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Theorem isnvi 21482
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.  CVec OLD
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.  CVec OLD
3 isnvi.8 . . 3  |-  N : X
--> RR
4 isnvi.9 . . . . . 6  |-  ( ( x  e.  X  /\  ( N `  x )  =  0 )  ->  x  =  Z )
54ex 423 . . . . 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 439 . . . . . 6  |-  ( ( x  e.  X  /\  y  e.  CC )  ->  ( N `  (
y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) ) )
87ralrimiva 2711 . . . . 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 2711 . . . . 5  |-  ( x  e.  X  ->  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) )
115, 8, 103jca 1133 . . . 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 2693 . . 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 21481 . . 3  |-  ( <. <. G ,  S >. ,  N >.  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 ) ) ) ) )
162, 3, 12, 15mpbir3an 1135 . 2  |-  <. <. G ,  S >. ,  N >.  e.  NrmCVec
171, 16eqeltri 2436 1  |-  U  e.  NrmCVec
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   A.wral 2628   <.cop 3732   class class class wbr 4125   ran crn 4793   -->wf 5354   ` cfv 5358  (class class class)co 5981   CCcc 8882   RRcr 8883   0cc0 8884    + caddc 8887    x. cmul 8889    <_ cle 9015   abscabs 11926  GIdcgi 21165   CVec OLDcvc 21414   NrmCVeccnv 21453
This theorem is referenced by:  cnnv  21558  hhnv  22057  hhssnv  22154
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-vc 21415  df-nv 21461
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