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Theorem elcnfn 25284
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 5688 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  w )  =  ( T `  w ) )
2 fveq1 5688 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
31, 2oveq12d 6107 . . . . . . . 8  |-  ( t  =  T  ->  (
( t `  w
)  -  ( t `
 x ) )  =  ( ( T `
 w )  -  ( T `  x ) ) )
43fveq2d 5693 . . . . . . 7  |-  ( t  =  T  ->  ( abs `  ( ( t `
 w )  -  ( t `  x
) ) )  =  ( abs `  (
( T `  w
)  -  ( T `
 x ) ) ) )
54breq1d 4300 . . . . . 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 2757 . . . 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 2755 . . 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 25249 . . 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 3117 . 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 9361 . . . 4  |-  CC  e.  _V
12 ax-hilex 24399 . . . 4  |-  ~H  e.  _V
1311, 12elmap 7239 . . 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 1369    e. wcel 1756   A.wral 2713   E.wrex 2714   class class class wbr 4290   -->wf 5412   ` cfv 5416  (class class class)co 6089    ^m cmap 7212   CCcc 9278    < clt 9416    - cmin 9593   RR+crp 10989   abscabs 12721   ~Hchil 24319   normhcno 24323    -h cmv 24325   ConFnccnfn 24353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-hilex 24399
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-map 7214  df-cnfn 25249
This theorem is referenced by:  cnfnc  25332  0cnfn  25382  lnfnconi  25457
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