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Theorem nvs 22152
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 2437 . . . . . . 7  |-  ( +v
`  U )  =  ( +v `  U
)
3 nvs.4 . . . . . . 7  |-  S  =  ( .s OLD `  U
)
4 eqid 2437 . . . . . . 7  |-  ( 0vec `  U )  =  (
0vec `  U )
5 nvs.6 . . . . . . 7  |-  N  =  ( normCV `  U )
61, 2, 3, 4, 5nvi 22094 . . . . . 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 972 . . . . 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 959 . . . . . 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 2782 . . . . 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 16 . . . 4  |-  ( U  e.  NrmCVec  ->  A. x  e.  X  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y )  x.  ( N `  x )
) )
11 oveq2 6090 . . . . . . 7  |-  ( x  =  B  ->  (
y S x )  =  ( y S B ) )
1211fveq2d 5733 . . . . . 6  |-  ( x  =  B  ->  ( N `  ( y S x ) )  =  ( N `  ( y S B ) ) )
13 fveq2 5729 . . . . . . 7  |-  ( x  =  B  ->  ( N `  x )  =  ( N `  B ) )
1413oveq2d 6098 . . . . . 6  |-  ( x  =  B  ->  (
( abs `  y
)  x.  ( N `
 x ) )  =  ( ( abs `  y )  x.  ( N `  B )
) )
1512, 14eqeq12d 2451 . . . . 5  |-  ( x  =  B  ->  (
( N `  (
y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) )  <->  ( N `  ( y S B ) )  =  ( ( abs `  y
)  x.  ( N `
 B ) ) ) )
16 oveq1 6089 . . . . . . 7  |-  ( y  =  A  ->  (
y S B )  =  ( A S B ) )
1716fveq2d 5733 . . . . . 6  |-  ( y  =  A  ->  ( N `  ( y S B ) )  =  ( N `  ( A S B ) ) )
18 fveq2 5729 . . . . . . 7  |-  ( y  =  A  ->  ( abs `  y )  =  ( abs `  A
) )
1918oveq1d 6097 . . . . . 6  |-  ( y  =  A  ->  (
( abs `  y
)  x.  ( N `
 B ) )  =  ( ( abs `  A )  x.  ( N `  B )
) )
2017, 19eqeq12d 2451 . . . . 5  |-  ( y  =  A  ->  (
( N `  (
y S B ) )  =  ( ( abs `  y )  x.  ( N `  B ) )  <->  ( N `  ( A S B ) )  =  ( ( abs `  A
)  x.  ( N `
 B ) ) ) )
2115, 20rspc2v 3059 . . . 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 31 . . 3  |-  ( ( B  e.  X  /\  A  e.  CC )  ->  ( U  e.  NrmCVec  -> 
( N `  ( A S B ) )  =  ( ( abs `  A )  x.  ( N `  B )
) ) )
23223impia 1151 . 2  |-  ( ( B  e.  X  /\  A  e.  CC  /\  U  e.  NrmCVec )  ->  ( N `  ( A S B ) )  =  ( ( abs `  A
)  x.  ( N `
 B ) ) )
24233com13 1159 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706   <.cop 3818   class class class wbr 4213   -->wf 5451   ` cfv 5455  (class class class)co 6082   CCcc 8989   RRcr 8990   0cc0 8991    + caddc 8994    x. cmul 8996    <_ cle 9122   abscabs 12040   CVec
OLDcvc 22025   NrmCVeccnv 22064   +vcpv 22065   BaseSetcba 22066   .s
OLDcns 22067   0veccn0v 22068   normCVcnmcv 22070
This theorem is referenced by:  nvsge0  22153  nvm1  22154  nvpi  22156  nvmtri  22161  smcnlem  22194  ipidsq  22210  minvecolem2  22378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-1st 6350  df-2nd 6351  df-vc 22026  df-nv 22072  df-va 22075  df-ba 22076  df-sm 22077  df-0v 22078  df-nmcv 22080
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