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Theorem nvmtri 25236
Description: Triangle inequality for the norm of a vector difference. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvmtri.1  |-  X  =  ( BaseSet `  U )
nvmtri.3  |-  M  =  ( -v `  U
)
nvmtri.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvmtri  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M B ) )  <_ 
( ( N `  A )  +  ( N `  B ) ) )

Proof of Theorem nvmtri
StepHypRef Expression
1 neg1cn 10628 . . . . 5  |-  -u 1  e.  CC
2 nvmtri.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
3 eqid 2460 . . . . . 6  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
42, 3nvscl 25183 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) B )  e.  X )
51, 4mp3an2 1307 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) B )  e.  X )
653adant2 1010 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) B )  e.  X )
7 eqid 2460 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
8 nvmtri.6 . . . 4  |-  N  =  ( normCV `  U )
92, 7, 8nvtri 25235 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  ( -u 1 ( .sOLD `  U ) B )  e.  X )  -> 
( N `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) B ) ) )  <_  (
( N `  A
)  +  ( N `
 ( -u 1
( .sOLD `  U ) B ) ) ) )
106, 9syld3an3 1268 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A
( +v `  U
) ( -u 1
( .sOLD `  U ) B ) ) )  <_  (
( N `  A
)  +  ( N `
 ( -u 1
( .sOLD `  U ) B ) ) ) )
11 nvmtri.3 . . . 4  |-  M  =  ( -v `  U
)
122, 7, 3, 11nvmval 25199 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .sOLD `  U ) B ) ) )
1312fveq2d 5861 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M B ) )  =  ( N `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) B ) ) ) )
142, 3, 8nvs 25227 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( N `  ( -u 1
( .sOLD `  U ) B ) )  =  ( ( abs `  -u 1
)  x.  ( N `
 B ) ) )
151, 14mp3an2 1307 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( N `  ( -u 1
( .sOLD `  U ) B ) )  =  ( ( abs `  -u 1
)  x.  ( N `
 B ) ) )
16 ax-1cn 9539 . . . . . . . . 9  |-  1  e.  CC
1716absnegi 13181 . . . . . . . 8  |-  ( abs `  -u 1 )  =  ( abs `  1
)
18 abs1 13080 . . . . . . . 8  |-  ( abs `  1 )  =  1
1917, 18eqtri 2489 . . . . . . 7  |-  ( abs `  -u 1 )  =  1
2019oveq1i 6285 . . . . . 6  |-  ( ( abs `  -u 1
)  x.  ( N `
 B ) )  =  ( 1  x.  ( N `  B
) )
212, 8nvcl 25224 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( N `  B )  e.  RR )
2221recnd 9611 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( N `  B )  e.  CC )
2322mulid2d 9603 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1  x.  ( N `
 B ) )  =  ( N `  B ) )
2420, 23syl5eq 2513 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( abs `  -u 1
)  x.  ( N `
 B ) )  =  ( N `  B ) )
2515, 24eqtr2d 2502 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( N `  B )  =  ( N `  ( -u 1 ( .sOLD `  U ) B ) ) )
26253adant2 1010 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  B )  =  ( N `  ( -u 1 ( .sOLD `  U ) B ) ) )
2726oveq2d 6291 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  A
)  +  ( N `
 B ) )  =  ( ( N `
 A )  +  ( N `  ( -u 1 ( .sOLD `  U ) B ) ) ) )
2810, 13, 273brtr4d 4470 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M B ) )  <_ 
( ( N `  A )  +  ( N `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   CCcc 9479   1c1 9482    + caddc 9484    x. cmul 9486    <_ cle 9618   -ucneg 9795   abscabs 13017   NrmCVeccnv 25139   +vcpv 25140   BaseSetcba 25141   .sOLDcns 25142   -vcnsb 25144   normCVcnmcv 25145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-grpo 24855  df-gid 24856  df-ginv 24857  df-gdiv 24858  df-ablo 24946  df-vc 25101  df-nv 25147  df-va 25150  df-ba 25151  df-sm 25152  df-0v 25153  df-vs 25154  df-nmcv 25155
This theorem is referenced by:  ubthlem2  25449
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