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Theorem isumrpcl 11214
Description: The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)
Hypotheses
Ref Expression
isumrpcl.1  |-  Z  =  ( ZZ>= `  M )
isumrpcl.2  |-  W  =  ( ZZ>= `  N )
isumrpcl.3  |-  ( ph  ->  N  e.  Z )
isumrpcl.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
isumrpcl.5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR+ )
isumrpcl.6  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
Assertion
Ref Expression
isumrpcl  |-  ( ph  -> 
sum_ k  e.  W  A  e.  RR+ )
Distinct variable groups:    k, F    k, M    k, N    ph, k    k, W    k, Z
Allowed substitution hint:    A( k)

Proof of Theorem isumrpcl
Dummy variables  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isumrpcl.2 . . 3  |-  W  =  ( ZZ>= `  N )
2 isumrpcl.3 . . . . 5  |-  ( ph  ->  N  e.  Z )
3 isumrpcl.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
42, 3syl6eleq 2208 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 9287 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 14 . . 3  |-  ( ph  ->  N  e.  ZZ )
7 uzss 9298 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  M ) )
84, 7syl 14 . . . . . 6  |-  ( ph  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
)
98, 1, 33sstr4g 3108 . . . . 5  |-  ( ph  ->  W  C_  Z )
109sselda 3065 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  k  e.  Z )
11 isumrpcl.4 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
1210, 11syldan 278 . . 3  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  =  A )
13 isumrpcl.5 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR+ )
1413rpred 9434 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR )
1510, 14syldan 278 . . 3  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  RR )
16 isumrpcl.6 . . . 4  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
1711, 13eqeltrd 2192 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR+ )
1817rpcnd 9436 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
193, 2, 18iserex 11059 . . . 4  |-  ( ph  ->  (  seq M (  +  ,  F )  e.  dom  ~~>  <->  seq N (  +  ,  F )  e.  dom  ~~>  ) )
2016, 19mpbid 146 . . 3  |-  ( ph  ->  seq N (  +  ,  F )  e. 
dom 
~~>  )
211, 6, 12, 15, 20isumrecl 11149 . 2  |-  ( ph  -> 
sum_ k  e.  W  A  e.  RR )
22 fveq2 5387 . . . 4  |-  ( k  =  N  ->  ( F `  k )  =  ( F `  N ) )
2322eleq1d 2184 . . 3  |-  ( k  =  N  ->  (
( F `  k
)  e.  RR+  <->  ( F `  N )  e.  RR+ ) )
2417ralrimiva 2480 . . 3  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  RR+ )
2523, 24, 2rspcdva 2766 . 2  |-  ( ph  ->  ( F `  N
)  e.  RR+ )
268sselda 3065 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  k  e.  ( ZZ>= `  M )
)
2726, 3syl6eleqr 2209 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  k  e.  Z )
2827, 17syldan 278 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  ( F `  k )  e.  RR+ )
29 rpaddcl 9416 . . . . 5  |-  ( ( k  e.  RR+  /\  y  e.  RR+ )  ->  (
k  +  y )  e.  RR+ )
3029adantl 273 . . . 4  |-  ( (
ph  /\  ( k  e.  RR+  /\  y  e.  RR+ ) )  ->  (
k  +  y )  e.  RR+ )
316, 28, 30seq3-1 10184 . . 3  |-  ( ph  ->  (  seq N (  +  ,  F ) `
 N )  =  ( F `  N
) )
32 uzid 9292 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
336, 32syl 14 . . . . 5  |-  ( ph  ->  N  e.  ( ZZ>= `  N ) )
3433, 1syl6eleqr 2209 . . . 4  |-  ( ph  ->  N  e.  W )
3515recnd 7758 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  CC )
361, 6, 12, 35, 20isumclim2 11142 . . . 4  |-  ( ph  ->  seq N (  +  ,  F )  ~~>  sum_ k  e.  W  A )
379sseld 3064 . . . . . . 7  |-  ( ph  ->  ( m  e.  W  ->  m  e.  Z ) )
38 fveq2 5387 . . . . . . . . 9  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
3938eleq1d 2184 . . . . . . . 8  |-  ( k  =  m  ->  (
( F `  k
)  e.  RR+  <->  ( F `  m )  e.  RR+ ) )
4039rspcv 2757 . . . . . . 7  |-  ( m  e.  Z  ->  ( A. k  e.  Z  ( F `  k )  e.  RR+  ->  ( F `
 m )  e.  RR+ ) )
4137, 24, 40syl6ci 1404 . . . . . 6  |-  ( ph  ->  ( m  e.  W  ->  ( F `  m
)  e.  RR+ )
)
4241imp 123 . . . . 5  |-  ( (
ph  /\  m  e.  W )  ->  ( F `  m )  e.  RR+ )
4342rpred 9434 . . . 4  |-  ( (
ph  /\  m  e.  W )  ->  ( F `  m )  e.  RR )
4442rpge0d 9438 . . . 4  |-  ( (
ph  /\  m  e.  W )  ->  0  <_  ( F `  m
) )
451, 34, 36, 43, 44climserle 11065 . . 3  |-  ( ph  ->  (  seq N (  +  ,  F ) `
 N )  <_  sum_ k  e.  W  A
)
4631, 45eqbrtrrd 3920 . 2  |-  ( ph  ->  ( F `  N
)  <_  sum_ k  e.  W  A )
4721, 25, 46rpgecld 9474 1  |-  ( ph  -> 
sum_ k  e.  W  A  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   A.wral 2391    C_ wss 3039   dom cdm 4507   ` cfv 5091  (class class class)co 5740   RRcr 7583    + caddc 7587    <_ cle 7765   ZZcz 9008   ZZ>=cuz 9278   RR+crp 9393    seqcseq 10169    ~~> cli 10998   sum_csu 11073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 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 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
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 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-fz 9742  df-fzo 9871  df-seqfrec 10170  df-exp 10244  df-ihash 10473  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-clim 10999  df-sumdc 11074
This theorem is referenced by:  effsumlt  11308  eirraplem  11390
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