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Theorem cnfnc 26672
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 26624 . . . 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 464 . . 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 6303 . . . . . . . 8  |-  ( z  =  A  ->  (
y  -h  z )  =  ( y  -h  A ) )
43fveq2d 5876 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( y  -h  z ) )  =  ( normh `  ( y  -h  A ) ) )
54breq1d 4463 . . . . . 6  |-  ( z  =  A  ->  (
( normh `  ( y  -h  z ) )  < 
x  <->  ( normh `  (
y  -h  A ) )  <  x ) )
6 fveq2 5872 . . . . . . . . 9  |-  ( z  =  A  ->  ( T `  z )  =  ( T `  A ) )
76oveq2d 6311 . . . . . . . 8  |-  ( z  =  A  ->  (
( T `  y
)  -  ( T `
 z ) )  =  ( ( T `
 y )  -  ( T `  A ) ) )
87fveq2d 5876 . . . . . . 7  |-  ( z  =  A  ->  ( abs `  ( ( T `
 y )  -  ( T `  z ) ) )  =  ( abs `  ( ( T `  y )  -  ( T `  A ) ) ) )
98breq1d 4463 . . . . . 6  |-  ( z  =  A  ->  (
( abs `  (
( T `  y
)  -  ( T `
 z ) ) )  <  w  <->  ( abs `  ( ( T `  y )  -  ( T `  A )
) )  <  w
) )
105, 9imbi12d 320 . . . . 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 2986 . . . 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 4457 . . . . . 6  |-  ( w  =  B  ->  (
( abs `  (
( T `  y
)  -  ( T `
 A ) ) )  <  w  <->  ( abs `  ( ( T `  y )  -  ( T `  A )
) )  <  B
) )
1312imbi2d 316 . . . . 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 2986 . . . 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 3228 . . 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 30 . 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 1194 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   class class class wbr 4453   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502    < clt 9640    - cmin 9817   RR+crp 11232   abscabs 13047   ~Hchil 25659   normhcno 25663    -h cmv 25665   ConFnccnfn 25693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-hilex 25739
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-cnfn 26589
This theorem is referenced by:  nmcfnexi  26793
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