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Theorem nvi 23997
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  =  ( .sOLD `  U )
nvi.5  |-  Z  =  ( 0vec `  U
)
nvi.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvi  |-  ( U  e.  NrmCVec  ->  ( <. G ,  S >.  e.  CVecOLD  /\  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 2443 . . . . . 6  |-  ( 1st `  U )  =  ( 1st `  U )
2 nvi.6 . . . . . 6  |-  N  =  ( normCV `  U )
31, 2nvop2 23991 . . . . 5  |-  ( U  e.  NrmCVec  ->  U  =  <. ( 1st `  U ) ,  N >. )
4 nvi.2 . . . . . . 7  |-  G  =  ( +v `  U
)
5 nvi.4 . . . . . . 7  |-  S  =  ( .sOLD `  U )
61, 4, 5nvvop 23992 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  =  <. G ,  S >. )
76opeq1d 4070 . . . . 5  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  N >.  = 
<. <. G ,  S >. ,  N >. )
83, 7eqtrd 2475 . . . 4  |-  ( U  e.  NrmCVec  ->  U  =  <. <. G ,  S >. ,  N >. )
9 id 22 . . . 4  |-  ( U  e.  NrmCVec  ->  U  e.  NrmCVec )
108, 9eqeltrrd 2518 . . 3  |-  ( U  e.  NrmCVec  ->  <. <. G ,  S >. ,  N >.  e.  NrmCVec )
11 nvi.1 . . . . 5  |-  X  =  ( BaseSet `  U )
1211, 4bafval 23987 . . . 4  |-  X  =  ran  G
13 eqid 2443 . . . 4  |-  (GId `  G )  =  (GId
`  G )
1412, 13isnv 23995 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  <->  ( <. G ,  S >.  e.  CVecOLD 
/\  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 196 . 2  |-  ( U  e.  NrmCVec  ->  ( <. G ,  S >.  e.  CVecOLD  /\  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 23989 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  G ) )
1817eqeq2d 2454 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( x  =  Z  <->  x  =  (GId `  G ) ) )
1918imbi2d 316 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( ( ( N `  x )  =  0  ->  x  =  Z )  <->  ( ( N `  x )  =  0  ->  x  =  (GId `  G )
) ) )
20193anbi1d 1293 . . . 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 2740 . . 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 1295 . 2  |-  ( U  e.  NrmCVec  ->  ( ( <. G ,  S >.  e. 
CVecOLD  /\  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.  CVecOLD 
/\  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 232 1  |-  ( U  e.  NrmCVec  ->  ( <. G ,  S >.  e.  CVecOLD  /\  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 setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720   <.cop 3888   class class class wbr 4297   -->wf 5419   ` cfv 5423  (class class class)co 6096   1stc1st 6580   CCcc 9285   RRcr 9286   0cc0 9287    + caddc 9290    x. cmul 9292    <_ cle 9424   abscabs 12728  GIdcgi 23679   CVecOLDcvc 23928   NrmCVeccnv 23967   +vcpv 23968   BaseSetcba 23969   .sOLDcns 23970   0veccn0v 23971   normCVcnmcv 23973
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-1st 6582  df-2nd 6583  df-vc 23929  df-nv 23975  df-va 23978  df-ba 23979  df-sm 23980  df-0v 23981  df-nmcv 23983
This theorem is referenced by:  nvvc  23998  nvf  24051  nvs  24055  nvz  24062  nvtri  24063
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