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Theorem nvtri 25099
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 2460 . . . . . . . . 9  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
43smfval 25024 . . . . . . . 8  |-  ( .sOLD `  U )  =  ( 2nd `  ( 1st `  U ) )
54eqcomi 2473 . . . . . . 7  |-  ( 2nd `  ( 1st `  U
) )  =  ( .sOLD `  U
)
6 eqid 2460 . . . . . . 7  |-  ( 0vec `  U )  =  (
0vec `  U )
7 nvtri.6 . . . . . . 7  |-  N  =  ( normCV `  U )
81, 2, 5, 6, 7nvi 25033 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( <. G , 
( 2nd `  ( 1st `  U ) )
>.  e.  CVecOLD  /\  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 1005 . . . . 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 993 . . . . . 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 2850 . . . . 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 16 . . . 4  |-  ( U  e.  NrmCVec  ->  A. x  e.  X  A. y  e.  X  ( N `  ( x G y ) )  <_  ( ( N `
 x )  +  ( N `  y
) ) )
13 oveq1 6282 . . . . . . 7  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
1413fveq2d 5861 . . . . . 6  |-  ( x  =  A  ->  ( N `  ( x G y ) )  =  ( N `  ( A G y ) ) )
15 fveq2 5857 . . . . . . 7  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
1615oveq1d 6290 . . . . . 6  |-  ( x  =  A  ->  (
( N `  x
)  +  ( N `
 y ) )  =  ( ( N `
 A )  +  ( N `  y
) ) )
1714, 16breq12d 4453 . . . . 5  |-  ( x  =  A  ->  (
( N `  (
x G y ) )  <_  ( ( N `  x )  +  ( N `  y ) )  <->  ( N `  ( A G y ) )  <_  (
( N `  A
)  +  ( N `
 y ) ) ) )
18 oveq2 6283 . . . . . . 7  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1918fveq2d 5861 . . . . . 6  |-  ( y  =  B  ->  ( N `  ( A G y ) )  =  ( N `  ( A G B ) ) )
20 fveq2 5857 . . . . . . 7  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
2120oveq2d 6291 . . . . . 6  |-  ( y  =  B  ->  (
( N `  A
)  +  ( N `
 y ) )  =  ( ( N `
 A )  +  ( N `  B
) ) )
2219, 21breq12d 4453 . . . . 5  |-  ( y  =  B  ->  (
( N `  ( A G y ) )  <_  ( ( N `
 A )  +  ( N `  y
) )  <->  ( N `  ( A G B ) )  <_  (
( N `  A
)  +  ( N `
 B ) ) ) )
2317, 22rspc2v 3216 . . . 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 32 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( U  e.  NrmCVec  -> 
( N `  ( A G B ) )  <_  ( ( N `
 A )  +  ( N `  B
) ) ) )
25243impia 1188 . 2  |-  ( ( A  e.  X  /\  B  e.  X  /\  U  e.  NrmCVec )  -> 
( N `  ( A G B ) )  <_  ( ( N `
 A )  +  ( N `  B
) ) )
26253comr 1199 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  <_ 
( ( N `  A )  +  ( N `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   <.cop 4026   class class class wbr 4440   -->wf 5575   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   CCcc 9479   RRcr 9480   0cc0 9481    + caddc 9484    x. cmul 9486    <_ cle 9618   abscabs 13017   CVecOLDcvc 24964   NrmCVeccnv 25003   +vcpv 25004   BaseSetcba 25005   .sOLDcns 25006   0veccn0v 25007   normCVcnmcv 25009
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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2812  df-rex 2813  df-reu 2814  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-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  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-ov 6278  df-oprab 6279  df-1st 6774  df-2nd 6775  df-vc 24965  df-nv 25011  df-va 25014  df-ba 25015  df-sm 25016  df-0v 25017  df-nmcv 25019
This theorem is referenced by:  nvmtri  25100  nvmtri2  25101  nvabs  25102  nvge0  25103  imsmetlem  25122  vacn  25130
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