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Theorem ulmcl 20299
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 20297 . . . 4  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
2 ulmval 20298 . . . 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 234 . 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 959 . . 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 2828 . 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 16 1  |-  ( F ( ~~> u `  S
) G  ->  G : S --> CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958   class class class wbr 4214   -->wf 5452   ` cfv 5456  (class class class)co 6083    ^m cmap 7020   CCcc 8990    < clt 9122    - cmin 9293   ZZcz 10284   ZZ>=cuz 10490   RR+crp 10614   abscabs 12041   ~~> uculm 20294
This theorem is referenced by:  ulmi  20304  ulmclm  20305  ulmres  20306  ulmshftlem  20307  ulmuni  20310  ulmcau  20313  ulmss  20315  ulmbdd  20316  ulmcn  20317  ulmdvlem1  20318  ulmdvlem3  20320  ulmdv  20321  mbfulm  20324  iblulm  20325  itgulm  20326  itgulm2  20327  pserulm  20340  lgamgulmlem6  24820  lgamgulm2  24822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-pm 7023  df-neg 9296  df-z 10285  df-uz 10491  df-ulm 20295
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