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Theorem climserle 13239
Description: The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
Hypotheses
Ref Expression
clim2ser.1  |-  Z  =  ( ZZ>= `  M )
climserle.2  |-  ( ph  ->  N  e.  Z )
climserle.3  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  A )
climserle.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
climserle.5  |-  ( (
ph  /\  k  e.  Z )  ->  0  <_  ( F `  k
) )
Assertion
Ref Expression
climserle  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  <_  A )
Distinct variable groups:    A, k    k, F    k, M    k, N    ph, k    k, Z

Proof of Theorem climserle
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 clim2ser.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 climserle.2 . 2  |-  ( ph  ->  N  e.  Z )
3 climserle.3 . 2  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  A )
42, 1syl6eleq 2547 . . . . 5  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzel2 10964 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
64, 5syl 16 . . . 4  |-  ( ph  ->  M  e.  ZZ )
7 climserle.4 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
81, 6, 7serfre 11933 . . 3  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> RR )
98ffvelrnda 5939 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  e.  RR )
101peano2uzs 11007 . . . . 5  |-  ( j  e.  Z  ->  (
j  +  1 )  e.  Z )
11 fveq2 5786 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  ( F `  k )  =  ( F `  ( j  +  1 ) ) )
1211breq2d 4399 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
0  <_  ( F `  k )  <->  0  <_  ( F `  ( j  +  1 ) ) ) )
1312imbi2d 316 . . . . . . 7  |-  ( k  =  ( j  +  1 )  ->  (
( ph  ->  0  <_ 
( F `  k
) )  <->  ( ph  ->  0  <_  ( F `  ( j  +  1 ) ) ) ) )
14 climserle.5 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  0  <_  ( F `  k
) )
1514expcom 435 . . . . . . 7  |-  ( k  e.  Z  ->  ( ph  ->  0  <_  ( F `  k )
) )
1613, 15vtoclga 3129 . . . . . 6  |-  ( ( j  +  1 )  e.  Z  ->  ( ph  ->  0  <_  ( F `  ( j  +  1 ) ) ) )
1716impcom 430 . . . . 5  |-  ( (
ph  /\  ( j  +  1 )  e.  Z )  ->  0  <_  ( F `  (
j  +  1 ) ) )
1810, 17sylan2 474 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  0  <_  ( F `  (
j  +  1 ) ) )
1911eleq1d 2519 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  (
( F `  k
)  e.  RR  <->  ( F `  ( j  +  1 ) )  e.  RR ) )
2019imbi2d 316 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
( ph  ->  ( F `
 k )  e.  RR )  <->  ( ph  ->  ( F `  (
j  +  1 ) )  e.  RR ) ) )
217expcom 435 . . . . . . . 8  |-  ( k  e.  Z  ->  ( ph  ->  ( F `  k )  e.  RR ) )
2220, 21vtoclga 3129 . . . . . . 7  |-  ( ( j  +  1 )  e.  Z  ->  ( ph  ->  ( F `  ( j  +  1 ) )  e.  RR ) )
2322impcom 430 . . . . . 6  |-  ( (
ph  /\  ( j  +  1 )  e.  Z )  ->  ( F `  ( j  +  1 ) )  e.  RR )
2410, 23sylan2 474 . . . . 5  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  ( j  +  1 ) )  e.  RR )
259, 24addge01d 10025 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  (
0  <_  ( F `  ( j  +  1 ) )  <->  (  seq M (  +  ,  F ) `  j
)  <_  ( (  seq M (  +  ,  F ) `  j
)  +  ( F `
 ( j  +  1 ) ) ) ) )
2618, 25mpbid 210 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  <_  ( (  seq M (  +  ,  F ) `  j
)  +  ( F `
 ( j  +  1 ) ) ) )
27 simpr 461 . . . . 5  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
2827, 1syl6eleq 2547 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
29 seqp1 11919 . . . 4  |-  ( j  e.  ( ZZ>= `  M
)  ->  (  seq M (  +  ,  F ) `  (
j  +  1 ) )  =  ( (  seq M (  +  ,  F ) `  j )  +  ( F `  ( j  +  1 ) ) ) )
3028, 29syl 16 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  (
j  +  1 ) )  =  ( (  seq M (  +  ,  F ) `  j )  +  ( F `  ( j  +  1 ) ) ) )
3126, 30breqtrrd 4413 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  <_  (  seq M (  +  ,  F ) `  (
j  +  1 ) ) )
321, 2, 3, 9, 31climub 13238 1  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  <_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4387   ` cfv 5513  (class class class)co 6187   RRcr 9379   0cc0 9380   1c1 9381    + caddc 9383    <_ cle 9517   ZZcz 10744   ZZ>=cuz 10959    seqcseq 11904    ~~> cli 13061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457  ax-pre-sup 9458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-er 7198  df-pm 7314  df-en 7408  df-dom 7409  df-sdom 7410  df-sup 7789  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-div 10092  df-nn 10421  df-2 10478  df-3 10479  df-n0 10678  df-z 10745  df-uz 10960  df-rp 11090  df-fz 11536  df-fl 11740  df-seq 11905  df-exp 11964  df-cj 12687  df-re 12688  df-im 12689  df-sqr 12823  df-abs 12824  df-clim 13065  df-rlim 13066
This theorem is referenced by:  isumrpcl  13405  ege2le3  13474  prmreclem6  14081  ioombl1lem4  21155  rge0scvg  26510
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