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Theorem climserle 11385
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
clim2iser.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 variables  j  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clim2iser.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 climserle.2 . 2  |-  ( ph  ->  N  e.  Z )
3 climserle.3 . 2  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  A )
42, 1eleqtrdi 2282 . . . . 5  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzel2 9563 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
64, 5syl 14 . . . 4  |-  ( ph  ->  M  e.  ZZ )
7 climserle.4 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
81, 6, 7serfre 10506 . . 3  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> RR )
98ffvelcdmda 5672 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  e.  RR )
101peano2uzs 9614 . . . . 5  |-  ( j  e.  Z  ->  (
j  +  1 )  e.  Z )
11 fveq2 5534 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  ( F `  k )  =  ( F `  ( j  +  1 ) ) )
1211breq2d 4030 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
0  <_  ( F `  k )  <->  0  <_  ( F `  ( j  +  1 ) ) ) )
1312imbi2d 230 . . . . . . 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 116 . . . . . . 7  |-  ( k  e.  Z  ->  ( ph  ->  0  <_  ( F `  k )
) )
1613, 15vtoclga 2818 . . . . . 6  |-  ( ( j  +  1 )  e.  Z  ->  ( ph  ->  0  <_  ( F `  ( j  +  1 ) ) ) )
1716impcom 125 . . . . 5  |-  ( (
ph  /\  ( j  +  1 )  e.  Z )  ->  0  <_  ( F `  (
j  +  1 ) ) )
1810, 17sylan2 286 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  0  <_  ( F `  (
j  +  1 ) ) )
1911eleq1d 2258 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  (
( F `  k
)  e.  RR  <->  ( F `  ( j  +  1 ) )  e.  RR ) )
2019imbi2d 230 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
( ph  ->  ( F `
 k )  e.  RR )  <->  ( ph  ->  ( F `  (
j  +  1 ) )  e.  RR ) ) )
217expcom 116 . . . . . . . 8  |-  ( k  e.  Z  ->  ( ph  ->  ( F `  k )  e.  RR ) )
2220, 21vtoclga 2818 . . . . . . 7  |-  ( ( j  +  1 )  e.  Z  ->  ( ph  ->  ( F `  ( j  +  1 ) )  e.  RR ) )
2322impcom 125 . . . . . 6  |-  ( (
ph  /\  ( j  +  1 )  e.  Z )  ->  ( F `  ( j  +  1 ) )  e.  RR )
2410, 23sylan2 286 . . . . 5  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  ( j  +  1 ) )  e.  RR )
259, 24addge01d 8520 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  (
0  <_  ( F `  ( j  +  1 ) )  <->  (  seq M (  +  ,  F ) `  j
)  <_  ( (  seq M (  +  ,  F ) `  j
)  +  ( F `
 ( j  +  1 ) ) ) ) )
2618, 25mpbid 147 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  <_  ( (  seq M (  +  ,  F ) `  j
)  +  ( F `
 ( j  +  1 ) ) ) )
271eleq2i 2256 . . . . . 6  |-  ( j  e.  Z  <->  j  e.  ( ZZ>= `  M )
)
2827biimpi 120 . . . . 5  |-  ( j  e.  Z  ->  j  e.  ( ZZ>= `  M )
)
2928adantl 277 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
30 simpll 527 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ph )
311eleq2i 2256 . . . . . . 7  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
3231biimpri 133 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  Z )
3332adantl 277 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  Z )
3430, 33, 7syl2anc 411 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  RR )
35 readdcl 7967 . . . . 5  |-  ( ( k  e.  RR  /\  v  e.  RR )  ->  ( k  +  v )  e.  RR )
3635adantl 277 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  (
k  e.  RR  /\  v  e.  RR )
)  ->  ( k  +  v )  e.  RR )
3729, 34, 36seq3p1 10493 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  (
j  +  1 ) )  =  ( (  seq M (  +  ,  F ) `  j )  +  ( F `  ( j  +  1 ) ) ) )
3826, 37breqtrrd 4046 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  <_  (  seq M (  +  ,  F ) `  (
j  +  1 ) ) )
391, 2, 3, 9, 38climub 11384 1  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  <_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   class class class wbr 4018   ` cfv 5235  (class class class)co 5896   RRcr 7840   0cc0 7841   1c1 7842    + caddc 7844    <_ cle 8023   ZZcz 9283   ZZ>=cuz 9558    seqcseq 10476    ~~> cli 11318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960  ax-caucvg 7961
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-rp 9684  df-fz 10039  df-seqfrec 10477  df-exp 10551  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040  df-clim 11319
This theorem is referenced by:  isumrpcl  11534  ege2le3  11711
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