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Theorem nvmtri 22001
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 9992 . . . . 5  |-  -u 1  e.  CC
2 nvmtri.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
3 eqid 2380 . . . . . 6  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
42, 3nvscl 21948 . . . . 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 2380 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
8 nvmtri.6 . . . 4  |-  N  =  ( normCV `  U )
92, 7, 8nvtri 22000 . . 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 21964 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) B ) ) )
1312fveq2d 5665 . 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 21992 . . . . . 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 8974 . . . . . . . . 9  |-  1  e.  CC
1716absnegi 12123 . . . . . . . 8  |-  ( abs `  -u 1 )  =  ( abs `  1
)
18 abs1 12022 . . . . . . . 8  |-  ( abs `  1 )  =  1
1917, 18eqtri 2400 . . . . . . 7  |-  ( abs `  -u 1 )  =  1
2019oveq1i 6023 . . . . . 6  |-  ( ( abs `  -u 1
)  x.  ( N `
 B ) )  =  ( 1  x.  ( N `  B
) )
212, 8nvcl 21989 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( N `  B )  e.  RR )
2221recnd 9040 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( N `  B )  e.  CC )
2322mulid2d 9032 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1  x.  ( N `
 B ) )  =  ( N `  B ) )
2420, 23syl5eq 2424 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( abs `  -u 1
)  x.  ( N `
 B ) )  =  ( N `  B ) )
2515, 24eqtr2d 2413 . . . 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 6029 . 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 4176 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 1717   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   CCcc 8914   1c1 8917    + caddc 8919    x. cmul 8921    <_ cle 9047   -ucneg 9217   abscabs 11959   NrmCVeccnv 21904   +vcpv 21905   BaseSetcba 21906   .s
OLDcns 21907   -vcnsb 21909   normCVcnmcv 21910
This theorem is referenced by:  ubthlem2  22214
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-grpo 21620  df-gid 21621  df-ginv 21622  df-gdiv 21623  df-ablo 21711  df-vc 21866  df-nv 21912  df-va 21915  df-ba 21916  df-sm 21917  df-0v 21918  df-vs 21919  df-nmcv 21920
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