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Theorem ulmclm 20334
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 5757 . . . . . . . . . . 11  |-  ( z  =  A  ->  (
( F `  k
) `  z )  =  ( ( F `
 k ) `  A ) )
4 fveq2 5757 . . . . . . . . . . 11  |-  ( z  =  A  ->  ( G `  z )  =  ( G `  A ) )
53, 4oveq12d 6128 . . . . . . . . . 10  |-  ( z  =  A  ->  (
( ( F `  k ) `  z
)  -  ( G `
 z ) )  =  ( ( ( F `  k ) `
 A )  -  ( G `  A ) ) )
65fveq2d 5761 . . . . . . . . 9  |-  ( z  =  A  ->  ( abs `  ( ( ( F `  k ) `
 z )  -  ( G `  z ) ) )  =  ( abs `  ( ( ( F `  k
) `  A )  -  ( G `  A ) ) ) )
76breq1d 4247 . . . . . . . 8  |-  ( z  =  A  ->  (
( abs `  (
( ( F `  k ) `  z
)  -  ( G `
 z ) ) )  <  x  <->  ( abs `  ( ( ( F `
 k ) `  A )  -  ( G `  A )
) )  <  x
) )
87rspcv 3054 . . . . . . 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 2791 . . . . 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 2823 . . . 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 2791 . . 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 2443 . . . 4  |-  ( (
ph  /\  ( k  e.  Z  /\  z  e.  S ) )  -> 
( ( F `  k ) `  z
)  =  ( ( F `  k ) `
 z ) )
17 eqidd 2443 . . . 4  |-  ( (
ph  /\  z  e.  S )  ->  ( G `  z )  =  ( G `  z ) )
18 ulmcl 20328 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  G : S --> CC )
191, 18syl 16 . . . 4  |-  ( ph  ->  G : S --> CC )
20 ulmscl 20326 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
211, 20syl 16 . . . 4  |-  ( ph  ->  S  e.  _V )
2213, 14, 15, 16, 17, 19, 21ulm2 20332 . . 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 2447 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k ) `  A ) )
2619, 2ffvelrnd 5900 . . . 4  |-  ( ph  ->  ( G `  A
)  e.  CC )
2715ffvelrnda 5899 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  ( CC  ^m  S
) )
28 elmapi 7067 . . . . . 6  |-  ( ( F `  k )  e.  ( CC  ^m  S )  ->  ( F `  k ) : S --> CC )
2927, 28syl 16 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k ) : S --> CC )
302adantr 453 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  S )
3129, 30ffvelrnd 5900 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
) `  A )  e.  CC )
3213, 14, 23, 25, 26, 31clim2c 12330 . . 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 261 . 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 360    = wceq 1653    e. wcel 1727   A.wral 2711   E.wrex 2712   _Vcvv 2962   class class class wbr 4237   -->wf 5479   ` cfv 5483  (class class class)co 6110    ^m cmap 7047   CCcc 9019    < clt 9151    - cmin 9322   ZZcz 10313   ZZ>=cuz 10519   RR+crp 10643   abscabs 12070    ~~> cli 12309   ~~> uculm 20323
This theorem is referenced by:  ulmuni  20339  ulmdvlem3  20349  mbfulm  20353  pserulm  20369  lgamgulm2  24851  lgamcvglem  24855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-pre-lttri 9095  ax-pre-lttrn 9096
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-er 6934  df-map 7049  df-pm 7050  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-neg 9325  df-z 10314  df-uz 10520  df-clim 12313  df-ulm 20324
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