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Theorem elcnfn 26474
Description: Property defining a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elcnfn  |-  ( T  e.  ConFn 
<->  ( T : ~H --> CC  /\  A. x  e. 
~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( abs `  (
( T `  w
)  -  ( T `
 x ) ) )  <  y ) ) )
Distinct variable group:    x, w, y, z, T

Proof of Theorem elcnfn
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 fveq1 5863 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  w )  =  ( T `  w ) )
2 fveq1 5863 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
31, 2oveq12d 6300 . . . . . . . 8  |-  ( t  =  T  ->  (
( t `  w
)  -  ( t `
 x ) )  =  ( ( T `
 w )  -  ( T `  x ) ) )
43fveq2d 5868 . . . . . . 7  |-  ( t  =  T  ->  ( abs `  ( ( t `
 w )  -  ( t `  x
) ) )  =  ( abs `  (
( T `  w
)  -  ( T `
 x ) ) ) )
54breq1d 4457 . . . . . 6  |-  ( t  =  T  ->  (
( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y  <->  ( abs `  ( ( T `  w )  -  ( T `  x )
) )  <  y
) )
65imbi2d 316 . . . . 5  |-  ( t  =  T  ->  (
( ( normh `  (
w  -h  x ) )  <  z  -> 
( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y )  <-> 
( ( normh `  (
w  -h  x ) )  <  z  -> 
( abs `  (
( T `  w
)  -  ( T `
 x ) ) )  <  y ) ) )
76rexralbidv 2981 . . . 4  |-  ( t  =  T  ->  ( E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y )  <->  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( abs `  (
( T `  w
)  -  ( T `
 x ) ) )  <  y ) ) )
872ralbidv 2908 . . 3  |-  ( t  =  T  ->  ( A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e. 
~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y )  <->  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e. 
~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( abs `  (
( T `  w
)  -  ( T `
 x ) ) )  <  y ) ) )
9 df-cnfn 26439 . . 3  |-  ConFn  =  {
t  e.  ( CC 
^m  ~H )  |  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y ) }
108, 9elrab2 3263 . 2  |-  ( T  e.  ConFn 
<->  ( T  e.  ( CC  ^m  ~H )  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e. 
~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( abs `  (
( T `  w
)  -  ( T `
 x ) ) )  <  y ) ) )
11 cnex 9569 . . . 4  |-  CC  e.  _V
12 ax-hilex 25589 . . . 4  |-  ~H  e.  _V
1311, 12elmap 7444 . . 3  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
1413anbi1i 695 . 2  |-  ( ( T  e.  ( CC 
^m  ~H )  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( abs `  (
( T `  w
)  -  ( T `
 x ) ) )  <  y ) )  <->  ( T : ~H
--> CC  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
( normh `  ( w  -h  x ) )  < 
z  ->  ( abs `  ( ( T `  w )  -  ( T `  x )
) )  <  y
) ) )
1510, 14bitri 249 1  |-  ( T  e.  ConFn 
<->  ( T : ~H --> CC  /\  A. x  e. 
~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( abs `  (
( T `  w
)  -  ( T `
 x ) ) )  <  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   class class class wbr 4447   -->wf 5582   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   CCcc 9486    < clt 9624    - cmin 9801   RR+crp 11216   abscabs 13024   ~Hchil 25509   normhcno 25513    -h cmv 25515   ConFnccnfn 25543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-hilex 25589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-cnfn 26439
This theorem is referenced by:  cnfnc  26522  0cnfn  26572  lnfnconi  26647
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