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Theorem nvf 21279
Description: Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvf.1  |-  X  =  ( BaseSet `  U )
nvf.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvf  |-  ( U  e.  NrmCVec  ->  N : X --> RR )

Proof of Theorem nvf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvf.1 . . 3  |-  X  =  ( BaseSet `  U )
2 eqid 2316 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2316 . . 3  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 eqid 2316 . . 3  |-  ( 0vec `  U )  =  (
0vec `  U )
5 nvf.6 . . 3  |-  N  =  ( normCV `  U )
61, 2, 3, 4, 5nvi 21225 . 2  |-  ( U  e.  NrmCVec  ->  ( <. ( +v `  U ) ,  ( .s OLD `  U
) >.  e.  CVec OLD  /\  N : X --> RR  /\  A. x  e.  X  ( ( ( N `  x )  =  0  ->  x  =  (
0vec `  U )
)  /\  A. y  e.  CC  ( N `  ( y ( .s
OLD `  U )
x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x ( +v
`  U ) y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) ) )
76simp2d 968 1  |-  ( U  e.  NrmCVec  ->  N : X --> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1633    e. wcel 1701   A.wral 2577   <.cop 3677   class class class wbr 4060   -->wf 5288   ` cfv 5292  (class class class)co 5900   CCcc 8780   RRcr 8781   0cc0 8782    + caddc 8785    x. cmul 8787    <_ cle 8913   abscabs 11766   CVec
OLDcvc 21156   NrmCVeccnv 21195   +vcpv 21196   BaseSetcba 21197   .s
OLDcns 21198   0veccn0v 21199   normCVcnmcv 21201
This theorem is referenced by:  nvcl  21280  nvdm  21282  imsdf  21313  nmcvcn  21323  sspn  21367  hilnormi  21797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-1st 6164  df-2nd 6165  df-vc 21157  df-nv 21203  df-va 21206  df-ba 21207  df-sm 21208  df-0v 21209  df-nmcv 21211
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