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Theorem ulmf 19777
Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
Assertion
Ref Expression
ulmf  |-  ( F ( ~~> u `  S
) G  ->  E. n  e.  ZZ  F : (
ZZ>= `  n ) --> ( CC  ^m  S ) )
Distinct variable groups:    n, F    n, G    S, n

Proof of Theorem ulmf
Dummy variables  j 
k  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 simp1 955 . . 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
)  ->  F :
( ZZ>= `  n ) --> ( CC  ^m  S ) )
65reximi 2663 . 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
)  ->  E. n  e.  ZZ  F : (
ZZ>= `  n ) --> ( CC  ^m  S ) )
74, 6syl 15 1  |-  ( F ( ~~> u `  S
) G  ->  E. n  e.  ZZ  F : (
ZZ>= `  n ) --> ( CC  ^m  S ) )
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:  ulmpm  19778  ulmss  19790
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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