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Theorem nvi 21941
Description: The properties of a normed complex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvi.1  |-  X  =  ( BaseSet `  U )
nvi.2  |-  G  =  ( +v `  U
)
nvi.4  |-  S  =  ( .s OLD `  U
)
nvi.5  |-  Z  =  ( 0vec `  U
)
nvi.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvi  |-  ( U  e.  NrmCVec  ->  ( <. G ,  S >.  e.  CVec OLD  /\  N : X --> RR  /\  A. x  e.  X  ( ( ( N `  x )  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) ) )
Distinct variable groups:    x, y, G    x, N, y    x, U    x, S, y    x, X, y
Allowed substitution hints:    U( y)    Z( x, y)

Proof of Theorem nvi
StepHypRef Expression
1 eqid 2387 . . . . . 6  |-  ( 1st `  U )  =  ( 1st `  U )
2 nvi.6 . . . . . 6  |-  N  =  ( normCV `  U )
31, 2nvop2 21935 . . . . 5  |-  ( U  e.  NrmCVec  ->  U  =  <. ( 1st `  U ) ,  N >. )
4 nvi.2 . . . . . . 7  |-  G  =  ( +v `  U
)
5 nvi.4 . . . . . . 7  |-  S  =  ( .s OLD `  U
)
61, 4, 5nvvop 21936 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  =  <. G ,  S >. )
76opeq1d 3932 . . . . 5  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  N >.  = 
<. <. G ,  S >. ,  N >. )
83, 7eqtrd 2419 . . . 4  |-  ( U  e.  NrmCVec  ->  U  =  <. <. G ,  S >. ,  N >. )
9 id 20 . . . 4  |-  ( U  e.  NrmCVec  ->  U  e.  NrmCVec )
108, 9eqeltrrd 2462 . . 3  |-  ( U  e.  NrmCVec  ->  <. <. G ,  S >. ,  N >.  e.  NrmCVec )
11 nvi.1 . . . . 5  |-  X  =  ( BaseSet `  U )
1211, 4bafval 21931 . . . 4  |-  X  =  ran  G
13 eqid 2387 . . . 4  |-  (GId `  G )  =  (GId
`  G )
1412, 13isnv 21939 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  <->  ( <. G ,  S >.  e.  CVec OLD 
/\  N : X --> RR  /\  A. x  e.  X  ( ( ( N `  x )  =  0  ->  x  =  (GId `  G )
)  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) ) )
1510, 14sylib 189 . 2  |-  ( U  e.  NrmCVec  ->  ( <. G ,  S >.  e.  CVec OLD  /\  N : X --> RR  /\  A. x  e.  X  ( ( ( N `  x )  =  0  ->  x  =  (GId
`  G ) )  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) ) )
16 nvi.5 . . . . . . . 8  |-  Z  =  ( 0vec `  U
)
174, 160vfval 21933 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  G ) )
1817eqeq2d 2398 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( x  =  Z  <->  x  =  (GId `  G ) ) )
1918imbi2d 308 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( ( ( N `  x )  =  0  ->  x  =  Z )  <->  ( ( N `  x )  =  0  ->  x  =  (GId `  G )
) ) )
20193anbi1d 1258 . . . 4  |-  ( U  e.  NrmCVec  ->  ( ( ( ( N `  x
)  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y )  x.  ( N `  x )
)  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) )  <->  ( ( ( N `  x )  =  0  ->  x  =  (GId `  G )
)  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) ) )
2120ralbidv 2669 . . 3  |-  ( U  e.  NrmCVec  ->  ( A. x  e.  X  ( (
( N `  x
)  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y )  x.  ( N `  x )
)  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) )  <->  A. x  e.  X  ( ( ( N `
 x )  =  0  ->  x  =  (GId `  G ) )  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) ) )
22213anbi3d 1260 . 2  |-  ( U  e.  NrmCVec  ->  ( ( <. G ,  S >.  e. 
CVec OLD  /\  N : X
--> RR  /\  A. x  e.  X  ( (
( N `  x
)  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y )  x.  ( N `  x )
)  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) )  <->  ( <. G ,  S >.  e.  CVec OLD 
/\  N : X --> RR  /\  A. x  e.  X  ( ( ( N `  x )  =  0  ->  x  =  (GId `  G )
)  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) ) ) )
2315, 22mpbird 224 1  |-  ( U  e.  NrmCVec  ->  ( <. G ,  S >.  e.  CVec OLD  /\  N : X --> RR  /\  A. x  e.  X  ( ( ( N `  x )  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   <.cop 3760   class class class wbr 4153   -->wf 5390   ` cfv 5394  (class class class)co 6020   1stc1st 6286   CCcc 8921   RRcr 8922   0cc0 8923    + caddc 8926    x. cmul 8928    <_ cle 9054   abscabs 11966  GIdcgi 21623   CVec OLDcvc 21872   NrmCVeccnv 21911   +vcpv 21912   BaseSetcba 21913   .s
OLDcns 21914   0veccn0v 21915   normCVcnmcv 21917
This theorem is referenced by:  nvvc  21942  nvf  21995  nvs  21999  nvz  22006  nvtri  22007
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-1st 6288  df-2nd 6289  df-vc 21873  df-nv 21919  df-va 21922  df-ba 21923  df-sm 21924  df-0v 21925  df-nmcv 21927
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