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Theorem ulm0 21999
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 10990 . . . . . . . 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 2553 . . . . . 6  |-  ( ph  ->  M  e.  Z )
6 ne0i 3754 . . . . . 6  |-  ( M  e.  Z  ->  Z  =/=  (/) )
75, 6syl 16 . . . . 5  |-  ( ph  ->  Z  =/=  (/) )
87adantr 465 . . . 4  |-  ( (
ph  /\  S  =  (/) )  ->  Z  =/=  (/) )
9 ral0 3895 . . . . . . 7  |-  A. z  e.  (/)  ( abs `  (
( ( F `  k ) `  z
)  -  ( G `
 z ) ) )  <  x
10 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  S  =  (/) )  ->  S  =  (/) )
1110raleqdv 3029 . . . . . . 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 2831 . . . . 5  |-  ( (
ph  /\  S  =  (/) )  ->  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( ( ( F `  k
) `  z )  -  ( G `  z ) ) )  <  x )
1413ralrimivw 2831 . . . 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 3880 . . . 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 2831 . 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 2455 . . 3  |-  ( ( ( ph  /\  S  =  (/) )  /\  (
k  e.  Z  /\  z  e.  S )
)  ->  ( ( F `  k ) `  z )  =  ( ( F `  k
) `  z )
)
22 eqidd 2455 . . 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 4533 . . . 4  |-  (/)  e.  _V
2610, 25syl6eqel 2550 . . 3  |-  ( (
ph  /\  S  =  (/) )  ->  S  e.  _V )
274, 18, 20, 21, 22, 24, 26ulm2 21993 . 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 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   _Vcvv 3078   (/)c0 3748   class class class wbr 4403   -->wf 5525   ` cfv 5529  (class class class)co 6203    ^m cmap 7327   CCcc 9395    < clt 9533    - cmin 9710   ZZcz 10761   ZZ>=cuz 10976   RR+crp 11106   abscabs 12845   ~~> uculm 21984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-pre-lttri 9471  ax-pre-lttrn 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-neg 9713  df-z 10762  df-uz 10977  df-ulm 21985
This theorem is referenced by:  pserulm  22030
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