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Theorem clim2iser 10313
Description: The limit of an infinite series with an initial segment removed. (Contributed by Jim Kingdon, 20-Aug-2021.)
Hypotheses
Ref Expression
clim2ser.1  |-  Z  =  ( ZZ>= `  M )
clim2ser.2  |-  ( ph  ->  N  e.  Z )
clim2ser.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
clim2ser.5  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) 
~~>  A )
Assertion
Ref Expression
clim2iser  |-  ( ph  ->  seq ( N  + 
1 ) (  +  ,  F ,  CC ) 
~~>  ( A  -  (  seq M (  +  ,  F ,  CC ) `  N ) ) )
Distinct variable groups:    A, k    k, F    k, M    k, N    ph, k    k, Z

Proof of Theorem clim2iser
Dummy variables  j  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2082 . 2  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
2 clim2ser.2 . . . . 5  |-  ( ph  ->  N  e.  Z )
3 clim2ser.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
42, 3syl6eleq 2172 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 peano2uz 8752 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
64, 5syl 14 . . 3  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )
7 eluzelz 8709 . . 3  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ZZ )
86, 7syl 14 . 2  |-  ( ph  ->  ( N  +  1 )  e.  ZZ )
9 clim2ser.5 . 2  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) 
~~>  A )
10 eluzel2 8705 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
114, 10syl 14 . . . 4  |-  ( ph  ->  M  e.  ZZ )
12 clim2ser.4 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
133, 11, 12iserf 9549 . . 3  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) : Z --> CC )
1413, 2ffvelrnd 5335 . 2  |-  ( ph  ->  (  seq M (  +  ,  F ,  CC ) `  N )  e.  CC )
15 iseqex 9523 . . 3  |-  seq ( N  +  1 ) (  +  ,  F ,  CC )  e.  _V
1615a1i 9 . 2  |-  ( ph  ->  seq ( N  + 
1 ) (  +  ,  F ,  CC )  e.  _V )
1713adantr 270 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  seq M (  +  ,  F ,  CC ) : Z --> CC )
186, 3syl6eleqr 2173 . . . 4  |-  ( ph  ->  ( N  +  1 )  e.  Z )
193uztrn2 8717 . . . 4  |-  ( ( ( N  +  1 )  e.  Z  /\  j  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
j  e.  Z )
2018, 19sylan 277 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  j  e.  Z )
2117, 20ffvelrnd 5335 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  +  ,  F ,  CC ) `  j )  e.  CC )
22 addcl 7160 . . . . . 6  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  +  x
)  e.  CC )
2322adantl 271 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  +  x
)  e.  CC )
24 addass 7165 . . . . . 6  |-  ( ( k  e.  CC  /\  x  e.  CC  /\  y  e.  CC )  ->  (
( k  +  x
)  +  y )  =  ( k  +  ( x  +  y ) ) )
2524adantl 271 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  ( k  e.  CC  /\  x  e.  CC  /\  y  e.  CC ) )  -> 
( ( k  +  x )  +  y )  =  ( k  +  ( x  +  y ) ) )
26 simpr 108 . . . . 5  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  j  e.  ( ZZ>= `  ( N  +  1 ) ) )
27 cnex 7159 . . . . . 6  |-  CC  e.  _V
2827a1i 9 . . . . 5  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  CC  e.  _V )
294adantr 270 . . . . 5  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  ( ZZ>= `  M )
)
303eleq2i 2146 . . . . . . 7  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
3130, 12sylan2br 282 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3231adantlr 461 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3323, 25, 26, 28, 29, 32iseqsplit 9554 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  +  ,  F ,  CC ) `  j )  =  ( (  seq M (  +  ,  F ,  CC ) `  N )  +  (  seq ( N  +  1 ) (  +  ,  F ,  CC ) `  j
) ) )
3433oveq1d 5558 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( (  seq M (  +  ,  F ,  CC ) `  j )  -  (  seq M (  +  ,  F ,  CC ) `  N ) )  =  ( ( (  seq M (  +  ,  F ,  CC ) `  N )  +  (  seq ( N  + 
1 ) (  +  ,  F ,  CC ) `  j )
)  -  (  seq M (  +  ,  F ,  CC ) `  N ) ) )
3514adantr 270 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  +  ,  F ,  CC ) `  N )  e.  CC )
363uztrn2 8717 . . . . . . . 8  |-  ( ( ( N  +  1 )  e.  Z  /\  k  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
k  e.  Z )
3718, 36sylan 277 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  Z )
3837, 12syldan 276 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  k )  e.  CC )
391, 8, 38iserf 9549 . . . . 5  |-  ( ph  ->  seq ( N  + 
1 ) (  +  ,  F ,  CC ) : ( ZZ>= `  ( N  +  1 ) ) --> CC )
4039ffvelrnda 5334 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq ( N  +  1
) (  +  ,  F ,  CC ) `  j )  e.  CC )
4135, 40pncan2d 7488 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( (
(  seq M (  +  ,  F ,  CC ) `  N )  +  (  seq ( N  +  1 ) (  +  ,  F ,  CC ) `  j
) )  -  (  seq M (  +  ,  F ,  CC ) `  N ) )  =  (  seq ( N  +  1 ) (  +  ,  F ,  CC ) `  j ) )
4234, 41eqtr2d 2115 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq ( N  +  1
) (  +  ,  F ,  CC ) `  j )  =  ( (  seq M (  +  ,  F ,  CC ) `  j )  -  (  seq M
(  +  ,  F ,  CC ) `  N
) ) )
431, 8, 9, 14, 16, 21, 42climsubc1 10308 1  |-  ( ph  ->  seq ( N  + 
1 ) (  +  ,  F ,  CC ) 
~~>  ( A  -  (  seq M (  +  ,  F ,  CC ) `  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285    e. wcel 1434   _Vcvv 2602   class class class wbr 3793   -->wf 4928   ` cfv 4932  (class class class)co 5543   CCcc 7041   1c1 7044    + caddc 7046    - cmin 7346   ZZcz 8432   ZZ>=cuz 8700    seqcseq 9521    ~~> cli 10255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156  ax-arch 7157  ax-caucvg 7158
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-2 8165  df-3 8166  df-4 8167  df-n0 8356  df-z 8433  df-uz 8701  df-rp 8816  df-fz 9106  df-iseq 9522  df-iexp 9573  df-cj 9867  df-re 9868  df-im 9869  df-rsqrt 10022  df-abs 10023  df-clim 10256
This theorem is referenced by:  iiserex  10315
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