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Theorem cnfnc 22526
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 22478 . . . 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 450 . . 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 5882 . . . . . . . 8  |-  ( z  =  A  ->  (
y  -h  z )  =  ( y  -h  A ) )
43fveq2d 5545 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( y  -h  z ) )  =  ( normh `  ( y  -h  A ) ) )
54breq1d 4049 . . . . . 6  |-  ( z  =  A  ->  (
( normh `  ( y  -h  z ) )  < 
x  <->  ( normh `  (
y  -h  A ) )  <  x ) )
6 fveq2 5541 . . . . . . . . 9  |-  ( z  =  A  ->  ( T `  z )  =  ( T `  A ) )
76oveq2d 5890 . . . . . . . 8  |-  ( z  =  A  ->  (
( T `  y
)  -  ( T `
 z ) )  =  ( ( T `
 y )  -  ( T `  A ) ) )
87fveq2d 5545 . . . . . . 7  |-  ( z  =  A  ->  ( abs `  ( ( T `
 y )  -  ( T `  z ) ) )  =  ( abs `  ( ( T `  y )  -  ( T `  A ) ) ) )
98breq1d 4049 . . . . . 6  |-  ( z  =  A  ->  (
( abs `  (
( T `  y
)  -  ( T `
 z ) ) )  <  w  <->  ( abs `  ( ( T `  y )  -  ( T `  A )
) )  <  w
) )
105, 9imbi12d 311 . . . . 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 2600 . . . 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 4043 . . . . . 6  |-  ( w  =  B  ->  (
( abs `  (
( T `  y
)  -  ( T `
 A ) ) )  <  w  <->  ( abs `  ( ( T `  y )  -  ( T `  A )
) )  <  B
) )
1312imbi2d 307 . . . . 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 2600 . . . 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 2903 . . 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 26 . 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 1149 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 358    /\ w3a 934    = 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   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:  nmcfnexi  22647
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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