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Theorem isumrpcl 12613
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 variable  m is 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 2525 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 10486 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 16 . . 3  |-  ( ph  ->  N  e.  ZZ )
7 uzss 10496 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  M ) )
84, 7syl 16 . . . . . 6  |-  ( ph  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
)
98, 1, 33sstr4g 3381 . . . . 5  |-  ( ph  ->  W  C_  Z )
109sselda 3340 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  k  e.  Z )
11 isumrpcl.4 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
1210, 11syldan 457 . . 3  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  =  A )
13 isumrpcl.5 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR+ )
1413rpred 10638 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR )
1510, 14syldan 457 . . 3  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  RR )
16 isumrpcl.6 . . . 4  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
1711, 13eqeltrd 2509 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR+ )
1817rpcnd 10640 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
193, 2, 18iserex 12440 . . . 4  |-  ( ph  ->  (  seq  M (  +  ,  F )  e.  dom  ~~>  <->  seq  N (  +  ,  F )  e.  dom  ~~>  ) )
2016, 19mpbid 202 . . 3  |-  ( ph  ->  seq  N (  +  ,  F )  e. 
dom 
~~>  )
211, 6, 12, 15, 20isumrecl 12539 . 2  |-  ( ph  -> 
sum_ k  e.  W  A  e.  RR )
2217ralrimiva 2781 . . 3  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  RR+ )
23 fveq2 5720 . . . . 5  |-  ( k  =  N  ->  ( F `  k )  =  ( F `  N ) )
2423eleq1d 2501 . . . 4  |-  ( k  =  N  ->  (
( F `  k
)  e.  RR+  <->  ( F `  N )  e.  RR+ ) )
2524rspcv 3040 . . 3  |-  ( N  e.  Z  ->  ( A. k  e.  Z  ( F `  k )  e.  RR+  ->  ( F `
 N )  e.  RR+ ) )
262, 22, 25sylc 58 . 2  |-  ( ph  ->  ( F `  N
)  e.  RR+ )
27 seq1 11326 . . . 4  |-  ( N  e.  ZZ  ->  (  seq  N (  +  ,  F ) `  N
)  =  ( F `
 N ) )
286, 27syl 16 . . 3  |-  ( ph  ->  (  seq  N (  +  ,  F ) `
 N )  =  ( F `  N
) )
29 uzid 10490 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
306, 29syl 16 . . . . 5  |-  ( ph  ->  N  e.  ( ZZ>= `  N ) )
3130, 1syl6eleqr 2526 . . . 4  |-  ( ph  ->  N  e.  W )
3215recnd 9104 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  CC )
331, 6, 12, 32, 20isumclim2 12532 . . . 4  |-  ( ph  ->  seq  N (  +  ,  F )  ~~>  sum_ k  e.  W  A )
349sseld 3339 . . . . . . 7  |-  ( ph  ->  ( m  e.  W  ->  m  e.  Z ) )
35 fveq2 5720 . . . . . . . . . 10  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
3635eleq1d 2501 . . . . . . . . 9  |-  ( k  =  m  ->  (
( F `  k
)  e.  RR+  <->  ( F `  m )  e.  RR+ ) )
3736rspcv 3040 . . . . . . . 8  |-  ( m  e.  Z  ->  ( A. k  e.  Z  ( F `  k )  e.  RR+  ->  ( F `
 m )  e.  RR+ ) )
3822, 37syl5com 28 . . . . . . 7  |-  ( ph  ->  ( m  e.  Z  ->  ( F `  m
)  e.  RR+ )
)
3934, 38syld 42 . . . . . 6  |-  ( ph  ->  ( m  e.  W  ->  ( F `  m
)  e.  RR+ )
)
4039imp 419 . . . . 5  |-  ( (
ph  /\  m  e.  W )  ->  ( F `  m )  e.  RR+ )
4140rpred 10638 . . . 4  |-  ( (
ph  /\  m  e.  W )  ->  ( F `  m )  e.  RR )
4240rpge0d 10642 . . . 4  |-  ( (
ph  /\  m  e.  W )  ->  0  <_  ( F `  m
) )
431, 31, 33, 41, 42climserle 12446 . . 3  |-  ( ph  ->  (  seq  N (  +  ,  F ) `
 N )  <_  sum_ k  e.  W  A
)
4428, 43eqbrtrrd 4226 . 2  |-  ( ph  ->  ( F `  N
)  <_  sum_ k  e.  W  A )
4521, 26, 44rpgecld 10673 1  |-  ( ph  -> 
sum_ k  e.  W  A  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   dom cdm 4870   ` cfv 5446   RRcr 8979    + caddc 8983    <_ cle 9111   ZZcz 10272   ZZ>=cuz 10478   RR+crp 10602    seq cseq 11313    ~~> cli 12268   sum_csu 12469
This theorem is referenced by:  effsumlt  12702  eirrlem  12793  aaliou3lem3  20251
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-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
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-int 4043  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-se 4534  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-isom 5455  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-1o 6716  df-oadd 6720  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7469  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-fz 11034  df-fzo 11126  df-fl 11192  df-seq 11314  df-exp 11373  df-hash 11609  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-clim 12272  df-rlim 12273  df-sum 12470
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