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Theorem clim2ser2 11235
Description: The limit of an infinite series with an initial segment added. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
Hypotheses
Ref Expression
clim2ser.1  |-  Z  =  ( ZZ>= `  M )
clim2ser.2  |-  ( ph  ->  N  e.  Z )
clim2ser.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
clim2ser2.5  |-  ( ph  ->  seq ( N  + 
1 ) (  +  ,  F )  ~~>  A )
Assertion
Ref Expression
clim2ser2  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  ( A  +  (  seq M
(  +  ,  F
) `  N )
) )
Distinct variable groups:    A, k    k, F    k, M    k, N    ph, k    k, Z

Proof of Theorem clim2ser2
Dummy variables  j  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2157 . 2  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
2 clim2ser.2 . . . . 5  |-  ( ph  ->  N  e.  Z )
3 clim2ser.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
42, 3eleqtrdi 2250 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 peano2uz 9494 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
64, 5syl 14 . . 3  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )
7 eluzelz 9448 . . 3  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ZZ )
86, 7syl 14 . 2  |-  ( ph  ->  ( N  +  1 )  e.  ZZ )
9 clim2ser2.5 . 2  |-  ( ph  ->  seq ( N  + 
1 ) (  +  ,  F )  ~~>  A )
10 eluzel2 9444 . . . . 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, 12serf 10373 . . 3  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> CC )
1413, 2ffvelrnd 5603 . 2  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  e.  CC )
15 seqex 10346 . . 3  |-  seq M
(  +  ,  F
)  e.  _V
1615a1i 9 . 2  |-  ( ph  ->  seq M (  +  ,  F )  e. 
_V )
176, 3eleqtrrdi 2251 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  Z )
183uztrn2 9456 . . . . . 6  |-  ( ( ( N  +  1 )  e.  Z  /\  k  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
k  e.  Z )
1917, 18sylan 281 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  Z )
2019, 12syldan 280 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  k )  e.  CC )
211, 8, 20serf 10373 . . 3  |-  ( ph  ->  seq ( N  + 
1 ) (  +  ,  F ) : ( ZZ>= `  ( N  +  1 ) ) --> CC )
2221ffvelrnda 5602 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq ( N  +  1
) (  +  ,  F ) `  j
)  e.  CC )
2314adantr 274 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  +  ,  F ) `  N
)  e.  CC )
24 addcl 7857 . . . . 5  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  +  x
)  e.  CC )
2524adantl 275 . . . 4  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  +  x
)  e.  CC )
26 addass 7862 . . . . 5  |-  ( ( k  e.  CC  /\  x  e.  CC  /\  y  e.  CC )  ->  (
( k  +  x
)  +  y )  =  ( k  +  ( x  +  y ) ) )
2726adantl 275 . . . 4  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  ( k  e.  CC  /\  x  e.  CC  /\  y  e.  CC ) )  -> 
( ( k  +  x )  +  y )  =  ( k  +  ( x  +  y ) ) )
28 simpr 109 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  j  e.  ( ZZ>= `  ( N  +  1 ) ) )
294adantr 274 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  ( ZZ>= `  M )
)
303eleq2i 2224 . . . . . 6  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
3130, 12sylan2br 286 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3231adantlr 469 . . . 4  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3325, 27, 28, 29, 32seq3split 10378 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  +  ,  F ) `  j
)  =  ( (  seq M (  +  ,  F ) `  N )  +  (  seq ( N  + 
1 ) (  +  ,  F ) `  j ) ) )
3423, 22, 33comraddd 8032 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  +  ,  F ) `  j
)  =  ( (  seq ( N  + 
1 ) (  +  ,  F ) `  j )  +  (  seq M (  +  ,  F ) `  N ) ) )
351, 8, 9, 14, 16, 22, 34climaddc1 11226 1  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  ( A  +  (  seq M
(  +  ,  F
) `  N )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    = wceq 1335    e. wcel 2128   _Vcvv 2712   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   CCcc 7730   1c1 7733    + caddc 7735   ZZcz 9167   ZZ>=cuz 9439    seqcseq 10344    ~~> cli 11175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-rp 9561  df-fz 9913  df-seqfrec 10345  df-exp 10419  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-clim 11176
This theorem is referenced by:  iserex  11236
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