ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  limccl Unicode version

Theorem limccl 12834
Description: Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
Assertion
Ref Expression
limccl  |-  ( F lim
CC  B )  C_  CC

Proof of Theorem limccl
Dummy variables  d  e  f  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . 4  |-  ( w  e.  ( F lim CC  B )  ->  w  e.  ( F lim CC  B
) )
2 df-limced 12831 . . . . . 6  |- lim CC  =  ( f  e.  ( CC  ^pm  CC ) ,  x  e.  CC  |->  { y  e.  CC  |  ( ( f : dom  f --> CC 
/\  dom  f  C_  CC )  /\  (
x  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  f ( ( z #  x  /\  ( abs `  ( z  -  x ) )  < 
d )  ->  ( abs `  ( ( f `
 z )  -  y ) )  < 
e ) ) ) } )
32elmpocl1 5976 . . . . 5  |-  ( w  e.  ( F lim CC  B )  ->  F  e.  ( CC  ^pm  CC ) )
4 limcrcl 12833 . . . . . 6  |-  ( w  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
54simp3d 996 . . . . 5  |-  ( w  e.  ( F lim CC  B )  ->  B  e.  CC )
6 cnex 7767 . . . . . . 7  |-  CC  e.  _V
76rabex 4079 . . . . . 6  |-  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) }  e.  _V
87a1i 9 . . . . 5  |-  ( w  e.  ( F lim CC  B )  ->  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) }  e.  _V )
9 simpl 108 . . . . . . . . . 10  |-  ( ( f  =  F  /\  x  =  B )  ->  f  =  F )
109dmeqd 4748 . . . . . . . . . 10  |-  ( ( f  =  F  /\  x  =  B )  ->  dom  f  =  dom  F )
119, 10feq12d 5269 . . . . . . . . 9  |-  ( ( f  =  F  /\  x  =  B )  ->  ( f : dom  f
--> CC  <->  F : dom  F --> CC ) )
1210sseq1d 3130 . . . . . . . . 9  |-  ( ( f  =  F  /\  x  =  B )  ->  ( dom  f  C_  CC 
<->  dom  F  C_  CC ) )
1311, 12anbi12d 465 . . . . . . . 8  |-  ( ( f  =  F  /\  x  =  B )  ->  ( ( f : dom  f --> CC  /\  dom  f  C_  CC )  <-> 
( F : dom  F --> CC  /\  dom  F  C_  CC ) ) )
14 simpr 109 . . . . . . . . . 10  |-  ( ( f  =  F  /\  x  =  B )  ->  x  =  B )
1514eleq1d 2209 . . . . . . . . 9  |-  ( ( f  =  F  /\  x  =  B )  ->  ( x  e.  CC  <->  B  e.  CC ) )
1614breq2d 3948 . . . . . . . . . . . . . 14  |-  ( ( f  =  F  /\  x  =  B )  ->  ( z #  x  <->  z #  B
) )
1714oveq2d 5797 . . . . . . . . . . . . . . . 16  |-  ( ( f  =  F  /\  x  =  B )  ->  ( z  -  x
)  =  ( z  -  B ) )
1817fveq2d 5432 . . . . . . . . . . . . . . 15  |-  ( ( f  =  F  /\  x  =  B )  ->  ( abs `  (
z  -  x ) )  =  ( abs `  ( z  -  B
) ) )
1918breq1d 3946 . . . . . . . . . . . . . 14  |-  ( ( f  =  F  /\  x  =  B )  ->  ( ( abs `  (
z  -  x ) )  <  d  <->  ( abs `  ( z  -  B
) )  <  d
) )
2016, 19anbi12d 465 . . . . . . . . . . . . 13  |-  ( ( f  =  F  /\  x  =  B )  ->  ( ( z #  x  /\  ( abs `  (
z  -  x ) )  <  d )  <-> 
( z #  B  /\  ( abs `  ( z  -  B ) )  <  d ) ) )
219fveq1d 5430 . . . . . . . . . . . . . . 15  |-  ( ( f  =  F  /\  x  =  B )  ->  ( f `  z
)  =  ( F `
 z ) )
2221fvoveq1d 5803 . . . . . . . . . . . . . 14  |-  ( ( f  =  F  /\  x  =  B )  ->  ( abs `  (
( f `  z
)  -  y ) )  =  ( abs `  ( ( F `  z )  -  y
) ) )
2322breq1d 3946 . . . . . . . . . . . . 13  |-  ( ( f  =  F  /\  x  =  B )  ->  ( ( abs `  (
( f `  z
)  -  y ) )  <  e  <->  ( abs `  ( ( F `  z )  -  y
) )  <  e
) )
2420, 23imbi12d 233 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  x  =  B )  ->  ( ( ( z #  x  /\  ( abs `  ( z  -  x
) )  <  d
)  ->  ( abs `  ( ( f `  z )  -  y
) )  <  e
)  <->  ( ( z #  B  /\  ( abs `  ( z  -  B
) )  <  d
)  ->  ( abs `  ( ( F `  z )  -  y
) )  <  e
) ) )
2510, 24raleqbidv 2641 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  x  =  B )  ->  ( A. z  e. 
