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Theorem climserle 11006
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 2207 . . . . 5  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzel2 9233 . . . . 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 10141 . . 3  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> RR )
98ffvelrnda 5509 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  e.  RR )
101peano2uzs 9281 . . . . 5  |-  ( j  e.  Z  ->  (
j  +  1 )  e.  Z )
11 fveq2 5375 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  ( F `  k )  =  ( F `  ( j  +  1 ) ) )
1211breq2d 3907 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
0  <_  ( F `  k )  <->  0  <_  ( F `  ( j  +  1 ) ) ) )
1312imbi2d 229 . . . . . . 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 115 . . . . . . 7  |-  ( k  e.  Z  ->  ( ph  ->  0  <_  ( F `  k )
) )
1613, 15vtoclga 2723 . . . . . 6  |-  ( ( j  +  1 )  e.  Z  ->  ( ph  ->  0  <_  ( F `  ( j  +  1 ) ) ) )
1716impcom 124 . . . . 5  |-  ( (
ph  /\  ( j  +  1 )  e.  Z )  ->  0  <_  ( F `  (
j  +  1 ) ) )
1810, 17sylan2 282 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  0  <_  ( F `  (
j  +  1 ) ) )
1911eleq1d 2183 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  (
( F `  k
)  e.  RR  <->  ( F `  ( j  +  1 ) )  e.  RR ) )
2019imbi2d 229 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
( ph  ->  ( F `
 k )  e.  RR )  <->  ( ph  ->  ( F `  (
j  +  1 ) )  e.  RR ) ) )
217expcom 115 . . . . . . . 8  |-  ( k  e.  Z  ->  ( ph  ->  ( F `  k )  e.  RR ) )
2220, 21vtoclga 2723 . . . . . . 7  |-  ( ( j  +  1 )  e.  Z  ->  ( ph  ->  ( F `  ( j  +  1 ) )  e.  RR ) )
2322impcom 124 . . . . . 6  |-  ( (
ph  /\  ( j  +  1 )  e.  Z )  ->  ( F `  ( j  +  1 ) )  e.  RR )
2410, 23sylan2 282 . . . . 5  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  ( j  +  1 ) )  e.  RR )
259, 24addge01d 8213 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  (
0  <_  ( F `  ( j  +  1 ) )  <->  (  seq M (  +  ,  F ) `  j
)  <_  ( (  seq M (  +  ,  F ) `  j
)  +  ( F `
 ( j  +  1 ) ) ) ) )
2618, 25mpbid 146 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  <_  ( (  seq M (  +  ,  F ) `  j
)  +  ( F `
 ( j  +  1 ) ) ) )
271eleq2i 2181 . . . . . 6  |-  ( j  e.  Z  <->  j  e.  ( ZZ>= `  M )
)
2827biimpi 119 . . . . 5  |-  ( j  e.  Z  ->  j  e.  ( ZZ>= `  M )
)
2928adantl 273 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
30 simpll 501 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ph )
311eleq2i 2181 . . . . . . 7  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
3231biimpri 132 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  Z )
3332adantl 273 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  Z )
3430, 33, 7syl2anc 406 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  RR )
35 readdcl 7670 . . . . 5  |-  ( ( k  e.  RR  /\  v  e.  RR )  ->  ( k  +  v )  e.  RR )
3635adantl 273 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  (
k  e.  RR  /\  v  e.  RR )
)  ->  ( k  +  v )  e.  RR )
3729, 34, 36seq3p1 10128 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  (
j  +  1 ) )  =  ( (  seq M (  +  ,  F ) `  j )  +  ( F `  ( j  +  1 ) ) ) )
3826, 37breqtrrd 3921 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  <_  (  seq M (  +  ,  F ) `  (
j  +  1 ) ) )
391, 2, 3, 9, 38climub 11005 1  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  <_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   RRcr 7546   0cc0 7547   1c1 7548    + caddc 7550    <_ cle 7725   ZZcz 8958   ZZ>=cuz 9228    seqcseq 10111    ~~> cli 10939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664  ax-caucvg 7665
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-3 8690  df-4 8691  df-n0 8882  df-z 8959  df-uz 9229  df-rp 9344  df-fz 9684  df-seqfrec 10112  df-exp 10186  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663  df-clim 10940
This theorem is referenced by:  isumrpcl  11155  ege2le3  11228
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