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Theorem ulmclm 20256
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 5687 . . . . . . . . . . 11  |-  ( z  =  A  ->  (
( F `  k
) `  z )  =  ( ( F `
 k ) `  A ) )
4 fveq2 5687 . . . . . . . . . . 11  |-  ( z  =  A  ->  ( G `  z )  =  ( G `  A ) )
53, 4oveq12d 6058 . . . . . . . . . 10  |-  ( z  =  A  ->  (
( ( F `  k ) `  z
)  -  ( G `
 z ) )  =  ( ( ( F `  k ) `
 A )  -  ( G `  A ) ) )
65fveq2d 5691 . . . . . . . . 9  |-  ( z  =  A  ->  ( abs `  ( ( ( F `  k ) `
 z )  -  ( G `  z ) ) )  =  ( abs `  ( ( ( F `  k
) `  A )  -  ( G `  A ) ) ) )
76breq1d 4182 . . . . . . . 8  |-  ( z  =  A  ->  (
( abs `  (
( ( F `  k ) `  z
)  -  ( G `
 z ) ) )  <  x  <->  ( abs `  ( ( ( F `
 k ) `  A )  -  ( G `  A )
) )  <  x
) )
87rspcv 3008 . . . . . . 7  |-  ( A  e.  S  ->  ( A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x  ->  ( abs `  ( ( ( F `  k ) `
 A )  -  ( G `  A ) ) )  <  x
) )
92, 8syl 16 . . . . . 6  |-  ( ph  ->  ( A. z  e.  S  ( abs `  (
( ( F `  k ) `  z
)  -  ( G `
 z ) ) )  <  x  -> 
( abs `  (
( ( F `  k ) `  A
)  -  ( G `
 A ) ) )  <  x ) )
109ralimdv 2745 . . . . 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 2777 . . . 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 2745 . . 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 2405 . . . 4  |-  ( (
ph  /\  ( k  e.  Z  /\  z  e.  S ) )  -> 
( ( F `  k ) `  z
)  =  ( ( F `  k ) `
 z ) )
17 eqidd 2405 . . . 4  |-  ( (
ph  /\  z  e.  S )  ->  ( G `  z )  =  ( G `  z ) )
18 ulmcl 20250 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  G : S --> CC )
191, 18syl 16 . . . 4  |-  ( ph  ->  G : S --> CC )
20 ulmscl 20248 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
211, 20syl 16 . . . 4  |-  ( ph  ->  S  e.  _V )
2213, 14, 15, 16, 17, 19, 21ulm2 20254 . . 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 2409 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k ) `  A ) )
2619, 2ffvelrnd 5830 . . . 4  |-  ( ph  ->  ( G `  A
)  e.  CC )
2715ffvelrnda 5829 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  ( CC  ^m  S
) )
28 elmapi 6997 . . . . . 6  |-  ( ( F `  k )  e.  ( CC  ^m  S )  ->  ( F `  k ) : S --> CC )
2927, 28syl 16 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k ) : S --> CC )
302adantr 452 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  S )
3129, 30ffvelrnd 5830 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
) `  A )  e.  CC )
3213, 14, 23, 25, 26, 31clim2c 12254 . . 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 ) )
3312, 22, 323imtr4d 260 . 2  |-  ( ph  ->  ( F ( ~~> u `  S ) G  ->  H 
~~>  ( G `  A
) ) )
341, 33mpd 15 1  |-  ( ph  ->  H  ~~>  ( G `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916   class class class wbr 4172   -->wf 5409   ` cfv 5413  (class class class)co 6040    ^m cmap 6977   CCcc 8944    < clt 9076    - cmin 9247   ZZcz 10238   ZZ>=cuz 10444   RR+crp 10568   abscabs 11994    ~~> cli 12233   ~~> uculm 20245
This theorem is referenced by:  ulmuni  20261  ulmdvlem3  20271  mbfulm  20275  pserulm  20291  lgamgulm2  24773  lgamcvglem  24777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-pre-lttri 9020  ax-pre-lttrn 9021
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-neg 9250  df-z 10239  df-uz 10445  df-clim 12237  df-ulm 20246
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