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Theorem ulmf2 21965
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 21964 . . . 4  |-  ( F ( ~~> u `  S
) G  ->  F  e.  ( ( CC  ^m  S )  ^pm  ZZ ) )
2 ovex 6215 . . . . . 6  |-  ( CC 
^m  S )  e. 
_V
3 zex 10756 . . . . . 6  |-  ZZ  e.  _V
42, 3elpm2 7344 . . . . 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 5608 . . . 4  |-  ( F  Fn  Z  ->  dom  F  =  Z )
98adantr 465 . . 3  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  dom  F  =  Z )
109feq2d 5645 . 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 1370    e. wcel 1758    C_ wss 3426   class class class wbr 4390   dom cdm 4938    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190    ^m cmap 7314    ^pm cpm 7315   CCcc 9381   ZZcz 10747   ~~> uculm 21957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-map 7316  df-pm 7317  df-neg 9699  df-z 10748  df-uz 10963  df-ulm 21958
This theorem is referenced by:  ulmdvlem1  21981  ulmdvlem2  21982  ulmdvlem3  21983  mtestbdd  21986  mbfulm  21987  iblulm  21988  itgulm  21989  itgulm2  21990  lgamgulm2  27156  lgamcvglem  27160
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