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Theorem ulmi 21736
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 21731 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  G : S --> CC )
92, 8syl 16 . . . 4  |-  ( ph  ->  G : S --> CC )
10 ulmscl 21729 . . . . 5  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
112, 10syl 16 . . . 4  |-  ( ph  ->  S  e.  _V )
123, 4, 5, 6, 7, 9, 11ulm2 21735 . . 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 4284 . . . . 5  |-  ( x  =  C  ->  (
( abs `  ( B  -  A )
)  <  x  <->  ( abs `  ( B  -  A
) )  <  C
) )
1514ralbidv 2725 . . . 4  |-  ( x  =  C  ->  ( A. z  e.  S  ( abs `  ( B  -  A ) )  <  x  <->  A. z  e.  S  ( abs `  ( B  -  A
) )  <  C
) )
1615rexralbidv 2749 . . 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 3058 . 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 1362    e. wcel 1755   A.wral 2705   E.wrex 2706   _Vcvv 2962   class class class wbr 4280   -->wf 5402   ` cfv 5406  (class class class)co 6080    ^m cmap 7202   CCcc 9268    < clt 9406    - cmin 9583   ZZcz 10634   ZZ>=cuz 10849   RR+crp 10979   abscabs 12707   ~~> uculm 21726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-pre-lttri 9344  ax-pre-lttrn 9345
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-po 4628  df-so 4629  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-neg 9586  df-z 10635  df-uz 10850  df-ulm 21727
This theorem is referenced by:  ulmshftlem  21739  ulmcau  21745  ulmbdd  21748  ulmcn  21749  iblulm  21757  itgulm  21758
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