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Theorem ulmi 23073
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 23068 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  G : S --> CC )
92, 8syl 17 . . . 4  |-  ( ph  ->  G : S --> CC )
10 ulmscl 23066 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
112, 10syl 17 . . . 4  |-  ( ph  ->  S  e.  _V )
123, 4, 5, 6, 7, 9, 11ulm2 23072 . . 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 210 . 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 4399 . . . . 5  |-  ( x  =  C  ->  (
( abs `  ( B  -  A )
)  <  x  <->  ( abs `  ( B  -  A
) )  <  C
) )
1514ralbidv 2843 . . . 4  |-  ( x  =  C  ->  ( A. z  e.  S  ( abs `  ( B  -  A ) )  <  x  <->  A. z  e.  S  ( abs `  ( B  -  A
) )  <  C
) )
1615rexralbidv 2926 . . 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 3156 . 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 59 1  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) A. z  e.  S  ( abs `  ( B  -  A ) )  < 
C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   _Vcvv 3059   class class class wbr 4395   -->wf 5565   ` cfv 5569  (class class class)co 6278    ^m cmap 7457   CCcc 9520    < clt 9658    - cmin 9841   ZZcz 10905   ZZ>=cuz 11127   RR+crp 11265   abscabs 13216   ~~> uculm 23063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-pre-lttri 9596  ax-pre-lttrn 9597
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-neg 9844  df-z 10906  df-uz 11128  df-ulm 23064
This theorem is referenced by:  ulmshftlem  23076  ulmcau  23082  ulmbdd  23085  ulmcn  23086  iblulm  23094  itgulm  23095
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