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Theorem ulmi 19765
Description: The uniform limit property. (Contributed by Mario Carneiro, 27-Feb-2015.)
Hypotheses
Ref Expression
ulm2.z  |-  Z  =  ( ZZ>= `  M )
ulm2.m  |-  ( ph  ->  M  e.  ZZ )
ulm2.f  |-  ( ph  ->  F : Z --> ( CC 
^m  S ) )
ulm2.b  |-  ( (
ph  /\  ( k  e.  Z  /\  z  e.  S ) )  -> 
( ( F `  k ) `  z
)  =  B )
ulm2.a  |-  ( (
ph  /\  z  e.  S )  ->  ( G `  z )  =  A )
ulmi.u  |-  ( ph  ->  F ( ~~> u `  S ) G )
ulmi.c  |-  ( ph  ->  C  e.  RR+ )
Assertion
Ref Expression
ulmi  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) A. z  e.  S  ( abs `  ( B  -  A ) )  < 
C )
Distinct variable groups:    j, k,
z, F    j, G, k, z    j, M, k, z    ph, j, k, z    A, j, k    C, j, k, z    S, j, k, z    j, Z
Allowed substitution hints:    A( z)    B( z, j, k)    Z( z, k)

Proof of Theorem ulmi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ulmi.c . 2  |-  ( ph  ->  C  e.  RR+ )
2 ulmi.u . . 3  |-  ( ph  ->  F ( ~~> u `  S ) G )
3 ulm2.z . . . 4  |-  Z  =  ( ZZ>= `  M )
4 ulm2.m . . . 4  |-  ( ph  ->  M  e.  ZZ )
5 ulm2.f . . . 4  |-  ( ph  ->  F : Z --> ( CC 
^m  S ) )
6 ulm2.b . . . 4  |-  ( (
ph  /\  ( k  e.  Z  /\  z  e.  S ) )  -> 
( ( F `  k ) `  z
)  =  B )
7 ulm2.a . . . 4  |-  ( (
ph  /\  z  e.  S )  ->  ( G `  z )  =  A )
8 ulmcl 19760 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  G : S --> CC )
92, 8syl 15 . . . 4  |-  ( ph  ->  G : S --> CC )
10 ulmscl 19758 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
112, 10syl 15 . . . 4  |-  ( ph  ->  S  e.  _V )
123, 4, 5, 6, 7, 9, 11ulm2 19764 . . 3  |-  ( ph  ->  ( F ( ~~> u `  S ) G  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( B  -  A ) )  <  x ) )
132, 12mpbid 201 . 2  |-  ( ph  ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( B  -  A
) )  <  x
)
14 breq2 4027 . . . . 5  |-  ( x  =  C  ->  (
( abs `  ( B  -  A )
)  <  x  <->  ( abs `  ( B  -  A
) )  <  C
) )
1514ralbidv 2563 . . . 4  |-  ( x  =  C  ->  ( A. z  e.  S  ( abs `  ( B  -  A ) )  <  x  <->  A. z  e.  S  ( abs `  ( B  -  A
) )  <  C
) )
1615rexralbidv 2587 . . 3  |-  ( x  =  C  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) A. z  e.  S  ( abs `  ( B  -  A ) )  < 
x  <->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) A. z  e.  S  ( abs `  ( B  -  A ) )  < 
C ) )
1716rspcv 2880 . 2  |-  ( C  e.  RR+  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( B  -  A ) )  <  x  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A. z  e.  S  ( abs `  ( B  -  A ) )  <  C ) )
181, 13, 17sylc 56 1  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) A. z  e.  S  ( abs `  ( B  -  A ) )  < 
C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788   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:  ulmshftlem  19768  ulmcau  19772  ulmbdd  19775  ulmcn  19776  iblulm  19783  itgulm  19784
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
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