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Theorem nvmtri 22121
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 10031 . . . . 5  |-  -u 1  e.  CC
2 nvmtri.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
3 eqid 2412 . . . . . 6  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
42, 3nvscl 22068 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
51, 4mp3an2 1267 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
653adant2 976 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
7 eqid 2412 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
8 nvmtri.6 . . . 4  |-  N  =  ( normCV `  U )
92, 7, 8nvtri 22120 . . 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 1229 . 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 22084 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) B ) ) )
1312fveq2d 5699 . 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 22112 . . . . . 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 1267 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( N `  ( -u 1
( .s OLD `  U
) B ) )  =  ( ( abs `  -u 1 )  x.  ( N `  B
) ) )
16 ax-1cn 9012 . . . . . . . . 9  |-  1  e.  CC
1716absnegi 12166 . . . . . . . 8  |-  ( abs `  -u 1 )  =  ( abs `  1
)
18 abs1 12065 . . . . . . . 8  |-  ( abs `  1 )  =  1
1917, 18eqtri 2432 . . . . . . 7  |-  ( abs `  -u 1 )  =  1
2019oveq1i 6058 . . . . . 6  |-  ( ( abs `  -u 1
)  x.  ( N `
 B ) )  =  ( 1  x.  ( N `  B
) )
212, 8nvcl 22109 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( N `  B )  e.  RR )
2221recnd 9078 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( N `  B )  e.  CC )
2322mulid2d 9070 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1  x.  ( N `
 B ) )  =  ( N `  B ) )
2420, 23syl5eq 2456 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( abs `  -u 1
)  x.  ( N `
 B ) )  =  ( N `  B ) )
2515, 24eqtr2d 2445 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( N `  B )  =  ( N `  ( -u 1 ( .s
OLD `  U ) B ) ) )
26253adant2 976 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  B )  =  ( N `  ( -u 1 ( .s
OLD `  U ) B ) ) )
2726oveq2d 6064 . 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 4210 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   CCcc 8952   1c1 8955    + caddc 8957    x. cmul 8959    <_ cle 9085   -ucneg 9256   abscabs 12002   NrmCVeccnv 22024   +vcpv 22025   BaseSetcba 22026   .s
OLDcns 22027   -vcnsb 22029   normCVcnmcv 22030
This theorem is referenced by:  ubthlem2  22334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-seq 11287  df-exp 11346  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-grpo 21740  df-gid 21741  df-ginv 21742  df-gdiv 21743  df-ablo 21831  df-vc 21986  df-nv 22032  df-va 22035  df-ba 22036  df-sm 22037  df-0v 22038  df-vs 22039  df-nmcv 22040
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