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Theorem nvs 25691
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 2457 . . . . . . 7  |-  ( +v
`  U )  =  ( +v `  U
)
3 nvs.4 . . . . . . 7  |-  S  =  ( .sOLD `  U )
4 eqid 2457 . . . . . . 7  |-  ( 0vec `  U )  =  (
0vec `  U )
5 nvs.6 . . . . . . 7  |-  N  =  ( normCV `  U )
61, 2, 3, 4, 5nvi 25633 . . . . . 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 1010 . . . . 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 997 . . . . . 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 2850 . . . . 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 6304 . . . . . . 7  |-  ( x  =  B  ->  (
y S x )  =  ( y S B ) )
1211fveq2d 5876 . . . . . 6  |-  ( x  =  B  ->  ( N `  ( y S x ) )  =  ( N `  ( y S B ) ) )
13 fveq2 5872 . . . . . . 7  |-  ( x  =  B  ->  ( N `  x )  =  ( N `  B ) )
1413oveq2d 6312 . . . . . 6  |-  ( x  =  B  ->  (
( abs `  y
)  x.  ( N `
 x ) )  =  ( ( abs `  y )  x.  ( N `  B )
) )
1512, 14eqeq12d 2479 . . . . 5  |-  ( x  =  B  ->  (
( N `  (
y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) )  <->  ( N `  ( y S B ) )  =  ( ( abs `  y
)  x.  ( N `
 B ) ) ) )
16 oveq1 6303 . . . . . . 7  |-  ( y  =  A  ->  (
y S B )  =  ( A S B ) )
1716fveq2d 5876 . . . . . 6  |-  ( y  =  A  ->  ( N `  ( y S B ) )  =  ( N `  ( A S B ) ) )
18 fveq2 5872 . . . . . . 7  |-  ( y  =  A  ->  ( abs `  y )  =  ( abs `  A
) )
1918oveq1d 6311 . . . . . 6  |-  ( y  =  A  ->  (
( abs `  y
)  x.  ( N `
 B ) )  =  ( ( abs `  A )  x.  ( N `  B )
) )
2017, 19eqeq12d 2479 . . . . 5  |-  ( y  =  A  ->  (
( N `  (
y S B ) )  =  ( ( abs `  y )  x.  ( N `  B ) )  <->  ( N `  ( A S B ) )  =  ( ( abs `  A
)  x.  ( N `
 B ) ) ) )
2115, 20rspc2v 3219 . . . 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 1193 . 2  |-  ( ( B  e.  X  /\  A  e.  CC  /\  U  e.  NrmCVec )  ->  ( N `  ( A S B ) )  =  ( ( abs `  A
)  x.  ( N `
 B ) ) )
24233com13 1201 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   <.cop 4038   class class class wbr 4456   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509    + caddc 9512    x. cmul 9514    <_ cle 9646   abscabs 13078   CVecOLDcvc 25564   NrmCVeccnv 25603   +vcpv 25604   BaseSetcba 25605   .sOLDcns 25606   0veccn0v 25607   normCVcnmcv 25609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-1st 6799  df-2nd 6800  df-vc 25565  df-nv 25611  df-va 25614  df-ba 25615  df-sm 25616  df-0v 25617  df-nmcv 25619
This theorem is referenced by:  nvsge0  25692  nvm1  25693  nvpi  25695  nvmtri  25700  smcnlem  25733  ipidsq  25749  minvecolem2  25917
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