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Theorem ulmpm 20301
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 20300 . 2  |-  ( F ( ~~> u `  S
) G  ->  E. n  e.  ZZ  F : (
ZZ>= `  n ) --> ( CC  ^m  S ) )
2 uzssz 10507 . . . 4  |-  ( ZZ>= `  n )  C_  ZZ
3 ovex 6108 . . . . 5  |-  ( CC 
^m  S )  e. 
_V
4 zex 10293 . . . . 5  |-  ZZ  e.  _V
5 elpm2r 7036 . . . . 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 665 . . . 4  |-  ( ( F : ( ZZ>= `  n ) --> ( CC 
^m  S )  /\  ( ZZ>= `  n )  C_  ZZ )  ->  F  e.  ( ( CC  ^m  S )  ^pm  ZZ ) )
72, 6mpan2 654 . . 3  |-  ( F : ( ZZ>= `  n
) --> ( CC  ^m  S )  ->  F  e.  ( ( CC  ^m  S )  ^pm  ZZ ) )
87rexlimivw 2828 . 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 set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   E.wrex 2708   _Vcvv 2958    C_ wss 3322   class class class wbr 4214   -->wf 5452   ` cfv 5456  (class class class)co 6083    ^m cmap 7020    ^pm cpm 7021   CCcc 8990   ZZcz 10284   ZZ>=cuz 10490   ~~> uculm 20294
This theorem is referenced by:  ulmf2  20302
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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