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Theorem ulmi 19781
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 19776 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  G : S --> CC )
92, 8syl 15 . . . 4  |-  ( ph  ->  G : S --> CC )
10 ulmscl 19774 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
112, 10syl 15 . . . 4  |-  ( ph  ->  S  e.  _V )
123, 4, 5, 6, 7, 9, 11ulm2 19780 . . 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 4043 . . . . 5  |-  ( x  =  C  ->  (
( abs `  ( B  -  A )
)  <  x  <->  ( abs `  ( B  -  A
) )  <  C
) )
1514ralbidv 2576 . . . 4  |-  ( x  =  C  ->  ( A. z  e.  S  ( abs `  ( B  -  A ) )  <  x  <->  A. z  e.  S  ( abs `  ( B  -  A
) )  <  C
) )
1615rexralbidv 2600 . . 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 2893 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801   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:  ulmshftlem  19784  ulmcau  19788  ulmbdd  19791  ulmcn  19792  iblulm  19799  itgulm  19800
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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