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Theorem nvtri 22151
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 2435 . . . . . . . . 9  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
43smfval 22076 . . . . . . . 8  |-  ( .s
OLD `  U )  =  ( 2nd `  ( 1st `  U ) )
54eqcomi 2439 . . . . . . 7  |-  ( 2nd `  ( 1st `  U
) )  =  ( .s OLD `  U
)
6 eqid 2435 . . . . . . 7  |-  ( 0vec `  U )  =  (
0vec `  U )
7 nvtri.6 . . . . . . 7  |-  N  =  ( normCV `  U )
81, 2, 5, 6, 7nvi 22085 . . . . . 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 2773 . . . . 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 6080 . . . . . . 7  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
1413fveq2d 5724 . . . . . 6  |-  ( x  =  A  ->  ( N `  ( x G y ) )  =  ( N `  ( A G y ) ) )
15 fveq2 5720 . . . . . . 7  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
1615oveq1d 6088 . . . . . 6  |-  ( x  =  A  ->  (
( N `  x
)  +  ( N `
 y ) )  =  ( ( N `
 A )  +  ( N `  y
) ) )
1714, 16breq12d 4217 . . . . 5  |-  ( x  =  A  ->  (
( N `  (
x G y ) )  <_  ( ( N `  x )  +  ( N `  y ) )  <->  ( N `  ( A G y ) )  <_  (
( N `  A
)  +  ( N `
 y ) ) ) )
18 oveq2 6081 . . . . . . 7  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1918fveq2d 5724 . . . . . 6  |-  ( y  =  B  ->  ( N `  ( A G y ) )  =  ( N `  ( A G B ) ) )
20 fveq2 5720 . . . . . . 7  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
2120oveq2d 6089 . . . . . 6  |-  ( y  =  B  ->  (
( N `  A
)  +  ( N `
 y ) )  =  ( ( N `
 A )  +  ( N `  B
) ) )
2219, 21breq12d 4217 . . . . 5  |-  ( y  =  B  ->  (
( N `  ( A G y ) )  <_  ( ( N `
 A )  +  ( N `  y
) )  <->  ( N `  ( A G B ) )  <_  (
( N `  A
)  +  ( N `
 B ) ) ) )
2317, 22rspc2v 3050 . . . 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 1652    e. wcel 1725   A.wral 2697   <.cop 3809   class class class wbr 4204   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   CCcc 8980   RRcr 8981   0cc0 8982    + caddc 8985    x. cmul 8987    <_ cle 9113   abscabs 12031   CVec
OLDcvc 22016   NrmCVeccnv 22055   +vcpv 22056   BaseSetcba 22057   .s
OLDcns 22058   0veccn0v 22059   normCVcnmcv 22061
This theorem is referenced by:  nvmtri  22152  nvmtri2  22153  nvabs  22154  nvge0  22155  imsmetlem  22174  vacn  22182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-1st 6341  df-2nd 6342  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071
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