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Theorem ulmshft 19785
Description: A sequence of functions converges iff the shifted sequence converges. (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypotheses
Ref Expression
ulmshft.z  |-  Z  =  ( ZZ>= `  M )
ulmshft.w  |-  W  =  ( ZZ>= `  ( M  +  K ) )
ulmshft.m  |-  ( ph  ->  M  e.  ZZ )
ulmshft.k  |-  ( ph  ->  K  e.  ZZ )
ulmshft.f  |-  ( ph  ->  F : Z --> ( CC 
^m  S ) )
ulmshft.h  |-  ( ph  ->  H  =  ( n  e.  W  |->  ( F `
 ( n  -  K ) ) ) )
Assertion
Ref Expression
ulmshft  |-  ( ph  ->  ( F ( ~~> u `  S ) G  <->  H ( ~~> u `  S ) G ) )
Distinct variable groups:    ph, n    n, W    n, F    n, K    S, n
Allowed substitution hints:    G( n)    H( n)    M( n)    Z( n)

Proof of Theorem ulmshft
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 ulmshft.z . . 3  |-  Z  =  ( ZZ>= `  M )
2 ulmshft.w . . 3  |-  W  =  ( ZZ>= `  ( M  +  K ) )
3 ulmshft.m . . 3  |-  ( ph  ->  M  e.  ZZ )
4 ulmshft.k . . 3  |-  ( ph  ->  K  e.  ZZ )
5 ulmshft.f . . 3  |-  ( ph  ->  F : Z --> ( CC 
^m  S ) )
6 ulmshft.h . . 3  |-  ( ph  ->  H  =  ( n  e.  W  |->  ( F `
 ( n  -  K ) ) ) )
71, 2, 3, 4, 5, 6ulmshftlem 19784 . 2  |-  ( ph  ->  ( F ( ~~> u `  S ) G  ->  H ( ~~> u `  S ) G ) )
8 eqid 2296 . . 3  |-  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )
93, 4zaddcld 10137 . . 3  |-  ( ph  ->  ( M  +  K
)  e.  ZZ )
104znegcld 10135 . . 3  |-  ( ph  -> 
-u K  e.  ZZ )
115adantr 451 . . . . . 6  |-  ( (
ph  /\  n  e.  W )  ->  F : Z --> ( CC  ^m  S ) )
123adantr 451 . . . . . . . 8  |-  ( (
ph  /\  n  e.  W )  ->  M  e.  ZZ )
134adantr 451 . . . . . . . 8  |-  ( (
ph  /\  n  e.  W )  ->  K  e.  ZZ )
14 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  W )  ->  n  e.  W )
1514, 2syl6eleq 2386 . . . . . . . 8  |-  ( (
ph  /\  n  e.  W )  ->  n  e.  ( ZZ>= `  ( M  +  K ) ) )
16 eluzsub 10273 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ  /\  n  e.  ( ZZ>= `  ( M  +  K ) ) )  ->  ( n  -  K )  e.  (
ZZ>= `  M ) )
1712, 13, 15, 16syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  n  e.  W )  ->  (
n  -  K )  e.  ( ZZ>= `  M
) )
1817, 1syl6eleqr 2387 . . . . . 6  |-  ( (
ph  /\  n  e.  W )  ->  (
n  -  K )  e.  Z )
19 ffvelrn 5679 . . . . . 6  |-  ( ( F : Z --> ( CC 
^m  S )  /\  ( n  -  K
)  e.  Z )  ->  ( F `  ( n  -  K
) )  e.  ( CC  ^m  S ) )
2011, 18, 19syl2anc 642 . . . . 5  |-  ( (
ph  /\  n  e.  W )  ->  ( F `  ( n  -  K ) )  e.  ( CC  ^m  S
) )
21 eqid 2296 . . . . 5  |-  ( n  e.  W  |->  ( F `
 ( n  -  K ) ) )  =  ( n  e.  W  |->  ( F `  ( n  -  K
) ) )
2220, 21fmptd 5700 . . . 4  |-  ( ph  ->  ( n  e.  W  |->  ( F `  (
n  -  K ) ) ) : W --> ( CC  ^m  S ) )
236feq1d 5395 . . . 4  |-  ( ph  ->  ( H : W --> ( CC  ^m  S )  <-> 
( n  e.  W  |->  ( F `  (
n  -  K ) ) ) : W --> ( CC  ^m  S ) ) )
2422, 23mpbird 223 . . 3  |-  ( ph  ->  H : W --> ( CC 
^m  S ) )
25 simpr 447 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  Z )
2625, 1syl6eleq 2386 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  ( ZZ>= `  M )
)
27 eluzelz 10254 . . . . . . . . . 10  |-  ( m  e.  ( ZZ>= `  M
)  ->  m  e.  ZZ )
2826, 27syl 15 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  ZZ )
2928zcnd 10134 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  CC )
304zcnd 10134 . . . . . . . . 9  |-  ( ph  ->  K  e.  CC )
3130adantr 451 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  K  e.  CC )
3229, 31subnegd 9180 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  (
m  -  -u K
)  =  ( m  +  K ) )
3332fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  m  e.  Z )  ->  ( H `  ( m  -  -u K ) )  =  ( H `  ( m  +  K
) ) )
346adantr 451 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  H  =  ( n  e.  W  |->  ( F `  ( n  -  K
) ) ) )
3534fveq1d 5543 . . . . . 6  |-  ( (
ph  /\  m  e.  Z )  ->  ( H `  ( m  +  K ) )  =  ( ( n  e.  W  |->  ( F `  ( n  -  K
) ) ) `  ( m  +  K
) ) )
364adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  Z )  ->  K  e.  ZZ )
37 eluzadd 10272 . . . . . . . . . 10  |-  ( ( m  e.  ( ZZ>= `  M )  /\  K  e.  ZZ )  ->  (
m  +  K )  e.  ( ZZ>= `  ( M  +  K )
) )
3826, 36, 37syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  Z )  ->  (
m  +  K )  e.  ( ZZ>= `  ( M  +  K )
) )
3938, 2syl6eleqr 2387 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  (
m  +  K )  e.  W )
40 oveq1 5881 . . . . . . . . . 10  |-  ( n  =  ( m  +  K )  ->  (
n  -  K )  =  ( ( m  +  K )  -  K ) )
4140fveq2d 5545 . . . . . . . . 9  |-  ( n  =  ( m  +  K )  ->  ( F `  ( n  -  K ) )  =  ( F `  (
( m  +  K
)  -  K ) ) )
42 fvex 5555 . . . . . . . . 9  |-  ( F `
 ( ( m  +  K )  -  K ) )  e. 
