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Theorem cnfnc 25356
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 25308 . . . 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 6120 . . . . . . . 8  |-  ( z  =  A  ->  (
y  -h  z )  =  ( y  -h  A ) )
43fveq2d 5716 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( y  -h  z ) )  =  ( normh `  ( y  -h  A ) ) )
54breq1d 4323 . . . . . 6  |-  ( z  =  A  ->  (
( normh `  ( y  -h  z ) )  < 
x  <->  ( normh `  (
y  -h  A ) )  <  x ) )
6 fveq2 5712 . . . . . . . . 9  |-  ( z  =  A  ->  ( T `  z )  =  ( T `  A ) )
76oveq2d 6128 . . . . . . . 8  |-  ( z  =  A  ->  (
( T `  y
)  -  ( T `
 z ) )  =  ( ( T `
 y )  -  ( T `  A ) ) )
87fveq2d 5716 . . . . . . 7  |-  ( z  =  A  ->  ( abs `  ( ( T `
 y )  -  ( T `  z ) ) )  =  ( abs `  ( ( T `  y )  -  ( T `  A ) ) ) )
98breq1d 4323 . . . . . 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 2780 . . . 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 4317 . . . . . 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 2780 . . . 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 3100 . . 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 1185 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 965    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737   class class class wbr 4313   -->wf 5435   ` cfv 5439  (class class class)co 6112   CCcc 9301    < clt 9439    - cmin 9616   RR+crp 11012   abscabs 12744   ~Hchil 24343   normhcno 24347    -h cmv 24349   ConFnccnfn 24377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-hilex 24423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-map 7237  df-cnfn 25273
This theorem is referenced by:  nmcfnexi  25477
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