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Theorem nvi 21170
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 2283 . . . . . 6  |-  ( 1st `  U )  =  ( 1st `  U )
2 nvi.6 . . . . . 6  |-  N  =  ( normCV `  U )
31, 2nvop2 21164 . . . . 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 21165 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  =  <. G ,  S >. )
76opeq1d 3802 . . . . 5  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  N >.  = 
<. <. G ,  S >. ,  N >. )
83, 7eqtrd 2315 . . . 4  |-  ( U  e.  NrmCVec  ->  U  =  <. <. G ,  S >. ,  N >. )
9 id 19 . . . 4  |-  ( U  e.  NrmCVec  ->  U  e.  NrmCVec )
108, 9eqeltrrd 2358 . . 3  |-  ( U  e.  NrmCVec  ->  <. <. G ,  S >. ,  N >.  e.  NrmCVec )
11 nvi.1 . . . . 5  |-  X  =  ( BaseSet `  U )
1211, 4bafval 21160 . . . 4  |-  X  =  ran  G
13 eqid 2283 . . . 4  |-  (GId `  G )  =  (GId
`  G )
1412, 13isnv 21168 . . 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 188 . 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 21162 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  G ) )
1817eqeq2d 2294 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( x  =  Z  <->  x  =  (GId `  G ) ) )
1918imbi2d 307 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( ( ( N `  x )  =  0  ->  x  =  Z )  <->  ( ( N `  x )  =  0  ->  x  =  (GId `  G )
) ) )
20193anbi1d 1256 . . . 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 2563 . . 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 1258 . 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 223 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 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    <_ cle 8868   abscabs 11719  GIdcgi 20854   CVec OLDcvc 21101   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   0veccn0v 21144   normCVcnmcv 21146
This theorem is referenced by:  nvvc  21171  nvf  21224  nvs  21228  nvz  21235  nvtri  21236
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-1st 6122  df-2nd 6123  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156
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