dom  f ( ( z #  x  /\  ( abs `  ( z  -  x ) )  < 
d )  ->  ( abs `  ( ( f `
 z )  -  y ) )  < 
e )  <->  A. z  e.  dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) )
2625rexbidv 2439 . . . . . . . . . 10  |-  ( ( f  =  F  /\  x  =  B )  ->  ( E. d  e.  RR+  A. z  e.  dom  f ( ( z #  x  /\  ( abs `  ( z  -  x
) )  <  d
)  ->  ( abs `  ( ( f `  z )  -  y
) )  <  e
)  <->  E. d  e.  RR+  A. z  e.  dom  F
( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  d )  ->  ( abs `  (
( F `  z
)  -  y ) )  <  e ) ) )
2726ralbidv 2438 . . . . . . . . 9  |-  ( ( f  =  F  /\  x  =  B )  ->  ( A. e  e.  RR+  E. d  e.  RR+  A. z  e.  dom  f
( ( z #  x  /\  ( abs `  (
z  -  x ) )  <  d )  ->  ( abs `  (
( f `  z
)  -  y ) )  <  e )  <->  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) )
2815, 27anbi12d 465 . . . . . . . 8  |-  ( ( f  =  F  /\  x  =  B )  ->  ( ( x  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e.  dom  f ( ( z #  x  /\  ( abs `  ( z  -  x
) )  <  d
)  ->  ( abs `  ( ( f `  z )  -  y
) )  <  e
) )  <->  ( B  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e.  dom  F ( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  d )  ->  ( abs `  (
( F `  z
)  -  y ) )  <  e ) ) ) )
2913, 28anbi12d 465 . . . . . . 7  |-  ( ( f  =  F  /\  x  =  B )  ->  ( ( ( f : dom  f --> CC 
/\  dom  f  C_  CC )  /\  (
x  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  f ( ( z #  x  /\  ( abs `  ( z  -  x ) )  < 
d )  ->  ( abs `  ( ( f `
 z )  -  y ) )  < 
e ) ) )  <-> 
( ( F : dom  F --> CC  /\  dom  F 
C_  CC )  /\  ( B  e.  CC  /\ 
A. e  e.  RR+  E. d  e.  RR+  A. z  e.  dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) ) )
3029rabbidv 2678 . . . . . 6  |-  ( ( f  =  F  /\  x  =  B )  ->  { y  e.  CC  |  ( ( f : dom  f --> CC 
/\  dom  f  C_  CC )  /\  (
x  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  f ( ( z #  x  /\  ( abs `  ( z  -  x ) )  < 
d )  ->  ( abs `  ( ( f `
 z )  -  y ) )  < 
e ) ) ) }  =  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) } )
3130, 2ovmpoga 5907 . . . . 5  |-  ( ( F  e.  ( CC 
^pm  CC )  /\  B  e.  CC  /\  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) }  e.  _V )  ->  ( F lim CC  B
)  =  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) } )
323, 5, 8, 31syl3anc 1217 . . . 4  |-  ( w  e.  ( F lim CC  B )  ->  ( F lim CC  B )  =  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\ 
A. e  e.  RR+  E. d  e.  RR+  A. z  e.  dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) } )
331, 32eleqtrd 2219 . . 3  |-  ( w  e.  ( F lim CC  B )  ->  w  e.  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\ 
A. e  e.  RR+  E. d  e.  RR+  A. z  e.  dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) } )
34 elrabi 2840 . . 3  |-  ( w  e.  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) }  ->  w  e.  CC )
3533, 34syl 14 . 2  |-  ( w  e.  ( F lim CC  B )  ->  w  e.  CC )
3635ssriv 3105 1  |-  ( F lim
CC  B )  C_  CC
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421   _Vcvv 2689    C_ wss 3075   class class class wbr 3936   dom cdm 4546   -->wf 5126   ` cfv 5130  (class class class)co 5781    ^pm cpm 6550   CCcc 7641    < clt 7823    - cmin 7956   # cap 8366   RR+crp 9469   abscabs 10800   lim CC climc 12829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-pm 6552  df-limced 12831
This theorem is referenced by:  reldvg  12854  dvfvalap  12856  dvcl  12858
  Copyright terms: Public domain W3C validator