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Theorem isnvi 22053
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 424 . . . . 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 440 . . . . . 6  |-  ( ( x  e.  X  /\  y  e.  CC )  ->  ( N `  (
y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) ) )
87ralrimiva 2757 . . . . 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 2757 . . . . 5  |-  ( x  e.  X  ->  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) )
115, 8, 103jca 1134 . . . 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 2739 . . 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 22052 . . 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 1136 . 2  |-  <. <. G ,  S >. ,  N >.  e.  NrmCVec
171, 16eqeltri 2482 1  |-  U  e.  NrmCVec
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674   <.cop 3785   class class class wbr 4180   ran crn 4846   -->wf 5417   ` cfv 5421  (class class class)co 6048   CCcc 8952   RRcr 8953   0cc0 8954    + caddc 8957    x. cmul 8959    <_ cle 9085   abscabs 12002  GIdcgi 21736   CVec OLDcvc 21985   NrmCVeccnv 22024
This theorem is referenced by:  cnnv  22129  hhnv  22628  hhssnv  22725
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-vc 21986  df-nv 22032
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