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Theorem clim2ser2 12145
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 2296 . 2  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
2 clim2ser.2 . . . . 5  |-  ( ph  ->  N  e.  Z )
3 clim2ser.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
42, 3syl6eleq 2386 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 peano2uz 10288 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
64, 5syl 15 . . 3  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )
7 eluzelz 10254 . . 3  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ZZ )
86, 7syl 15 . 2  |-  ( ph  ->  ( N  +  1 )  e.  ZZ )
9 clim2ser2.5 . 2  |-  ( ph  ->  seq  ( N  + 
1 ) (  +  ,  F )  ~~>  A )
10 eluzel2 10251 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
114, 10syl 15 . . . 4  |-  ( ph  ->  M  e.  ZZ )
12 clim2ser.4 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
133, 11, 12serf 11090 . . 3  |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> CC )
14 ffvelrn 5679 . . 3  |-  ( (  seq  M (  +  ,  F ) : Z --> CC  /\  N  e.  Z )  ->  (  seq  M (  +  ,  F ) `  N
)  e.  CC )
1513, 2, 14syl2anc 642 . 2  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 N )  e.  CC )
16 seqex 11064 . . 3  |-  seq  M
(  +  ,  F
)  e.  _V
1716a1i 10 . 2  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
_V )
186, 3syl6eleqr 2387 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  Z )
193uztrn2 10261 . . . . . 6  |-  ( ( ( N  +  1 )  e.  Z  /\  k  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
k  e.  Z )
2018, 19sylan 457 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  Z )
2120, 12syldan 456 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  k )  e.  CC )
221, 8, 21serf 11090 . . 3  |-  ( ph  ->  seq  ( N  + 
1 ) (  +  ,  F ) : ( ZZ>= `  ( N  +  1 ) ) --> CC )
23 ffvelrn 5679 . . 3  |-  ( (  seq  ( N  + 
1 ) (  +  ,  F ) : ( ZZ>= `  ( N  +  1 ) ) --> CC  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  ( N  +  1
) (  +  ,  F ) `  j
)  e.  CC )
2422, 23sylan 457 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  ( N  +  1
) (  +  ,  F ) `  j
)  e.  CC )
25 addcl 8835 . . . . 5  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  +  x
)  e.  CC )
2625adantl 452 . . . 4  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  +  x
)  e.  CC )
27 addass 8840 . . . . 5  |-  ( ( k  e.  CC  /\  x  e.  CC  /\  y  e.  CC )  ->  (
( k  +  x
)  +  y )  =  ( k  +  ( x  +  y ) ) )
2827adantl 452 . . . 4  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  ( k  e.  CC  /\  x  e.  CC  /\  y  e.  CC ) )  -> 
( ( k  +  x )  +  y )  =  ( k  +  ( x  +  y ) ) )
29 simpr 447 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  j  e.  ( ZZ>= `  ( N  +  1 ) ) )
304adantr 451 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  ( ZZ>= `  M )
)
31 elfzuz 10810 . . . . . . 7  |-  ( k  e.  ( M ... j )  ->  k  e.  ( ZZ>= `  M )
)
3231, 3syl6eleqr 2387 . . . . . 6  |-  ( k  e.  ( M ... j )  ->  k  e.  Z )
3332, 12sylan2 460 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... j ) )  ->  ( F `  k )  e.  CC )
3433adantlr 695 . . . 4  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  k  e.  ( M ... j
) )  ->  ( F `  k )  e.  CC )
3526, 28, 29, 30, 34seqsplit 11095 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  j )  =  ( (  seq 
M (  +  ,  F ) `  N
)  +  (  seq  ( N  +  1 ) (  +  ,  F ) `  j
) ) )
3615adantr 451 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  N )  e.  CC )
3736, 24addcomd 9030 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( (  seq  M (  +  ,  F ) `  N
)  +  (  seq  ( N  +  1 ) (  +  ,  F ) `  j
) )  =  ( (  seq  ( N  +  1 ) (  +  ,  F ) `
 j )  +  (  seq  M (  +  ,  F ) `
 N ) ) )
3835, 37eqtrd 2328 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  j )  =  ( (  seq  ( N  +  1 ) (  +  ,  F ) `  j
)  +  (  seq 
M (  +  ,  F ) `  N
) ) )
391, 8, 9, 15, 17, 24, 38climaddc1 12124 1  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  ( A  +  (  seq  M
(  +  ,  F
) `  N )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062    ~~> cli 11974
This theorem is referenced by:  iserex  12146  abelthlem6  19828  abelthlem9  19832  leibpi  20254
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-inf2 7358  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  ax-pre-sup 8831
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-rmo 2564  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-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978
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