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Theorem cnfnc 23433
Description: Basic continuity property of a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
cnfnc  |-  ( ( T  e.  ConFn  /\  A  e.  ~H  /\  B  e.  RR+ )  ->  E. x  e.  RR+  A. y  e. 
~H  ( ( normh `  ( y  -h  A
) )  <  x  ->  ( abs `  (
( T `  y
)  -  ( T `
 A ) ) )  <  B ) )
Distinct variable groups:    x, y, A    x, B, y    x, T, y

Proof of Theorem cnfnc
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elcnfn 23385 . . . 4  |-  ( T  e.  ConFn 
<->  ( T : ~H --> CC  /\  A. z  e. 
~H  A. w  e.  RR+  E. x  e.  RR+  A. y  e.  ~H  ( ( normh `  ( y  -h  z
) )  <  x  ->  ( abs `  (
( T `  y
)  -  ( T `
 z ) ) )  <  w ) ) )
21simprbi 451 . . 3  |-  ( T  e.  ConFn  ->  A. z  e.  ~H  A. w  e.  RR+  E. x  e.  RR+  A. y  e.  ~H  (
( normh `  ( y  -h  z ) )  < 
x  ->  ( abs `  ( ( T `  y )  -  ( T `  z )
) )  <  w
) )
3 oveq2 6089 . . . . . . . 8  |-  ( z  =  A  ->  (
y  -h  z )  =  ( y  -h  A ) )
43fveq2d 5732 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( y  -h  z ) )  =  ( normh `  ( y  -h  A ) ) )
54breq1d 4222 . . . . . 6  |-  ( z  =  A  ->  (
( normh `  ( y  -h  z ) )  < 
x  <->  ( normh `  (
y  -h  A ) )  <  x ) )
6 fveq2 5728 . . . . . . . . 9  |-  ( z  =  A  ->  ( T `  z )  =  ( T `  A ) )
76oveq2d 6097 . . . . . . . 8  |-  ( z  =  A  ->  (
( T `  y
)  -  ( T `
 z ) )  =  ( ( T `
 y )  -  ( T `  A ) ) )
87fveq2d 5732 . . . . . . 7  |-  ( z  =  A  ->  ( abs `  ( ( T `
 y )  -  ( T `  z ) ) )  =  ( abs `  ( ( T `  y )  -  ( T `  A ) ) ) )
98breq1d 4222 . . . . . 6  |-  ( z  =  A  ->  (
( abs `  (
( T `  y
)  -  ( T `
 z ) ) )  <  w  <->  ( abs `  ( ( T `  y )  -  ( T `  A )
) )  <  w
) )
105, 9imbi12d 312 . . . . 5  |-  ( z  =  A  ->  (
( ( normh `  (
y  -h  z ) )  <  x  -> 
( abs `  (
( T `  y
)  -  ( T `
 z ) ) )  <  w )  <-> 
( ( normh `  (
y  -h  A ) )  <  x  -> 
( abs `  (
( T `  y
)  -  ( T `
 A ) ) )  <  w ) ) )
1110rexralbidv 2749 . . . 4  |-  ( z  =  A  ->  ( E. x  e.  RR+  A. y  e.  ~H  ( ( normh `  ( y  -h  z
) )  <  x  ->  ( abs `  (
( T `  y
)  -  ( T `
 z ) ) )  <  w )  <->  E. x  e.  RR+  A. y  e.  ~H  ( ( normh `  ( y  -h  A
) )  <  x  ->  ( abs `  (
( T `  y
)  -  ( T `
 A ) ) )  <  w ) ) )
12 breq2 4216 . . . . . 6  |-  ( w  =  B  ->  (
( abs `  (
( T `  y
)  -  ( T `
 A ) ) )  <  w  <->  ( abs `  ( ( T `  y )  -  ( T `  A )
) )  <  B
) )
1312imbi2d 308 . . . . 5  |-  ( w  =  B  ->  (
( ( normh `  (
y  -h  A ) )  <  x  -> 
( abs `  (
( T `  y
)  -  ( T `
 A ) ) )  <  w )  <-> 
( ( normh `  (
y  -h  A ) )  <  x  -> 
( abs `  (
( T `  y
)  -  ( T `
 A ) ) )  <  B ) ) )
1413rexralbidv 2749 . . . 4  |-  ( w  =  B  ->  ( E. x  e.  RR+  A. y  e.  ~H  ( ( normh `  ( y  -h  A
) )  <  x  ->  ( abs `  (
( T `  y
)  -  ( T `
 A ) ) )  <  w )  <->  E. x  e.  RR+  A. y  e.  ~H  ( ( normh `  ( y  -h  A
) )  <  x  ->  ( abs `  (
( T `  y
)  -  ( T `
 A ) ) )  <  B ) ) )
1511, 14rspc2v 3058 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  RR+ )  -> 
( A. z  e. 
~H  A. w  e.  RR+  E. x  e.  RR+  A. y  e.  ~H  ( ( normh `  ( y  -h  z
) )  <  x  ->  ( abs `  (
( T `  y
)  -  ( T `
 z ) ) )  <  w )  ->  E. x  e.  RR+  A. y  e.  ~H  (
( normh `  ( y  -h  A ) )  < 
x  ->  ( abs `  ( ( T `  y )  -  ( T `  A )
) )  <  B
) ) )
162, 15syl5com 28 . 2  |-  ( T  e.  ConFn  ->  ( ( A  e.  ~H  /\  B  e.  RR+ )  ->  E. x  e.  RR+  A. y  e. 
~H  ( ( normh `  ( y  -h  A
) )  <  x  ->  ( abs `  (
( T `  y
)  -  ( T `
 A ) ) )  <  B ) ) )
17163impib 1151 1  |-  ( ( T  e.  ConFn  /\  A  e.  ~H  /\  B  e.  RR+ )  ->  E. x  e.  RR+  A. y  e. 
~H  ( ( normh `  ( y  -h  A
) )  <  x  ->  ( abs `  (
( T `  y
)  -  ( T `
 A ) ) )  <  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = 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   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:  nmcfnexi  23554
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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