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Theorem climserle 12448
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 2525 . . . . 5  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzel2 10485 . . . . 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 11344 . . 3  |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> RR )
98ffvelrnda 5862 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  ,  F ) `  j
)  e.  RR )
101peano2uzs 10523 . . . . 5  |-  ( j  e.  Z  ->  (
j  +  1 )  e.  Z )
11 fveq2 5720 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  ( F `  k )  =  ( F `  ( j  +  1 ) ) )
1211breq2d 4216 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
0  <_  ( F `  k )  <->  0  <_  ( F `  ( j  +  1 ) ) ) )
1312imbi2d 308 . . . . . . 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 425 . . . . . . 7  |-  ( k  e.  Z  ->  ( ph  ->  0  <_  ( F `  k )
) )
1613, 15vtoclga 3009 . . . . . 6  |-  ( ( j  +  1 )  e.  Z  ->  ( ph  ->  0  <_  ( F `  ( j  +  1 ) ) ) )
1716impcom 420 . . . . 5  |-  ( (
ph  /\  ( j  +  1 )  e.  Z )  ->  0  <_  ( F `  (
j  +  1 ) ) )
1810, 17sylan2 461 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  0  <_  ( F `  (
j  +  1 ) ) )
1911eleq1d 2501 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  (
( F `  k
)  e.  RR  <->  ( F `  ( j  +  1 ) )  e.  RR ) )
2019imbi2d 308 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
( ph  ->  ( F `
 k )  e.  RR )  <->  ( ph  ->  ( F `  (
j  +  1 ) )  e.  RR ) ) )
217expcom 425 . . . . . . . 8  |-  ( k  e.  Z  ->  ( ph  ->  ( F `  k )  e.  RR ) )
2220, 21vtoclga 3009 . . . . . . 7  |-  ( ( j  +  1 )  e.  Z  ->  ( ph  ->  ( F `  ( j  +  1 ) )  e.  RR ) )
2322impcom 420 . . . . . 6  |-  ( (
ph  /\  ( j  +  1 )  e.  Z )  ->  ( F `  ( j  +  1 ) )  e.  RR )
2410, 23sylan2 461 . . . . 5  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  ( j  +  1 ) )  e.  RR )
259, 24addge01d 9606 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  (
0  <_  ( F `  ( j  +  1 ) )  <->  (  seq  M (  +  ,  F
) `  j )  <_  ( (  seq  M
(  +  ,  F
) `  j )  +  ( F `  ( j  +  1 ) ) ) ) )
2618, 25mpbid 202 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  ,  F ) `  j
)  <_  ( (  seq  M (  +  ,  F ) `  j
)  +  ( F `
 ( j  +  1 ) ) ) )
27 simpr 448 . . . . 5  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
2827, 1syl6eleq 2525 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
29 seqp1 11330 . . . 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 4230 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  ,  F ) `  j
)  <_  (  seq  M (  +  ,  F
) `  ( j  +  1 ) ) )
321, 2, 3, 9, 31climub 12447 1  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 N )  <_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    <_ cle 9113   ZZcz 10274   ZZ>=cuz 10480    seq cseq 11315    ~~> cli 12270
This theorem is referenced by:  isumrpcl  12615  ege2le3  12684  prmreclem6  13281  ioombl1lem4  19447  rge0scvg  24327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fl 11194  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275
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