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Theorem nvi 22085
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 2435 . . . . . 6  |-  ( 1st `  U )  =  ( 1st `  U )
2 nvi.6 . . . . . 6  |-  N  =  ( normCV `  U )
31, 2nvop2 22079 . . . . 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 22080 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  =  <. G ,  S >. )
76opeq1d 3982 . . . . 5  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  N >.  = 
<. <. G ,  S >. ,  N >. )
83, 7eqtrd 2467 . . . 4  |-  ( U  e.  NrmCVec  ->  U  =  <. <. G ,  S >. ,  N >. )
9 id 20 . . . 4  |-  ( U  e.  NrmCVec  ->  U  e.  NrmCVec )
108, 9eqeltrrd 2510 . . 3  |-  ( U  e.  NrmCVec  ->  <. <. G ,  S >. ,  N >.  e.  NrmCVec )
11 nvi.1 . . . . 5  |-  X  =  ( BaseSet `  U )
1211, 4bafval 22075 . . . 4  |-  X  =  ran  G
13 eqid 2435 . . . 4  |-  (GId `  G )  =  (GId
`  G )
1412, 13isnv 22083 . . 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 22077 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  G ) )
1817eqeq2d 2446 . . . . . 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 2717 . . 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 1652    e. wcel 1725   A.wral 2697   <.cop 3809   class class class wbr 4204   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   CCcc 8980   RRcr 8981   0cc0 8982    + caddc 8985    x. cmul 8987    <_ cle 9113   abscabs 12031  GIdcgi 21767   CVec OLDcvc 22016   NrmCVeccnv 22055   +vcpv 22056   BaseSetcba 22057   .s
OLDcns 22058   0veccn0v 22059   normCVcnmcv 22061
This theorem is referenced by:  nvvc  22086  nvf  22139  nvs  22143  nvz  22150  nvtri  22151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-1st 6341  df-2nd 6342  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071
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