ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  clim2ser2 Unicode version

Theorem clim2ser2 11301
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 2170 . 2  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
2 clim2ser.2 . . . . 5  |-  ( ph  ->  N  e.  Z )
3 clim2ser.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
42, 3eleqtrdi 2263 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 peano2uz 9542 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
64, 5syl 14 . . 3  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )
7 eluzelz 9496 . . 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 9492 . . . . 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 10430 . . 3  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> CC )
1413, 2ffvelrnd 5632 . 2  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  e.  CC )
15 seqex 10403 . . 3  |-  seq M
(  +  ,  F
)  e.  _V
1615a1i 9 . 2  |-  ( ph  ->  seq M (  +  ,  F )  e. 
_V )
176, 3eleqtrrdi 2264 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  Z )
183uztrn2 9504 . . . . . 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 10430 . . 3  |-  ( ph  ->  seq ( N  + 
1 ) (  +  ,  F ) : ( ZZ>= `  ( N  +  1 ) ) --> CC )
2221ffvelrnda 5631 . 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 7899 . . . . 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 7904 . . . . 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 2237 . . . . . 6  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
3130, 12sylan2br 286 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3231adantlr 474 . . . 4  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3325, 27, 28, 29, 32seq3split 10435 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  +  ,  F ) `  j
)  =  ( (  seq M (  +  ,  F ) `  N )  +  (  seq ( N  + 
1 ) (  +  ,  F ) `  j ) ) )
3423, 22, 33comraddd 8076 . 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 11292 1  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  ( A  +  (  seq M
(  +  ,  F
) `  N )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 973    = wceq 1348    e. wcel 2141   _Vcvv 2730   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   1c1 7775    + caddc 7777   ZZcz 9212   ZZ>=cuz 9487    seqcseq 10401    ~~> cli 11241
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-fz 9966  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242
This theorem is referenced by:  iserex  11302
  Copyright terms: Public domain W3C validator