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Theorem isnvi 25179
Description: Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
isnvi.5  |-  X  =  ran  G
isnvi.6  |-  Z  =  (GId `  G )
isnvi.7  |-  <. G ,  S >.  e.  CVecOLD
isnvi.8  |-  N : X
--> RR
isnvi.9  |-  ( ( x  e.  X  /\  ( N `  x )  =  0 )  ->  x  =  Z )
isnvi.10  |-  ( ( y  e.  CC  /\  x  e.  X )  ->  ( N `  (
y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) ) )
isnvi.11  |-  ( ( x  e.  X  /\  y  e.  X )  ->  ( N `  (
x G y ) )  <_  ( ( N `  x )  +  ( N `  y ) ) )
isnvi.12  |-  U  = 
<. <. G ,  S >. ,  N >.
Assertion
Ref Expression
isnvi  |-  U  e.  NrmCVec
Distinct variable groups:    x, y, G    x, N, y    x, S, y    x, X, y
Allowed substitution hints:    U( x, y)    Z( x, y)

Proof of Theorem isnvi
StepHypRef Expression
1 isnvi.12 . 2  |-  U  = 
<. <. G ,  S >. ,  N >.
2 isnvi.7 . . 3  |-  <. G ,  S >.  e.  CVecOLD
3 isnvi.8 . . 3  |-  N : X
--> RR
4 isnvi.9 . . . . . 6  |-  ( ( x  e.  X  /\  ( N `  x )  =  0 )  ->  x  =  Z )
54ex 434 . . . . 5  |-  ( x  e.  X  ->  (
( N `  x
)  =  0  ->  x  =  Z )
)
6 isnvi.10 . . . . . . 7  |-  ( ( y  e.  CC  /\  x  e.  X )  ->  ( N `  (
y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) ) )
76ancoms 453 . . . . . 6  |-  ( ( x  e.  X  /\  y  e.  CC )  ->  ( N `  (
y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) ) )
87ralrimiva 2878 . . . . 5  |-  ( x  e.  X  ->  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) ) )
9 isnvi.11 . . . . . 6  |-  ( ( x  e.  X  /\  y  e.  X )  ->  ( N `  (
x G y ) )  <_  ( ( N `  x )  +  ( N `  y ) ) )
109ralrimiva 2878 . . . . 5  |-  ( x  e.  X  ->  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) )
115, 8, 103jca 1176 . . . 4  |-  ( 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 ) ) ) )
1211rgen 2824 . . 3  |-  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 ) ) )
13 isnvi.5 . . . 4  |-  X  =  ran  G
14 isnvi.6 . . . 4  |-  Z  =  (GId `  G )
1513, 14isnv 25178 . . 3  |-  ( <. <. G ,  S >. ,  N >.  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 ) ) ) ) )
162, 3, 12, 15mpbir3an 1178 . 2  |-  <. <. G ,  S >. ,  N >.  e.  NrmCVec
171, 16eqeltri 2551 1  |-  U  e.  NrmCVec
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   <.cop 4033   class class class wbr 4447   ran crn 5000   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488    + caddc 9491    x. cmul 9493    <_ cle 9625   abscabs 13024  GIdcgi 24862   CVecOLDcvc 25111   NrmCVeccnv 25150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-vc 25112  df-nv 25158
This theorem is referenced by:  cnnv  25255  hhnv  25755  hhssnv  25853
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