | 19 | oveq1i 7441 | . . . . . 6 ⊢ ((abs‘-1) · (𝑁‘𝐵)) = (1 · (𝑁‘𝐵)) | 
| 21 | 2, 8 | nvcl 30680 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ ℝ) | 
| 22 | 21 | recnd 11289 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ ℂ) | 
| 23 | 22 | mullidd 11279 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (1 · (𝑁‘𝐵)) = (𝑁‘𝐵)) | 
| 24 | 20, 23 | eqtrid 2789 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → ((abs‘-1) · (𝑁‘𝐵)) = (𝑁‘𝐵)) | 
| 25 | 15, 24 | eqtr2d 2778 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) = (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵))) | 
| 26 | 25 | 3adant2 1132 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) = (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵))) | 
| 27 | 26 | oveq2d 7447 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴) + (𝑁‘𝐵)) = ((𝑁‘𝐴) + (𝑁‘(-1( ·𝑠OLD ‘𝑈)𝐵)))) | 
| 28 | 10, 13, 27 | 3brtr4d 5175 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑀𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 · cmul 11160 ≤ cle 11296 -cneg 11493 abscabs 15273 NrmCVeccnv 30603 +𝑣 cpv 30604 BaseSetcba 30605 ·𝑠OLD cns 30606 −𝑣 cnsb 30608 normCVcnmcv 30609 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-grpo 30512 df-gid 30513 df-ginv 30514 df-gdiv 30515 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-vs 30618 df-nmcv 30619 | 
| This theorem is referenced by: ubthlem2 30890 | 
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