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Theorem ulmf 20298
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 20295 . . . 4  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
2 ulmval 20296 . . . 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 16 . . 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 233 . 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 957 . . 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 2813 . 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 16 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 177    /\ w3a 936    e. wcel 1725   A.wral 2705   E.wrex 2706   _Vcvv 2956   class class class wbr 4212   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   CCcc 8988    < clt 9120    - cmin 9291   ZZcz 10282   ZZ>=cuz 10488   RR+crp 10612   abscabs 12039   ~~> uculm 20292
This theorem is referenced by:  ulmpm  20299  ulmuni  20308  ulmss  20313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-pm 7021  df-neg 9294  df-z 10283  df-uz 10489  df-ulm 20293
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