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Theorem ulmclm 19981
Description: A uniform limit of functions converges pointwise. (Contributed by Mario Carneiro, 27-Feb-2015.)
Hypotheses
Ref Expression
ulmclm.z  |-  Z  =  ( ZZ>= `  M )
ulmclm.m  |-  ( ph  ->  M  e.  ZZ )
ulmclm.f  |-  ( ph  ->  F : Z --> ( CC 
^m  S ) )
ulmclm.a  |-  ( ph  ->  A  e.  S )
ulmclm.h  |-  ( ph  ->  H  e.  W )
ulmclm.e  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
) `  A )  =  ( H `  k ) )
ulmclm.u  |-  ( ph  ->  F ( ~~> u `  S ) G )
Assertion
Ref Expression
ulmclm  |-  ( ph  ->  H  ~~>  ( G `  A ) )
Distinct variable groups:    A, k    k, F    k, G    ph, k    k, H    k, M    S, k    k, Z
Allowed substitution hint:    W( k)

Proof of Theorem ulmclm
Dummy variables  j  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ulmclm.u . 2  |-  ( ph  ->  F ( ~~> u `  S ) G )
2 ulmclm.a . . . . . . 7  |-  ( ph  ->  A  e.  S )
3 fveq2 5632 . . . . . . . . . . 11  |-  ( z  =  A  ->  (
( F `  k
) `  z )  =  ( ( F `
 k ) `  A ) )
4 fveq2 5632 . . . . . . . . . . 11  |-  ( z  =  A  ->  ( G `  z )  =  ( G `  A ) )
53, 4oveq12d 5999 . . . . . . . . . 10  |-  ( z  =  A  ->  (
( ( F `  k ) `  z
)  -  ( G `
 z ) )  =  ( ( ( F `  k ) `
 A )  -  ( G `  A ) ) )
65fveq2d 5636 . . . . . . . . 9  |-  ( z  =  A  ->  ( abs `  ( ( ( F `  k ) `
 z )  -  ( G `  z ) ) )  =  ( abs `  ( ( ( F `  k
) `  A )  -  ( G `  A ) ) ) )
76breq1d 4135 . . . . . . . 8  |-  ( z  =  A  ->  (
( abs `  (
( ( F `  k ) `  z
)  -  ( G `
 z ) ) )  <  x  <->  ( abs `  ( ( ( F `
 k ) `  A )  -  ( G `  A )
) )  <  x
) )
87rspcv 2965 . . . . . . 7  |-  ( A  e.  S  ->  ( A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x  ->  ( abs `  ( ( ( F `  k ) `
 A )  -  ( G `  A ) ) )  <  x
) )
92, 8syl 15 . . . . . 6  |-  ( ph  ->  ( A. z  e.  S  ( abs `  (
( ( F `  k ) `  z
)  -  ( G `
 z ) ) )  <  x  -> 
( abs `  (
( ( F `  k ) `  A
)  -  ( G `
 A ) ) )  <  x ) )
109ralimdv 2707 . . . . 5  |-  ( ph  ->  ( A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x  ->  A. k  e.  ( ZZ>= `  j )
( abs `  (
( ( F `  k ) `  A
)  -  ( G `
 A ) ) )  <  x ) )
1110reximdv 2739 . . . 4  |-  ( ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( ( F `  k ) `  A
)  -  ( G `
 A ) ) )  <  x ) )
1211ralimdv 2707 . . 3  |-  ( ph  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) A. z  e.  S  ( abs `  ( ( ( F `  k ) `
 z )  -  ( G `  z ) ) )  <  x  ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( ( F `
 k ) `  A )  -  ( G `  A )
) )  <  x
) )
13 ulmclm.z . . . 4  |-  Z  =  ( ZZ>= `  M )
14 ulmclm.m . . . 4  |-  ( ph  ->  M  e.  ZZ )
15 ulmclm.f . . . 4  |-  ( ph  ->  F : Z --> ( CC 
^m  S ) )
16 eqidd 2367 . . . 4  |-  ( (
ph  /\  ( k  e.  Z  /\  z  e.  S ) )  -> 
( ( F `  k ) `  z
)  =  ( ( F `  k ) `
 z ) )
17 eqidd 2367 . . . 4  |-  ( (
ph  /\  z  e.  S )  ->  ( G `  z )  =  ( G `  z ) )
18 ulmcl 19975 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  G : S --> CC )
191, 18syl 15 . . . 4  |-  ( ph  ->  G : S --> CC )
20 ulmscl 19973 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
211, 20syl 15 . . . 4  |-  ( ph  ->  S  e.  _V )
2213, 14, 15, 16, 17, 19, 21ulm2 19979 . . 3  |-  ( ph  ->  ( F ( ~~> u `  S ) G  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x ) )
23 ulmclm.h . . . 4  |-  ( ph  ->  H  e.  W )
24 ulmclm.e . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
) `  A )  =  ( H `  k ) )
2524eqcomd 2371 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k ) `  A ) )
26 ffvelrn 5770 . . . . 5  |-  ( ( G : S --> CC  /\  A  e.  S )  ->  ( G `  A
)  e.  CC )
2719, 2, 26syl2anc 642 . . . 4  |-  ( ph  ->  ( G `  A
)  e.  CC )
28 ffvelrn 5770 . . . . . . 7  |-  ( ( F : Z --> ( CC 
^m  S )  /\  k  e.  Z )  ->  ( F `  k
)  e.  ( CC 
^m  S ) )
2915, 28sylan 457 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  ( CC  ^m  S
) )
30 elmapi 6935 . . . . . 6  |-  ( ( F `  k )  e.  ( CC  ^m  S )  ->  ( F `  k ) : S --> CC )
3129, 30syl 15 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k ) : S --> CC )
322adantr 451 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  S )
33 ffvelrn 5770 . . . . 5  |-  ( ( ( F `  k
) : S --> CC  /\  A  e.  S )  ->  ( ( F `  k ) `  A
)  e.  CC )
3431, 32, 33syl2anc 642 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
) `  A )  e.  CC )
3513, 14, 23, 25, 27, 34clim2c 12186 . . 3  |-  ( ph  ->  ( H  ~~>  ( G `
 A )  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( ( F `  k ) `  A
)  -  ( G `
 A ) ) )  <  x ) )
3612, 22, 353imtr4d 259 . 2  |-  ( ph  ->  ( F ( ~~> u `  S ) G  ->  H 
~~>  ( G `  A
) ) )
371, 36mpd 14 1  |-  ( ph  ->  H  ~~>  ( G `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   A.wral 2628   E.wrex 2629   _Vcvv 2873   class class class wbr 4125   -->wf 5354   ` cfv 5358  (class class class)co 5981    ^m cmap 6915   CCcc 8882    < clt 9014    - cmin 9184   ZZcz 10175   ZZ>=cuz 10381   RR+crp 10505   abscabs 11926    ~~> cli 12165   ~~> uculm 19970
This theorem is referenced by:  ulmuni  19986  ulmdvlem3  19996  mbfulm  20000  pserulm  20016  lgamgulm2  24268  lgamcvglem  24272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-pre-lttri 8958  ax-pre-lttrn 8959
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-er 6802  df-map 6917  df-pm 6918  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-neg 9187  df-z 10176  df-uz 10382  df-clim 12169  df-ulm 19971
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