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Theorem ulm0 23333
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 11174 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
31, 2syl 17 . . . . . . 7  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
4 ulm0.z . . . . . . 7  |-  Z  =  ( ZZ>= `  M )
53, 4syl6eleqr 2521 . . . . . 6  |-  ( ph  ->  M  e.  Z )
6 ne0i 3767 . . . . . 6  |-  ( M  e.  Z  ->  Z  =/=  (/) )
75, 6syl 17 . . . . 5  |-  ( ph  ->  Z  =/=  (/) )
87adantr 466 . . . 4  |-  ( (
ph  /\  S  =  (/) )  ->  Z  =/=  (/) )
9 ral0 3902 . . . . . . 7  |-  A. z  e.  (/)  ( abs `  (
( ( F `  k ) `  z
)  -  ( G `
 z ) ) )  <  x
10 simpr 462 . . . . . . . 8  |-  ( (
ph  /\  S  =  (/) )  ->  S  =  (/) )
1110raleqdv 3031 . . . . . . 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 236 . . . . . 6  |-  ( (
ph  /\  S  =  (/) )  ->  A. z  e.  S  ( abs `  ( ( ( F `
 k ) `  z )  -  ( G `  z )
) )  <  x
)
1312ralrimivw 2840 . . . . 5  |-  ( (
ph  /\  S  =  (/) )  ->  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x )
1413ralrimivw 2840 . . . 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 3886 . . . 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 665 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x )
1716ralrimivw 2840 . 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 466 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  M  e.  ZZ )
19 ulm0.f . . . 4  |-  ( ph  ->  F : Z --> ( CC 
^m  S ) )
2019adantr 466 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  F : Z
--> ( CC  ^m  S
) )
21 eqidd 2423 . . 3  |-  ( ( ( ph  /\  S  =  (/) )  /\  (
k  e.  Z  /\  z  e.  S )
)  ->  ( ( F `  k ) `  z )  =  ( ( F `  k
) `  z )
)
22 eqidd 2423 . . 3  |-  ( ( ( ph  /\  S  =  (/) )  /\  z  e.  S )  ->  ( G `  z )  =  ( G `  z ) )
23 ulm0.g . . . 4  |-  ( ph  ->  G : S --> CC )
2423adantr 466 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  G : S
--> CC )
25 0ex 4553 . . . 4  |-  (/)  e.  _V
2610, 25syl6eqel 2518 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  S  e.  _V )
274, 18, 20, 21, 22, 24, 26ulm2 23327 . 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 235 1  |-  ( (
ph  /\  S  =  (/) )  ->  F ( ~~> u `  S ) G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   E.wrex 2776   _Vcvv 3081   (/)c0 3761   class class class wbr 4420   -->wf 5594   ` cfv 5598  (class class class)co 6302    ^m cmap 7477   CCcc 9538    < clt 9676    - cmin 9861   ZZcz 10938   ZZ>=cuz 11160   RR+crp 11303   abscabs 13286   ~~> uculm 23318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-pre-lttri 9614  ax-pre-lttrn 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-po 4771  df-so 4772  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-neg 9864  df-z 10939  df-uz 11161  df-ulm 23319
This theorem is referenced by:  pserulm  23364
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