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Theorem nvs 24048
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  =  ( .sOLD `  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 2441 . . . . . . 7  |-  ( +v
`  U )  =  ( +v `  U
)
3 nvs.4 . . . . . . 7  |-  S  =  ( .sOLD `  U )
4 eqid 2441 . . . . . . 7  |-  ( 0vec `  U )  =  (
0vec `  U )
5 nvs.6 . . . . . . 7  |-  N  =  ( normCV `  U )
61, 2, 3, 4, 5nvi 23990 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( <. ( +v `  U ) ,  S >.  e.  CVecOLD 
/\  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 1002 . . . . 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 989 . . . . . 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 2789 . . . . 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 6097 . . . . . . 7  |-  ( x  =  B  ->  (
y S x )  =  ( y S B ) )
1211fveq2d 5693 . . . . . 6  |-  ( x  =  B  ->  ( N `  ( y S x ) )  =  ( N `  ( y S B ) ) )
13 fveq2 5689 . . . . . . 7  |-  ( x  =  B  ->  ( N `  x )  =  ( N `  B ) )
1413oveq2d 6105 . . . . . 6  |-  ( x  =  B  ->  (
( abs `  y
)  x.  ( N `
 x ) )  =  ( ( abs `  y )  x.  ( N `  B )
) )
1512, 14eqeq12d 2455 . . . . 5  |-  ( x  =  B  ->  (
( N `  (
y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) )  <->  ( N `  ( y S B ) )  =  ( ( abs `  y
)  x.  ( N `
 B ) ) ) )
16 oveq1 6096 . . . . . . 7  |-  ( y  =  A  ->  (
y S B )  =  ( A S B ) )
1716fveq2d 5693 . . . . . 6  |-  ( y  =  A  ->  ( N `  ( y S B ) )  =  ( N `  ( A S B ) ) )
18 fveq2 5689 . . . . . . 7  |-  ( y  =  A  ->  ( abs `  y )  =  ( abs `  A
) )
1918oveq1d 6104 . . . . . 6  |-  ( y  =  A  ->  (
( abs `  y
)  x.  ( N `
 B ) )  =  ( ( abs `  A )  x.  ( N `  B )
) )
2017, 19eqeq12d 2455 . . . . 5  |-  ( y  =  A  ->  (
( N `  (
y S B ) )  =  ( ( abs `  y )  x.  ( N `  B ) )  <->  ( N `  ( A S B ) )  =  ( ( abs `  A
)  x.  ( N `
 B ) ) ) )
2115, 20rspc2v 3077 . . . 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 32 . . 3  |-  ( ( B  e.  X  /\  A  e.  CC )  ->  ( U  e.  NrmCVec  -> 
( N `  ( A S B ) )  =  ( ( abs `  A )  x.  ( N `  B )
) ) )
23223impia 1184 . 2  |-  ( ( B  e.  X  /\  A  e.  CC  /\  U  e.  NrmCVec )  ->  ( N `  ( A S B ) )  =  ( ( abs `  A
)  x.  ( N `
 B ) ) )
24233com13 1192 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( N `  ( A S B ) )  =  ( ( abs `  A
)  x.  ( N `
 B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2713   <.cop 3881   class class class wbr 4290   -->wf 5412   ` cfv 5416  (class class class)co 6089   CCcc 9278   RRcr 9279   0cc0 9280    + caddc 9283    x. cmul 9285    <_ cle 9417   abscabs 12721   CVecOLDcvc 23921   NrmCVeccnv 23960   +vcpv 23961   BaseSetcba 23962   .sOLDcns 23963   0veccn0v 23964   normCVcnmcv 23966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-1st 6575  df-2nd 6576  df-vc 23922  df-nv 23968  df-va 23971  df-ba 23972  df-sm 23973  df-0v 23974  df-nmcv 23976
This theorem is referenced by:  nvsge0  24049  nvm1  24050  nvpi  24052  nvmtri  24057  smcnlem  24090  ipidsq  24106  minvecolem2  24274
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