MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ulm0 Unicode version

Theorem ulm0 19770
Description: Every function converges uniformly on the empty set. (Contributed by Mario Carneiro, 3-Mar-2015.)
Hypotheses
Ref Expression
ulm0.z  |-  Z  =  ( ZZ>= `  M )
ulm0.m  |-  ( ph  ->  M  e.  ZZ )
ulm0.f  |-  ( ph  ->  F : Z --> ( CC 
^m  S ) )
ulm0.g  |-  ( ph  ->  G : S --> CC )
Assertion
Ref Expression
ulm0  |-  ( (
ph  /\  S  =  (/) )  ->  F ( ~~> u `  S ) G )

Proof of Theorem ulm0
Dummy variables  j 
k  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ulm0.m . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
2 uzid 10242 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
31, 2syl 15 . . . . . . 7  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
4 ulm0.z . . . . . . 7  |-  Z  =  ( ZZ>= `  M )
53, 4syl6eleqr 2374 . . . . . 6  |-  ( ph  ->  M  e.  Z )
6 ne0i 3461 . . . . . 6  |-  ( M  e.  Z  ->  Z  =/=  (/) )
75, 6syl 15 . . . . 5  |-  ( ph  ->  Z  =/=  (/) )
87adantr 451 . . . 4  |-  ( (
ph  /\  S  =  (/) )  ->  Z  =/=  (/) )
9 ral0 3558 . . . . . . 7  |-  A. z  e.  (/)  ( abs `  (
( ( F `  k ) `  z
)  -  ( G `
 z ) ) )  <  x
10 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  S  =  (/) )  ->  S  =  (/) )
1110raleqdv 2742 . . . . . . 7  |-  ( (
ph  /\  S  =  (/) )  ->  ( A. z  e.  S  ( abs `  ( ( ( F `  k ) `
 z )  -  ( G `  z ) ) )  <  x  <->  A. z  e.  (/)  ( abs `  ( ( ( F `
 k ) `  z )  -  ( G `  z )
) )  <  x
) )
129, 11mpbiri 224 . . . . . 6  |-  ( (
ph  /\  S  =  (/) )  ->  A. z  e.  S  ( abs `  ( ( ( F `
 k ) `  z )  -  ( G `  z )
) )  <  x
)
1312ralrimivw 2627 . . . . 5  |-  ( (
ph  /\  S  =  (/) )  ->  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x )
1413ralrimivw 2627 . . . 4  |-  ( (
ph  /\  S  =  (/) )  ->  A. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x )
15 r19.2z 3543 . . . 4  |-  ( ( Z  =/=  (/)  /\  A. 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 ) A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x )
168, 14, 15syl2anc 642 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x )
1716ralrimivw 2627 . 2  |-  ( (
ph  /\  S  =  (/) )  ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x )
181adantr 451 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  M  e.  ZZ )
19 ulm0.f . . . 4  |-  ( ph  ->  F : Z --> ( CC 
^m  S ) )
2019adantr 451 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  F : Z
--> ( CC  ^m  S
) )
21 eqidd 2284 . . 3  |-  ( ( ( ph  /\  S  =  (/) )  /\  (
k  e.  Z  /\  z  e.  S )
)  ->  ( ( F `  k ) `  z )  =  ( ( F `  k
) `  z )
)
22 eqidd 2284 . . 3  |-  ( ( ( ph  /\  S  =  (/) )  /\  z  e.  S )  ->  ( G `  z )  =  ( G `  z ) )
23 ulm0.g . . . 4  |-  ( ph  ->  G : S --> CC )
2423adantr 451 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  G : S
--> CC )
25 0ex 4150 . . . 4  |-  (/)  e.  _V
2610, 25syl6eqel 2371 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  S  e.  _V )
274, 18, 20, 21, 22, 24, 26ulm2 19764 . 2  |-  ( (
ph  /\  S  =  (/) )  ->  ( 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 ) )
2817, 27mpbird 223 1  |-  ( (
ph  /\  S  =  (/) )  ->  F ( ~~> u `  S ) G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788   (/)c0 3455   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   ~~> uculm 19755
This theorem is referenced by:  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-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-ulm 19756
  Copyright terms: Public domain W3C validator