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Theorem ulmf2 22646
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 22645 . . . 4  |-  ( F ( ~~> u `  S
) G  ->  F  e.  ( ( CC  ^m  S )  ^pm  ZZ ) )
2 ovex 6320 . . . . . 6  |-  ( CC 
^m  S )  e. 
_V
3 zex 10885 . . . . . 6  |-  ZZ  e.  _V
42, 3elpm2 7462 . . . . 5  |-  ( F  e.  ( ( CC 
^m  S )  ^pm  ZZ )  <->  ( F : dom  F --> ( CC  ^m  S )  /\  dom  F 
C_  ZZ ) )
54simplbi 460 . . . 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 466 . 2  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  F : dom  F --> ( CC  ^m  S
) )
8 fndm 5686 . . . 4  |-  ( F  Fn  Z  ->  dom  F  =  Z )
98adantr 465 . . 3  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  dom  F  =  Z )
109feq2d 5724 . 2  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  ( F : dom  F --> ( CC  ^m  S )  <->  F : Z
--> ( CC  ^m  S
) ) )
117, 10mpbid 210 1  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  F : Z --> ( CC  ^m  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3481   class class class wbr 4453   dom cdm 5005    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432    ^pm cpm 7433   CCcc 9502   ZZcz 10876   ~~> uculm 22638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-pm 7435  df-neg 9820  df-z 10877  df-uz 11095  df-ulm 22639
This theorem is referenced by:  ulmdvlem1  22662  ulmdvlem2  22663  ulmdvlem3  22664  mtestbdd  22667  mbfulm  22668  iblulm  22669  itgulm  22670  itgulm2  22671  lgamgulm2  28403  lgamcvglem  28407
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