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Theorem nvi 26078
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  =  ( .sOLD `  U )
nvi.5  |-  Z  =  ( 0vec `  U
)
nvi.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvi  |-  ( U  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 ) ) ) ) )
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 2429 . . . . . 6  |-  ( 1st `  U )  =  ( 1st `  U )
2 nvi.6 . . . . . 6  |-  N  =  ( normCV `  U )
31, 2nvop2 26072 . . . . 5  |-  ( U  e.  NrmCVec  ->  U  =  <. ( 1st `  U ) ,  N >. )
4 nvi.2 . . . . . . 7  |-  G  =  ( +v `  U
)
5 nvi.4 . . . . . . 7  |-  S  =  ( .sOLD `  U )
61, 4, 5nvvop 26073 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  =  <. G ,  S >. )
76opeq1d 4196 . . . . 5  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  N >.  = 
<. <. G ,  S >. ,  N >. )
83, 7eqtrd 2470 . . . 4  |-  ( U  e.  NrmCVec  ->  U  =  <. <. G ,  S >. ,  N >. )
9 id 23 . . . 4  |-  ( U  e.  NrmCVec  ->  U  e.  NrmCVec )
108, 9eqeltrrd 2518 . . 3  |-  ( U  e.  NrmCVec  ->  <. <. G ,  S >. ,  N >.  e.  NrmCVec )
11 nvi.1 . . . . 5  |-  X  =  ( BaseSet `  U )
1211, 4bafval 26068 . . . 4  |-  X  =  ran  G
13 eqid 2429 . . . 4  |-  (GId `  G )  =  (GId
`  G )
1412, 13isnv 26076 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  <->  ( <. G ,  S >.  e.  CVecOLD 
/\  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 199 . 2  |-  ( U  e.  NrmCVec  ->  ( <. G ,  S >.  e.  CVecOLD  /\  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 26070 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  G ) )
1817eqeq2d 2443 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( x  =  Z  <->  x  =  (GId `  G ) ) )
1918imbi2d 317 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( ( ( N `  x )  =  0  ->  x  =  Z )  <->  ( ( N `  x )  =  0  ->  x  =  (GId `  G )
) ) )
20193anbi1d 1339 . . . 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 2871 . . 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 1341 . 2  |-  ( U  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 ) ) ) )  <->  ( <. G ,  S >.  e.  CVecOLD 
/\  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 235 1  |-  ( U  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 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   <.cop 4008   class class class wbr 4426   -->wf 5597   ` cfv 5601  (class class class)co 6305   1stc1st 6805   CCcc 9536   RRcr 9537   0cc0 9538    + caddc 9541    x. cmul 9543    <_ cle 9675   abscabs 13276  GIdcgi 25760   CVecOLDcvc 26009   NrmCVeccnv 26048   +vcpv 26049   BaseSetcba 26050   .sOLDcns 26051   0veccn0v 26052   normCVcnmcv 26054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-1st 6807  df-2nd 6808  df-vc 26010  df-nv 26056  df-va 26059  df-ba 26060  df-sm 26061  df-0v 26062  df-nmcv 26064
This theorem is referenced by:  nvvc  26079  nvf  26132  nvs  26136  nvz  26143  nvtri  26144
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