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Theorem ulmpm 22604
Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
Assertion
Ref Expression
ulmpm  |-  ( F ( ~~> u `  S
) G  ->  F  e.  ( ( CC  ^m  S )  ^pm  ZZ ) )

Proof of Theorem ulmpm
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 ulmf 22603 . 2  |-  ( F ( ~~> u `  S
) G  ->  E. n  e.  ZZ  F : (
ZZ>= `  n ) --> ( CC  ^m  S ) )
2 uzssz 11102 . . . 4  |-  ( ZZ>= `  n )  C_  ZZ
3 ovex 6310 . . . . 5  |-  ( CC 
^m  S )  e. 
_V
4 zex 10874 . . . . 5  |-  ZZ  e.  _V
5 elpm2r 7437 . . . . 5  |-  ( ( ( ( CC  ^m  S )  e.  _V  /\  ZZ  e.  _V )  /\  ( F : (
ZZ>= `  n ) --> ( CC  ^m  S )  /\  ( ZZ>= `  n
)  C_  ZZ )
)  ->  F  e.  ( ( CC  ^m  S )  ^pm  ZZ ) )
63, 4, 5mpanl12 682 . . . 4  |-  ( ( F : ( ZZ>= `  n ) --> ( CC 
^m  S )  /\  ( ZZ>= `  n )  C_  ZZ )  ->  F  e.  ( ( CC  ^m  S )  ^pm  ZZ ) )
72, 6mpan2 671 . . 3  |-  ( F : ( ZZ>= `  n
) --> ( CC  ^m  S )  ->  F  e.  ( ( CC  ^m  S )  ^pm  ZZ ) )
87rexlimivw 2952 . 2  |-  ( E. n  e.  ZZ  F : ( ZZ>= `  n
) --> ( CC  ^m  S )  ->  F  e.  ( ( CC  ^m  S )  ^pm  ZZ ) )
91, 8syl 16 1  |-  ( F ( ~~> u `  S
) G  ->  F  e.  ( ( CC  ^m  S )  ^pm  ZZ ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   E.wrex 2815   _Vcvv 3113    C_ wss 3476   class class class wbr 4447   -->wf 5584   ` cfv 5588  (class class class)co 6285    ^m cmap 7421    ^pm cpm 7422   CCcc 9491   ZZcz 10865   ZZ>=cuz 11083   ~~> uculm 22597
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-map 7423  df-pm 7424  df-neg 9809  df-z 10866  df-uz 11084  df-ulm 22598
This theorem is referenced by:  ulmf2  22605
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