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Theorem nvtri 22009
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 2389 . . . . . . . . 9  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
43smfval 21934 . . . . . . . 8  |-  ( .s
OLD `  U )  =  ( 2nd `  ( 1st `  U ) )
54eqcomi 2393 . . . . . . 7  |-  ( 2nd `  ( 1st `  U
) )  =  ( .s OLD `  U
)
6 eqid 2389 . . . . . . 7  |-  ( 0vec `  U )  =  (
0vec `  U )
7 nvtri.6 . . . . . . 7  |-  N  =  ( normCV `  U )
81, 2, 5, 6, 7nvi 21943 . . . . . 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 971 . . . . 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 959 . . . . . 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 2726 . . . . 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 6029 . . . . . . 7  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
1413fveq2d 5674 . . . . . 6  |-  ( x  =  A  ->  ( N `  ( x G y ) )  =  ( N `  ( A G y ) ) )
15 fveq2 5670 . . . . . . 7  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
1615oveq1d 6037 . . . . . 6  |-  ( x  =  A  ->  (
( N `  x
)  +  ( N `
 y ) )  =  ( ( N `
 A )  +  ( N `  y
) ) )
1714, 16breq12d 4168 . . . . 5  |-  ( x  =  A  ->  (
( N `  (
x G y ) )  <_  ( ( N `  x )  +  ( N `  y ) )  <->  ( N `  ( A G y ) )  <_  (
( N `  A
)  +  ( N `
 y ) ) ) )
18 oveq2 6030 . . . . . . 7  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1918fveq2d 5674 . . . . . 6  |-  ( y  =  B  ->  ( N `  ( A G y ) )  =  ( N `  ( A G B ) ) )
20 fveq2 5670 . . . . . . 7  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
2120oveq2d 6038 . . . . . 6  |-  ( y  =  B  ->  (
( N `  A
)  +  ( N `
 y ) )  =  ( ( N `
 A )  +  ( N `  B
) ) )
2219, 21breq12d 4168 . . . . 5  |-  ( y  =  B  ->  (
( N `  ( A G y ) )  <_  ( ( N `
 A )  +  ( N `  y
) )  <->  ( N `  ( A G B ) )  <_  (
( N `  A
)  +  ( N `
 B ) ) ) )
2317, 22rspc2v 3003 . . . 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 30 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( U  e.  NrmCVec  -> 
( N `  ( A G B ) )  <_  ( ( N `
 A )  +  ( N `  B
) ) ) )
25243impia 1150 . 2  |-  ( ( A  e.  X  /\  B  e.  X  /\  U  e.  NrmCVec )  -> 
( N `  ( A G B ) )  <_  ( ( N `
 A )  +  ( N `  B
) ) )
26253comr 1161 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 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   <.cop 3762   class class class wbr 4155   -->wf 5392   ` cfv 5396  (class class class)co 6022   1stc1st 6288   2ndc2nd 6289   CCcc 8923   RRcr 8924   0cc0 8925    + caddc 8928    x. cmul 8930    <_ cle 9056   abscabs 11968   CVec
OLDcvc 21874   NrmCVeccnv 21913   +vcpv 21914   BaseSetcba 21915   .s
OLDcns 21916   0veccn0v 21917   normCVcnmcv 21919
This theorem is referenced by:  nvmtri  22010  nvmtri2  22011  nvabs  22012  nvge0  22013  imsmetlem  22032  vacn  22040
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-1st 6290  df-2nd 6291  df-vc 21875  df-nv 21921  df-va 21924  df-ba 21925  df-sm 21926  df-0v 21927  df-nmcv 21929
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