MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvtri Unicode version

Theorem nvtri 21236
Description: Triangle inequality for the norm of a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvtri.1  |-  X  =  ( BaseSet `  U )
nvtri.2  |-  G  =  ( +v `  U
)
nvtri.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvtri  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  <_ 
( ( N `  A )  +  ( N `  B ) ) )

Proof of Theorem nvtri
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvtri.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
2 nvtri.2 . . . . . . 7  |-  G  =  ( +v `  U
)
3 eqid 2283 . . . . . . . . 9  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
43smfval 21161 . . . . . . . 8  |-  ( .s
OLD `  U )  =  ( 2nd `  ( 1st `  U ) )
54eqcomi 2287 . . . . . . 7  |-  ( 2nd `  ( 1st `  U
) )  =  ( .s OLD `  U
)
6 eqid 2283 . . . . . . 7  |-  ( 0vec `  U )  =  (
0vec `  U )
7 nvtri.6 . . . . . . 7  |-  N  =  ( normCV `  U )
81, 2, 5, 6, 7nvi 21170 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( <. G , 
( 2nd `  ( 1st `  U ) )
>.  e.  CVec OLD  /\  N : X
--> RR  /\  A. x  e.  X  ( (
( N `  x
)  =  0  ->  x  =  ( 0vec `  U ) )  /\  A. y  e.  CC  ( N `  ( y
( 2nd `  ( 1st `  U ) ) x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) ) )
98simp3d 969 . . . . 5  |-  ( U  e.  NrmCVec  ->  A. x  e.  X  ( ( ( N `
 x )  =  0  ->  x  =  ( 0vec `  U )
)  /\  A. y  e.  CC  ( N `  ( y ( 2nd `  ( 1st `  U
) ) x ) )  =  ( ( abs `  y )  x.  ( N `  x ) )  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  ( ( N `
 x )  +  ( N `  y
) ) ) )
10 simp3 957 . . . . . 6  |-  ( ( ( ( N `  x )  =  0  ->  x  =  (
0vec `  U )
)  /\  A. y  e.  CC  ( N `  ( y ( 2nd `  ( 1st `  U
) ) x ) )  =  ( ( abs `  y )  x.  ( N `  x ) )  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  ( ( N `
 x )  +  ( N `  y
) ) )  ->  A. y  e.  X  ( N `  ( x G y ) )  <_  ( ( N `
 x )  +  ( N `  y
) ) )
1110ralimi 2618 . . . . 5  |-  ( A. x  e.  X  (
( ( N `  x )  =  0  ->  x  =  (
0vec `  U )
)  /\  A. y  e.  CC  ( N `  ( y ( 2nd `  ( 1st `  U
) ) x ) )  =  ( ( abs `  y )  x.  ( N `  x ) )  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  ( ( N `
 x )  +  ( N `  y
) ) )  ->  A. x  e.  X  A. y  e.  X  ( N `  ( x G y ) )  <_  ( ( N `
 x )  +  ( N `  y
) ) )
129, 11syl 15 . . . 4  |-  ( U  e.  NrmCVec  ->  A. x  e.  X  A. y  e.  X  ( N `  ( x G y ) )  <_  ( ( N `
 x )  +  ( N `  y
) ) )
13 oveq1 5865 . . . . . . 7  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
1413fveq2d 5529 . . . . . 6  |-  ( x  =  A  ->  ( N `  ( x G y ) )  =  ( N `  ( A G y ) ) )
15 fveq2 5525 . . . . . . 7  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
1615oveq1d 5873 . . . . . 6  |-  ( x  =  A  ->  (
( N `  x
)  +  ( N `
 y ) )  =  ( ( N `
 A )  +  ( N `  y
) ) )
1714, 16breq12d 4036 . . . . 5  |-  ( x  =  A  ->  (
( N `  (
x G y ) )  <_  ( ( N `  x )  +  ( N `  y ) )  <->  ( N `  ( A G y ) )  <_  (
( N `  A
)  +  ( N `
 y ) ) ) )
18 oveq2 5866 . . . . . . 7  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1918fveq2d 5529 . . . . . 6  |-  ( y  =  B  ->  ( N `  ( A G y ) )  =  ( N `  ( A G B ) ) )
20 fveq2 5525 . . . . . . 7  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
2120oveq2d 5874 . . . . . 6  |-  ( y  =  B  ->  (
( N `  A
)  +  ( N `
 y ) )  =  ( ( N `
 A )  +  ( N `  B
) ) )
2219, 21breq12d 4036 . . . . 5  |-  ( y  =  B  ->  (
( N `  ( A G y ) )  <_  ( ( N `
 A )  +  ( N `  y
) )  <->  ( N `  ( A G B ) )  <_  (
( N `  A
)  +  ( N `
 B ) ) ) )
2317, 22rspc2v 2890 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) )  ->  ( N `  ( A G B ) )  <_  ( ( N `  A )  +  ( N `  B ) ) ) )
2412, 23syl5 28 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( U  e.  NrmCVec  -> 
( N `  ( A G B ) )  <_  ( ( N `
 A )  +  ( N `  B
) ) ) )
25243impia 1148 . 2  |-  ( ( A  e.  X  /\  B  e.  X  /\  U  e.  NrmCVec )  -> 
( N `  ( A G B ) )  <_  ( ( N `
 A )  +  ( N `  B
) ) )
26253comr 1159 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  <_ 
( ( N `  A )  +  ( N `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    <_ cle 8868   abscabs 11719   CVec
OLDcvc 21101   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   0veccn0v 21144   normCVcnmcv 21146
This theorem is referenced by:  nvmtri  21237  nvmtri2  21238  nvabs  21239  nvge0  21240  imsmetlem  21259  vacn  21267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-1st 6122  df-2nd 6123  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156
  Copyright terms: Public domain W3C validator