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Theorem elcnfn 23385
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 5727 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  w )  =  ( T `  w ) )
2 fveq1 5727 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
31, 2oveq12d 6099 . . . . . . . 8  |-  ( t  =  T  ->  (
( t `  w
)  -  ( t `
 x ) )  =  ( ( T `
 w )  -  ( T `  x ) ) )
43fveq2d 5732 . . . . . . 7  |-  ( t  =  T  ->  ( abs `  ( ( t `
 w )  -  ( t `  x
) ) )  =  ( abs `  (
( T `  w
)  -  ( T `
 x ) ) ) )
54breq1d 4222 . . . . . 6  |-  ( t  =  T  ->  (
( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y  <->  ( abs `  ( ( T `  w )  -  ( T `  x )
) )  <  y
) )
65imbi2d 308 . . . . 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 2749 . . . 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 2747 . . 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 23350 . . 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 3094 . 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 9071 . . . 4  |-  CC  e.  _V
12 ax-hilex 22502 . . . 4  |-  ~H  e.  _V
1311, 12elmap 7042 . . 3  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
1413anbi1i 677 . 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 241 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   class class class wbr 4212   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   CCcc 8988    < clt 9120    - cmin 9291   RR+crp 10612   abscabs 12039   ~Hchil 22422   normhcno 22426    -h cmv 22428   ConFnccnfn 22456
This theorem is referenced by:  cnfnc  23433  0cnfn  23483  lnfnconi  23558
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-hilex 22502
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-cnfn 23350
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