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Theorem nvmtri 22161
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 10068 . . . . 5  |-  -u 1  e.  CC
2 nvmtri.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
3 eqid 2437 . . . . . 6  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
42, 3nvscl 22108 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
51, 4mp3an2 1268 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
653adant2 977 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
7 eqid 2437 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
8 nvmtri.6 . . . 4  |-  N  =  ( normCV `  U )
92, 7, 8nvtri 22160 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  ( -u 1 ( .s OLD `  U ) B )  e.  X )  -> 
( N `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) )  <_  (
( N `  A
)  +  ( N `
 ( -u 1
( .s OLD `  U
) B ) ) ) )
106, 9syld3an3 1230 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A
( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) )  <_  ( ( N `  A )  +  ( N `  ( -u 1 ( .s
OLD `  U ) B ) ) ) )
11 nvmtri.3 . . . 4  |-  M  =  ( -v `  U
)
122, 7, 3, 11nvmval 22124 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) B ) ) )
1312fveq2d 5733 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M B ) )  =  ( N `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) )
142, 3, 8nvs 22152 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( N `  ( -u 1
( .s OLD `  U
) B ) )  =  ( ( abs `  -u 1 )  x.  ( N `  B
) ) )
151, 14mp3an2 1268 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( N `  ( -u 1
( .s OLD `  U
) B ) )  =  ( ( abs `  -u 1 )  x.  ( N `  B
) ) )
16 ax-1cn 9049 . . . . . . . . 9  |-  1  e.  CC
1716absnegi 12204 . . . . . . . 8  |-  ( abs `  -u 1 )  =  ( abs `  1
)
18 abs1 12103 . . . . . . . 8  |-  ( abs `  1 )  =  1
1917, 18eqtri 2457 . . . . . . 7  |-  ( abs `  -u 1 )  =  1
2019oveq1i 6092 . . . . . 6  |-  ( ( abs `  -u 1
)  x.  ( N `
 B ) )  =  ( 1  x.  ( N `  B
) )
212, 8nvcl 22149 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( N `  B )  e.  RR )
2221recnd 9115 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( N `  B )  e.  CC )
2322mulid2d 9107 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1  x.  ( N `
 B ) )  =  ( N `  B ) )
2420, 23syl5eq 2481 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( abs `  -u 1
)  x.  ( N `
 B ) )  =  ( N `  B ) )
2515, 24eqtr2d 2470 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( N `  B )  =  ( N `  ( -u 1 ( .s
OLD `  U ) B ) ) )
26253adant2 977 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  B )  =  ( N `  ( -u 1 ( .s
OLD `  U ) B ) ) )
2726oveq2d 6098 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  A
)  +  ( N `
 B ) )  =  ( ( N `
 A )  +  ( N `  ( -u 1 ( .s OLD `  U ) B ) ) ) )
2810, 13, 273brtr4d 4243 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M B ) )  <_ 
( ( N `  A )  +  ( N `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   CCcc 8989   1c1 8992    + caddc 8994    x. cmul 8996    <_ cle 9122   -ucneg 9293   abscabs 12040   NrmCVeccnv 22064   +vcpv 22065   BaseSetcba 22066   .s
OLDcns 22067   -vcnsb 22069   normCVcnmcv 22070
This theorem is referenced by:  ubthlem2  22374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-grpo 21780  df-gid 21781  df-ginv 21782  df-gdiv 21783  df-ablo 21871  df-vc 22026  df-nv 22072  df-va 22075  df-ba 22076  df-sm 22077  df-0v 22078  df-vs 22079  df-nmcv 22080
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