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Theorem nvs 21228
Description: Proportionality property of the norm of a scalar product in a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvs.1  |-  X  =  ( BaseSet `  U )
nvs.4  |-  S  =  ( .s OLD `  U
)
nvs.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvs  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( N `  ( A S B ) )  =  ( ( abs `  A
)  x.  ( N `
 B ) ) )

Proof of Theorem nvs
Dummy variables  y  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvs.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
2 eqid 2283 . . . . . . 7  |-  ( +v
`  U )  =  ( +v `  U
)
3 nvs.4 . . . . . . 7  |-  S  =  ( .s OLD `  U
)
4 eqid 2283 . . . . . . 7  |-  ( 0vec `  U )  =  (
0vec `  U )
5 nvs.6 . . . . . . 7  |-  N  =  ( normCV `  U )
61, 2, 3, 4, 5nvi 21170 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( <. ( +v `  U ) ,  S >.  e.  CVec OLD 
/\  N : X --> RR  /\  A. x  e.  X  ( ( ( N `  x )  =  0  ->  x  =  ( 0vec `  U
) )  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y )  x.  ( N `  x )
)  /\  A. y  e.  X  ( N `  ( x ( +v
`  U ) y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) ) )
76simp3d 969 . . . . 5  |-  ( U  e.  NrmCVec  ->  A. x  e.  X  ( ( ( N `
 x )  =  0  ->  x  =  ( 0vec `  U )
)  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x ( +v
`  U ) y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) )
8 simp2 956 . . . . . 6  |-  ( ( ( ( N `  x )  =  0  ->  x  =  (
0vec `  U )
)  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x ( +v
`  U ) y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) )  ->  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) ) )
98ralimi 2618 . . . . 5  |-  ( A. x  e.  X  (
( ( N `  x )  =  0  ->  x  =  (
0vec `  U )
)  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x ( +v
`  U ) y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) )  ->  A. x  e.  X  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) ) )
107, 9syl 15 . . . 4  |-  ( U  e.  NrmCVec  ->  A. x  e.  X  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y )  x.  ( N `  x )
) )
11 oveq2 5866 . . . . . . 7  |-  ( x  =  B  ->  (
y S x )  =  ( y S B ) )
1211fveq2d 5529 . . . . . 6  |-  ( x  =  B  ->  ( N `  ( y S x ) )  =  ( N `  ( y S B ) ) )
13 fveq2 5525 . . . . . . 7  |-  ( x  =  B  ->  ( N `  x )  =  ( N `  B ) )
1413oveq2d 5874 . . . . . 6  |-  ( x  =  B  ->  (
( abs `  y
)  x.  ( N `
 x ) )  =  ( ( abs `  y )  x.  ( N `  B )
) )
1512, 14eqeq12d 2297 . . . . 5  |-  ( x  =  B  ->  (
( N `  (
y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) )  <->  ( N `  ( y S B ) )  =  ( ( abs `  y
)  x.  ( N `
 B ) ) ) )
16 oveq1 5865 . . . . . . 7  |-  ( y  =  A  ->  (
y S B )  =  ( A S B ) )
1716fveq2d 5529 . . . . . 6  |-  ( y  =  A  ->  ( N `  ( y S B ) )  =  ( N `  ( A S B ) ) )
18 fveq2 5525 . . . . . . 7  |-  ( y  =  A  ->  ( abs `  y )  =  ( abs `  A
) )
1918oveq1d 5873 . . . . . 6  |-  ( y  =  A  ->  (
( abs `  y
)  x.  ( N `
 B ) )  =  ( ( abs `  A )  x.  ( N `  B )
) )
2017, 19eqeq12d 2297 . . . . 5  |-  ( y  =  A  ->  (
( N `  (
y S B ) )  =  ( ( abs `  y )  x.  ( N `  B ) )  <->  ( N `  ( A S B ) )  =  ( ( abs `  A
)  x.  ( N `
 B ) ) ) )
2115, 20rspc2v 2890 . . . 4  |-  ( ( B  e.  X  /\  A  e.  CC )  ->  ( A. x  e.  X  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  ->  ( N `  ( A S B ) )  =  ( ( abs `  A )  x.  ( N `  B ) ) ) )
2210, 21syl5 28 . . 3  |-  ( ( B  e.  X  /\  A  e.  CC )  ->  ( U  e.  NrmCVec  -> 
( N `  ( A S B ) )  =  ( ( abs `  A )  x.  ( N `  B )
) ) )
23223impia 1148 . 2  |-  ( ( B  e.  X  /\  A  e.  CC  /\  U  e.  NrmCVec )  ->  ( N `  ( A S B ) )  =  ( ( abs `  A
)  x.  ( N `
 B ) ) )
24233com13 1156 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( N `  ( A S B ) )  =  ( ( abs `  A
)  x.  ( 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   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:  nvsge0  21229  nvm1  21230  nvpi  21232  nvmtri  21237  smcnlem  21270  ipidsq  21286  minvecolem2  21454
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
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