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Theorem ulmcl 19776
Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
Assertion
Ref Expression
ulmcl  |-  ( F ( ~~> u `  S
) G  ->  G : S --> CC )

Proof of Theorem ulmcl
Dummy variables  j 
k  n  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ulmscl 19774 . . . 4  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
2 ulmval 19775 . . . 4  |-  ( S  e.  _V  ->  ( F ( ~~> u `  S ) G  <->  E. n  e.  ZZ  ( F :
( ZZ>= `  n ) --> ( CC  ^m  S )  /\  G : S --> CC  /\  A. x  e.  RR+  E. j  e.  (
ZZ>= `  n ) A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `
 k ) `  z )  -  ( G `  z )
) )  <  x
) ) )
31, 2syl 15 . . 3  |-  ( F ( ~~> u `  S
) G  ->  ( F ( ~~> u `  S ) G  <->  E. n  e.  ZZ  ( F :
( ZZ>= `  n ) --> ( CC  ^m  S )  /\  G : S --> CC  /\  A. x  e.  RR+  E. j  e.  (
ZZ>= `  n ) A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `
 k ) `  z )  -  ( G `  z )
) )  <  x
) ) )
43ibi 232 . 2  |-  ( F ( ~~> u `  S
) G  ->  E. n  e.  ZZ  ( F :
( ZZ>= `  n ) --> ( CC  ^m  S )  /\  G : S --> CC  /\  A. x  e.  RR+  E. j  e.  (
ZZ>= `  n ) A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `
 k ) `  z )  -  ( G `  z )
) )  <  x
) )
5 simp2 956 . . 3  |-  ( ( F : ( ZZ>= `  n ) --> ( CC 
^m  S )  /\  G : S --> CC  /\  A. x  e.  RR+  E. j  e.  ( ZZ>= `  n ) A. k  e.  ( ZZ>=
`  j ) A. z  e.  S  ( abs `  ( ( ( F `  k ) `
 z )  -  ( G `  z ) ) )  <  x
)  ->  G : S
--> CC )
65rexlimivw 2676 . 2  |-  ( E. n  e.  ZZ  ( F : ( ZZ>= `  n
) --> ( CC  ^m  S )  /\  G : S --> CC  /\  A. x  e.  RR+  E. j  e.  ( ZZ>= `  n ) A. k  e.  ( ZZ>=
`  j ) A. z  e.  S  ( abs `  ( ( ( F `  k ) `
 z )  -  ( G `  z ) ) )  <  x
)  ->  G : S
--> CC )
74, 6syl 15 1  |-  ( F ( ~~> u `  S
) G  ->  G : S --> CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   CCcc 8751    < clt 8883    - cmin 9053   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   abscabs 11735   ~~> uculm 19771
This theorem is referenced by:  ulmi  19781  ulmclm  19782  ulmres  19783  ulmshftlem  19784  ulmcau  19788  ulmss  19790  ulmbdd  19791  ulmcn  19792  ulmdvlem1  19793  ulmdvlem3  19795  ulmdv  19796  mbfulm  19798  iblulm  19799  itgulm  19800  itgulm2  19801  pserulm  19814
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-pm 6791  df-neg 9056  df-z 10041  df-uz 10247  df-ulm 19772
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