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Theorem ulmclm 19766
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 5525 . . . . . . . . . . 11  |-  ( z  =  A  ->  (
( F `  k
) `  z )  =  ( ( F `
 k ) `  A ) )
4 fveq2 5525 . . . . . . . . . . 11  |-  ( z  =  A  ->  ( G `  z )  =  ( G `  A ) )
53, 4oveq12d 5876 . . . . . . . . . 10  |-  ( z  =  A  ->  (
( ( F `  k ) `  z
)  -  ( G `
 z ) )  =  ( ( ( F `  k ) `
 A )  -  ( G `  A ) ) )
65fveq2d 5529 . . . . . . . . 9  |-  ( z  =  A  ->  ( abs `  ( ( ( F `  k ) `
 z )  -  ( G `  z ) ) )  =  ( abs `  ( ( ( F `  k
) `  A )  -  ( G `  A ) ) ) )
76breq1d 4033 . . . . . . . 8  |-  ( z  =  A  ->  (
( abs `  (
( ( F `  k ) `  z
)  -  ( G `
 z ) ) )  <  x  <->  ( abs `  ( ( ( F `
 k ) `  A )  -  ( G `  A )
) )  <  x
) )
87rspcv 2880 . . . . . . 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 2622 . . . . 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 2654 . . . 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 2622 . . 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 2284 . . . 4  |-  ( (
ph  /\  ( k  e.  Z  /\  z  e.  S ) )  -> 
( ( F `  k ) `  z
)  =  ( ( F `  k ) `
 z ) )
17 eqidd 2284 . . . 4  |-  ( (
ph  /\  z  e.  S )  ->  ( G `  z )  =  ( G `  z ) )
18 ulmcl 19760 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  G : S --> CC )
191, 18syl 15 . . . 4  |-  ( ph  ->  G : S --> CC )
20 ulmscl 19758 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
211, 20syl 15 . . . 4  |-  ( ph  ->  S  e.  _V )
2213, 14, 15, 16, 17, 19, 21ulm2 19764 . . 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 2288 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k ) `  A ) )
26 ffvelrn 5663 . . . . 5  |-  ( ( G : S --> CC  /\  A  e.  S )  ->  ( G `  A
)  e.  CC )
2719, 2, 26syl2anc 642 . . . 4  |-  ( ph  ->  ( G `  A
)  e.  CC )
28 ffvelrn 5663 . . . . . . 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 6792 . . . . . 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 5663 . . . . 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 11979 . . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735    < clt 8867    - cmin 9037   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   abscabs 11719    ~~> cli 11958   ~~> uculm 19755
This theorem is referenced by:  ulmdvlem3  19779  mbfulm  19782  pserulm  19798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-neg 9040  df-z 10025  df-uz 10231  df-clim 11962  df-ulm 19756
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