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Theorem elcnfn 22478
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 5540 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  w )  =  ( T `  w ) )
2 fveq1 5540 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
31, 2oveq12d 5892 . . . . . . . 8  |-  ( t  =  T  ->  (
( t `  w
)  -  ( t `
 x ) )  =  ( ( T `
 w )  -  ( T `  x ) ) )
43fveq2d 5545 . . . . . . 7  |-  ( t  =  T  ->  ( abs `  ( ( t `
 w )  -  ( t `  x
) ) )  =  ( abs `  (
( T `  w
)  -  ( T `
 x ) ) ) )
54breq1d 4049 . . . . . 6  |-  ( t  =  T  ->  (
( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y  <->  ( abs `  ( ( T `  w )  -  ( T `  x )
) )  <  y
) )
65imbi2d 307 . . . . 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 2600 . . . 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 2598 . . 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 22443 . . 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 2938 . 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 8834 . . . 4  |-  CC  e.  _V
12 ax-hilex 21595 . . . 4  |-  ~H  e.  _V
1311, 12elmap 6812 . . 3  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
1413anbi1i 676 . 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 240 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   CCcc 8751    < clt 8883    - cmin 9053   RR+crp 10370   abscabs 11735   ~Hchil 21515   normhcno 21519    -h cmv 21521   ConFnccnfn 21549
This theorem is referenced by:  cnfnc  22526  0cnfn  22576  lnfnconi  22651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-cnfn 22443
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