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Theorem elcnfn 27520
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 5876 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  w )  =  ( T `  w ) )
2 fveq1 5876 . . . . . . . . 9  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
31, 2oveq12d 6319 . . . . . . . 8  |-  ( t  =  T  ->  (
( t `  w
)  -  ( t `
 x ) )  =  ( ( T `
 w )  -  ( T `  x ) ) )
43fveq2d 5881 . . . . . . 7  |-  ( t  =  T  ->  ( abs `  ( ( t `
 w )  -  ( t `  x
) ) )  =  ( abs `  (
( T `  w
)  -  ( T `
 x ) ) ) )
54breq1d 4430 . . . . . 6  |-  ( t  =  T  ->  (
( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y  <->  ( abs `  ( ( T `  w )  -  ( T `  x )
) )  <  y
) )
65imbi2d 317 . . . . 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 2947 . . . 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 2869 . . 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 27485 . . 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 3231 . 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 9620 . . . 4  |-  CC  e.  _V
12 ax-hilex 26637 . . . 4  |-  ~H  e.  _V
1311, 12elmap 7504 . . 3  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
1413anbi1i 699 . 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 252 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 187    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775   E.wrex 2776   class class class wbr 4420   -->wf 5593   ` cfv 5597  (class class class)co 6301    ^m cmap 7476   CCcc 9537    < clt 9675    - cmin 9860   RR+crp 11302   abscabs 13285   ~Hchil 26557   normhcno 26561    -h cmv 26563   ConFnccnfn 26591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-hilex 26637
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-map 7478  df-cnfn 27485
This theorem is referenced by:  cnfnc  27568  0cnfn  27618  lnfnconi  27693
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