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Theorem climserle 10734
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, 1syl6eleq 2180 . . . . 5  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzel2 9024 . . . . 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 9901 . . 3  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> RR )
98ffvelrnda 5434 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  e.  RR )
101peano2uzs 9072 . . . . 5  |-  ( j  e.  Z  ->  (
j  +  1 )  e.  Z )
11 fveq2 5305 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  ( F `  k )  =  ( F `  ( j  +  1 ) ) )
1211breq2d 3857 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
0  <_  ( F `  k )  <->  0  <_  ( F `  ( j  +  1 ) ) ) )
1312imbi2d 228 . . . . . . 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 114 . . . . . . 7  |-  ( k  e.  Z  ->  ( ph  ->  0  <_  ( F `  k )
) )
1613, 15vtoclga 2685 . . . . . 6  |-  ( ( j  +  1 )  e.  Z  ->  ( ph  ->  0  <_  ( F `  ( j  +  1 ) ) ) )
1716impcom 123 . . . . 5  |-  ( (
ph  /\  ( j  +  1 )  e.  Z )  ->  0  <_  ( F `  (
j  +  1 ) ) )
1810, 17sylan2 280 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  0  <_  ( F `  (
j  +  1 ) ) )
1911eleq1d 2156 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  (
( F `  k
)  e.  RR  <->  ( F `  ( j  +  1 ) )  e.  RR ) )
2019imbi2d 228 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
( ph  ->  ( F `
 k )  e.  RR )  <->  ( ph  ->  ( F `  (
j  +  1 ) )  e.  RR ) ) )
217expcom 114 . . . . . . . 8  |-  ( k  e.  Z  ->  ( ph  ->  ( F `  k )  e.  RR ) )
2220, 21vtoclga 2685 . . . . . . 7  |-  ( ( j  +  1 )  e.  Z  ->  ( ph  ->  ( F `  ( j  +  1 ) )  e.  RR ) )
2322impcom 123 . . . . . 6  |-  ( (
ph  /\  ( j  +  1 )  e.  Z )  ->  ( F `  ( j  +  1 ) )  e.  RR )
2410, 23sylan2 280 . . . . 5  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  ( j  +  1 ) )  e.  RR )
259, 24addge01d 8010 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  (
0  <_  ( F `  ( j  +  1 ) )  <->  (  seq M (  +  ,  F ) `  j
)  <_  ( (  seq M (  +  ,  F ) `  j
)  +  ( F `
 ( j  +  1 ) ) ) ) )
2618, 25mpbid 145 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  <_  ( (  seq M (  +  ,  F ) `  j
)  +  ( F `
 ( j  +  1 ) ) ) )
271eleq2i 2154 . . . . . 6  |-  ( j  e.  Z  <->  j  e.  ( ZZ>= `  M )
)
2827biimpi 118 . . . . 5  |-  ( j  e.  Z  ->  j  e.  ( ZZ>= `  M )
)
2928adantl 271 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
30 simpll 496 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ph )
311eleq2i 2154 . . . . . . 7  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
3231biimpri 131 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  Z )
3332adantl 271 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  Z )
3430, 33, 7syl2anc 403 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  RR )
35 readdcl 7468 . . . . 5  |-  ( ( k  e.  RR  /\  v  e.  RR )  ->  ( k  +  v )  e.  RR )
3635adantl 271 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  (
k  e.  RR  /\  v  e.  RR )
)  ->  ( k  +  v )  e.  RR )
3729, 34, 36seq3p1 9884 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  (
j  +  1 ) )  =  ( (  seq M (  +  ,  F ) `  j )  +  ( F `  ( j  +  1 ) ) ) )
3826, 37breqtrrd 3871 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  <_  (  seq M (  +  ,  F ) `  (
j  +  1 ) ) )
391, 2, 3, 9, 38climub 10733 1  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  <_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   RRcr 7349   0cc0 7350   1c1 7351    + caddc 7353    <_ cle 7523   ZZcz 8750   ZZ>=cuz 9019    seqcseq 9852    ~~> cli 10666
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-rp 9135  df-fz 9425  df-iseq 9853  df-seq3 9854  df-exp 9955  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-clim 10667
This theorem is referenced by:  isumrpcl  10888  ege2le3  10961
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