_V
4341, 21, 42fvmpt 5618 . . . . . . . 8  |-  ( ( m  +  K )  e.  W  ->  (
( n  e.  W  |->  ( F `  (
n  -  K ) ) ) `  (
m  +  K ) )  =  ( F `
 ( ( m  +  K )  -  K ) ) )
4439, 43syl 15 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  (
( n  e.  W  |->  ( F `  (
n  -  K ) ) ) `  (
m  +  K ) )  =  ( F `
 ( ( m  +  K )  -  K ) ) )
4529, 31pncand 9174 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  (
( m  +  K
)  -  K )  =  m )
4645fveq2d 5545 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  ( F `  ( (
m  +  K )  -  K ) )  =  ( F `  m ) )
4744, 46eqtrd 2328 . . . . . 6  |-  ( (
ph  /\  m  e.  Z )  ->  (
( n  e.  W  |->  ( F `  (
n  -  K ) ) ) `  (
m  +  K ) )  =  ( F `
 m ) )
4833, 35, 473eqtrd 2332 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  ( H `  ( m  -  -u K ) )  =  ( F `  m ) )
4948mpteq2dva 4122 . . . 4  |-  ( ph  ->  ( m  e.  Z  |->  ( H `  (
m  -  -u K
) ) )  =  ( m  e.  Z  |->  ( F `  m
) ) )
503zcnd 10134 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
5110zcnd 10134 . . . . . . . . 9  |-  ( ph  -> 
-u K  e.  CC )
5250, 30, 51addassd 8873 . . . . . . . 8  |-  ( ph  ->  ( ( M  +  K )  +  -u K )  =  ( M  +  ( K  +  -u K ) ) )
5330negidd 9163 . . . . . . . . 9  |-  ( ph  ->  ( K  +  -u K )  =  0 )
5453oveq2d 5890 . . . . . . . 8  |-  ( ph  ->  ( M  +  ( K  +  -u K
) )  =  ( M  +  0 ) )
5550addid1d 9028 . . . . . . . 8  |-  ( ph  ->  ( M  +  0 )  =  M )
5652, 54, 553eqtrd 2332 . . . . . . 7  |-  ( ph  ->  ( ( M  +  K )  +  -u K )  =  M )
5756fveq2d 5545 . . . . . 6  |-  ( ph  ->  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  ( ZZ>= `  M
) )
5857, 1syl6eqr 2346 . . . . 5  |-  ( ph  ->  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  Z )
59 mpteq1 4116 . . . . 5  |-  ( (
ZZ>= `  ( ( M  +  K )  + 
-u K ) )  =  Z  ->  (
m  e.  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  |->  ( H `  ( m  -  -u K ) ) )  =  ( m  e.  Z  |->  ( H `
 ( m  -  -u K ) ) ) )
6058, 59syl 15 . . . 4  |-  ( ph  ->  ( m  e.  (
ZZ>= `  ( ( M  +  K )  + 
-u K ) ) 
|->  ( H `  (
m  -  -u K
) ) )  =  ( m  e.  Z  |->  ( H `  (
m  -  -u K
) ) ) )
615feqmptd 5591 . . . 4  |-  ( ph  ->  F  =  ( m  e.  Z  |->  ( F `
 m ) ) )
6249, 60, 613eqtr4rd 2339 . . 3  |-  ( ph  ->  F  =  ( m  e.  ( ZZ>= `  (
( M  +  K
)  +  -u K
) )  |->  ( H `
 ( m  -  -u K ) ) ) )
632, 8, 9, 10, 24, 62ulmshftlem 19784 . 2  |-  ( ph  ->  ( H ( ~~> u `  S ) G  ->  F ( ~~> u `  S ) G ) )
647, 63impbid 183 1  |-  ( ph  ->  ( F ( ~~> u `  S ) G  <->  H ( ~~> u `  S ) G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   CCcc 8751   0cc0 8753    + caddc 8756    - cmin 9053   -ucneg 9054   ZZcz 10040   ZZ>=cuz 10246   ~~> uculm 19771
This theorem is referenced by:  pserdvlem2  19820
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-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-riota 6320  df-recs 6404  df-rdg 6439  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-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-ulm 19772
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