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Theorem ulm0 19786
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 10258 . . . . . . . 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 2387 . . . . . 6  |-  ( ph  ->  M  e.  Z )
6 ne0i 3474 . . . . . 6  |-  ( M  e.  Z  ->  Z  =/=  (/) )
75, 6syl 15 . . . . 5  |-  ( ph  ->  Z  =/=  (/) )
87adantr 451 . . . 4  |-  ( (
ph  /\  S  =  (/) )  ->  Z  =/=  (/) )
9 ral0 3571 . . . . . . 7  |-  A. z  e.  (/)  ( abs `  (
( ( F `  k ) `  z
)  -  ( G `
 z ) ) )  <  x
10 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  S  =  (/) )  ->  S  =  (/) )
1110raleqdv 2755 . . . . . . 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 2640 . . . . 5  |-  ( (
ph  /\  S  =  (/) )  ->  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x )
1413ralrimivw 2640 . . . 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 3556 . . . 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 2640 . 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 2297 . . 3  |-  ( ( ( ph  /\  S  =  (/) )  /\  (
k  e.  Z  /\  z  e.  S )
)  ->  ( ( F `  k ) `  z )  =  ( ( F `  k
) `  z )
)
22 eqidd 2297 . . 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 4166 . . . 4  |-  (/)  e.  _V
2610, 25syl6eqel 2384 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  S  e.  _V )
274, 18, 20, 21, 22, 24, 26ulm2 19780 . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801   (/)c0 3468   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   CCcc 8751    < clt 8883    - cmin 9053   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   abscabs 11735   ~~> uculm 19771
This theorem is referenced by:  pserulm  19814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-neg 9056  df-z 10041  df-uz 10247  df-ulm 19772
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