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Theorem ulmf2 20331
Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 18-Mar-2015.)
Assertion
Ref Expression
ulmf2  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  F : Z --> ( CC  ^m  S ) )

Proof of Theorem ulmf2
StepHypRef Expression
1 ulmpm 20330 . . . 4  |-  ( F ( ~~> u `  S
) G  ->  F  e.  ( ( CC  ^m  S )  ^pm  ZZ ) )
2 ovex 6135 . . . . . 6  |-  ( CC 
^m  S )  e. 
_V
3 zex 10322 . . . . . 6  |-  ZZ  e.  _V
42, 3elpm2 7074 . . . . 5  |-  ( F  e.  ( ( CC 
^m  S )  ^pm  ZZ )  <->  ( F : dom  F --> ( CC  ^m  S )  /\  dom  F 
C_  ZZ ) )
54simplbi 448 . . . 4  |-  ( F  e.  ( ( CC 
^m  S )  ^pm  ZZ )  ->  F : dom  F --> ( CC  ^m  S ) )
61, 5syl 16 . . 3  |-  ( F ( ~~> u `  S
) G  ->  F : dom  F --> ( CC 
^m  S ) )
76adantl 454 . 2  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  F : dom  F --> ( CC  ^m  S
) )
8 fndm 5573 . . . 4  |-  ( F  Fn  Z  ->  dom  F  =  Z )
98adantr 453 . . 3  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  dom  F  =  Z )
109feq2d 5610 . 2  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  ( F : dom  F --> ( CC  ^m  S )  <->  F : Z
--> ( CC  ^m  S
) ) )
117, 10mpbid 203 1  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  F : Z --> ( CC  ^m  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727    C_ wss 3306   class class class wbr 4237   dom cdm 4907    Fn wfn 5478   -->wf 5479   ` cfv 5483  (class class class)co 6110    ^m cmap 7047    ^pm cpm 7048   CCcc 9019   ZZcz 10313   ~~> uculm 20323
This theorem is referenced by:  ulmdvlem1  20347  ulmdvlem2  20348  ulmdvlem3  20349  mtestbdd  20352  mbfulm  20353  iblulm  20354  itgulm  20355  itgulm2  20356  lgamgulm2  24851  lgamcvglem  24855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-map 7049  df-pm 7050  df-neg 9325  df-z 10314  df-uz 10520  df-ulm 20324
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