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Theorem nvtri 24195
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 2451 . . . . . . . . 9  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
43smfval 24120 . . . . . . . 8  |-  ( .sOLD `  U )  =  ( 2nd `  ( 1st `  U ) )
54eqcomi 2464 . . . . . . 7  |-  ( 2nd `  ( 1st `  U
) )  =  ( .sOLD `  U
)
6 eqid 2451 . . . . . . 7  |-  ( 0vec `  U )  =  (
0vec `  U )
7 nvtri.6 . . . . . . 7  |-  N  =  ( normCV `  U )
81, 2, 5, 6, 7nvi 24129 . . . . . 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 1002 . . . . 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 990 . . . . . 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 2811 . . . . 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 6199 . . . . . . 7  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
1413fveq2d 5795 . . . . . 6  |-  ( x  =  A  ->  ( N `  ( x G y ) )  =  ( N `  ( A G y ) ) )
15 fveq2 5791 . . . . . . 7  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
1615oveq1d 6207 . . . . . 6  |-  ( x  =  A  ->  (
( N `  x
)  +  ( N `
 y ) )  =  ( ( N `
 A )  +  ( N `  y
) ) )
1714, 16breq12d 4405 . . . . 5  |-  ( x  =  A  ->  (
( N `  (
x G y ) )  <_  ( ( N `  x )  +  ( N `  y ) )  <->  ( N `  ( A G y ) )  <_  (
( N `  A
)  +  ( N `
 y ) ) ) )
18 oveq2 6200 . . . . . . 7  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1918fveq2d 5795 . . . . . 6  |-  ( y  =  B  ->  ( N `  ( A G y ) )  =  ( N `  ( A G B ) ) )
20 fveq2 5791 . . . . . . 7  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
2120oveq2d 6208 . . . . . 6  |-  ( y  =  B  ->  (
( N `  A
)  +  ( N `
 y ) )  =  ( ( N `
 A )  +  ( N `  B
) ) )
2219, 21breq12d 4405 . . . . 5  |-  ( y  =  B  ->  (
( N `  ( A G y ) )  <_  ( ( N `
 A )  +  ( N `  y
) )  <->  ( N `  ( A G B ) )  <_  (
( N `  A
)  +  ( N `
 B ) ) ) )
2317, 22rspc2v 3178 . . . 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 1185 . 2  |-  ( ( A  e.  X  /\  B  e.  X  /\  U  e.  NrmCVec )  -> 
( N `  ( A G B ) )  <_  ( ( N `
 A )  +  ( N `  B
) ) )
26253comr 1196 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 965    = wceq 1370    e. wcel 1758   A.wral 2795   <.cop 3983   class class class wbr 4392   -->wf 5514   ` cfv 5518  (class class class)co 6192   1stc1st 6677   2ndc2nd 6678   CCcc 9383   RRcr 9384   0cc0 9385    + caddc 9388    x. cmul 9390    <_ cle 9522   abscabs 12827   CVecOLDcvc 24060   NrmCVeccnv 24099   +vcpv 24100   BaseSetcba 24101   .sOLDcns 24102   0veccn0v 24103   normCVcnmcv 24105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-1st 6679  df-2nd 6680  df-vc 24061  df-nv 24107  df-va 24110  df-ba 24111  df-sm 24112  df-0v 24113  df-nmcv 24115
This theorem is referenced by:  nvmtri  24196  nvmtri2  24197  nvabs  24198  nvge0  24199  imsmetlem  24218  vacn  24226
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