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Theorem ulm0 22658
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 11104 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
4 ulm0.z . . . . . . 7  |-  Z  =  ( ZZ>= `  M )
53, 4syl6eleqr 2542 . . . . . 6  |-  ( ph  ->  M  e.  Z )
6 ne0i 3776 . . . . . 6  |-  ( M  e.  Z  ->  Z  =/=  (/) )
75, 6syl 16 . . . . 5  |-  ( ph  ->  Z  =/=  (/) )
87adantr 465 . . . 4  |-  ( (
ph  /\  S  =  (/) )  ->  Z  =/=  (/) )
9 ral0 3919 . . . . . . 7  |-  A. z  e.  (/)  ( abs `  (
( ( F `  k ) `  z
)  -  ( G `
 z ) ) )  <  x
10 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  S  =  (/) )  ->  S  =  (/) )
1110raleqdv 3046 . . . . . . 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 233 . . . . . 6  |-  ( (
ph  /\  S  =  (/) )  ->  A. z  e.  S  ( abs `  ( ( ( F `
 k ) `  z )  -  ( G `  z )
) )  <  x
)
1312ralrimivw 2858 . . . . 5  |-  ( (
ph  /\  S  =  (/) )  ->  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x )
1413ralrimivw 2858 . . . 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 3904 . . . 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 661 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x )
1716ralrimivw 2858 . 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 465 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  M  e.  ZZ )
19 ulm0.f . . . 4  |-  ( ph  ->  F : Z --> ( CC 
^m  S ) )
2019adantr 465 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  F : Z
--> ( CC  ^m  S
) )
21 eqidd 2444 . . 3  |-  ( ( ( ph  /\  S  =  (/) )  /\  (
k  e.  Z  /\  z  e.  S )
)  ->  ( ( F `  k ) `  z )  =  ( ( F `  k
) `  z )
)
22 eqidd 2444 . . 3  |-  ( ( ( ph  /\  S  =  (/) )  /\  z  e.  S )  ->  ( G `  z )  =  ( G `  z ) )
23 ulm0.g . . . 4  |-  ( ph  ->  G : S --> CC )
2423adantr 465 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  G : S
--> CC )
25 0ex 4567 . . . 4  |-  (/)  e.  _V
2610, 25syl6eqel 2539 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  S  e.  _V )
274, 18, 20, 21, 22, 24, 26ulm2 22652 . 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 232 1  |-  ( (
ph  /\  S  =  (/) )  ->  F ( ~~> u `  S ) G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   E.wrex 2794   _Vcvv 3095   (/)c0 3770   class class class wbr 4437   -->wf 5574   ` cfv 5578  (class class class)co 6281    ^m cmap 7422   CCcc 9493    < clt 9631    - cmin 9810   ZZcz 10870   ZZ>=cuz 11090   RR+crp 11229   abscabs 13046   ~~> uculm 22643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-pre-lttri 9569  ax-pre-lttrn 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-neg 9813  df-z 10871  df-uz 11091  df-ulm 22644
This theorem is referenced by:  pserulm  22689
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