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Theorem ulmi 21994
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 21989 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  G : S --> CC )
92, 8syl 16 . . . 4  |-  ( ph  ->  G : S --> CC )
10 ulmscl 21987 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
112, 10syl 16 . . . 4  |-  ( ph  ->  S  e.  _V )
123, 4, 5, 6, 7, 9, 11ulm2 21993 . . 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 4407 . . . . 5  |-  ( x  =  C  ->  (
( abs `  ( B  -  A )
)  <  x  <->  ( abs `  ( B  -  A
) )  <  C
) )
1514ralbidv 2846 . . . 4  |-  ( x  =  C  ->  ( A. z  e.  S  ( abs `  ( B  -  A ) )  <  x  <->  A. z  e.  S  ( abs `  ( B  -  A
) )  <  C
) )
1615rexralbidv 2881 . . 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 3175 . 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 60 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 369    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800   _Vcvv 3078   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:  ulmshftlem  21997  ulmcau  22003  ulmbdd  22006  ulmcn  22007  iblulm  22015  itgulm  22016
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