HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  cnfnc Unicode version

Theorem cnfnc 22510
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 22462 . . . 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 5866 . . . . . . . 8  |-  ( z  =  A  ->  (
y  -h  z )  =  ( y  -h  A ) )
43fveq2d 5529 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( y  -h  z ) )  =  ( normh `  ( y  -h  A ) ) )
54breq1d 4033 . . . . . 6  |-  ( z  =  A  ->  (
( normh `  ( y  -h  z ) )  < 
x  <->  ( normh `  (
y  -h  A ) )  <  x ) )
6 fveq2 5525 . . . . . . . . 9  |-  ( z  =  A  ->  ( T `  z )  =  ( T `  A ) )
76oveq2d 5874 . . . . . . . 8  |-  ( z  =  A  ->  (
( T `  y
)  -  ( T `
 z ) )  =  ( ( T `
 y )  -  ( T `  A ) ) )
87fveq2d 5529 . . . . . . 7  |-  ( z  =  A  ->  ( abs `  ( ( T `
 y )  -  ( T `  z ) ) )  =  ( abs `  ( ( T `  y )  -  ( T `  A ) ) ) )
98breq1d 4033 . . . . . 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 2587 . . . 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 4027 . . . . . 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 2587 . . . 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 2890 . . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735    < clt 8867    - cmin 9037   RR+crp 10354   abscabs 11719   ~Hchil 21499   normhcno 21503    -h cmv 21505   ConFnccnfn 21533
This theorem is referenced by:  nmcfnexi  22631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-cnfn 22427
  Copyright terms: Public domain W3C validator