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Theorem isumrpcl 12302
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 2373 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 10238 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 15 . . 3  |-  ( ph  ->  N  e.  ZZ )
7 uzss 10248 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  M ) )
84, 7syl 15 . . . . . 6  |-  ( ph  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
)
98, 1, 33sstr4g 3219 . . . . 5  |-  ( ph  ->  W  C_  Z )
109sselda 3180 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  k  e.  Z )
11 isumrpcl.4 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
1210, 11syldan 456 . . 3  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  =  A )
13 isumrpcl.5 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR+ )
1413rpred 10390 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR )
1510, 14syldan 456 . . 3  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  RR )
16 isumrpcl.6 . . . 4  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
1711, 13eqeltrd 2357 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR+ )
1817rpcnd 10392 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
193, 2, 18iserex 12130 . . . 4  |-  ( ph  ->  (  seq  M (  +  ,  F )  e.  dom  ~~>  <->  seq  N (  +  ,  F )  e.  dom  ~~>  ) )
2016, 19mpbid 201 . . 3  |-  ( ph  ->  seq  N (  +  ,  F )  e. 
dom 
~~>  )
211, 6, 12, 15, 20isumrecl 12228 . 2  |-  ( ph  -> 
sum_ k  e.  W  A  e.  RR )
2217ralrimiva 2626 . . 3  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  RR+ )
23 fveq2 5525 . . . . 5  |-  ( k  =  N  ->  ( F `  k )  =  ( F `  N ) )
2423eleq1d 2349 . . . 4  |-  ( k  =  N  ->  (
( F `  k
)  e.  RR+  <->  ( F `  N )  e.  RR+ ) )
2524rspcv 2880 . . 3  |-  ( N  e.  Z  ->  ( A. k  e.  Z  ( F `  k )  e.  RR+  ->  ( F `
 N )  e.  RR+ ) )
262, 22, 25sylc 56 . 2  |-  ( ph  ->  ( F `  N
)  e.  RR+ )
27 seq1 11059 . . . 4  |-  ( N  e.  ZZ  ->  (  seq  N (  +  ,  F ) `  N
)  =  ( F `
 N ) )
286, 27syl 15 . . 3  |-  ( ph  ->  (  seq  N (  +  ,  F ) `
 N )  =  ( F `  N
) )
29 uzid 10242 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
306, 29syl 15 . . . . 5  |-  ( ph  ->  N  e.  ( ZZ>= `  N ) )
3130, 1syl6eleqr 2374 . . . 4  |-  ( ph  ->  N  e.  W )
3215recnd 8861 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  CC )
331, 6, 12, 32, 20isumclim2 12221 . . . 4  |-  ( ph  ->  seq  N (  +  ,  F )  ~~>  sum_ k  e.  W  A )
349sseld 3179 . . . . . . 7  |-  ( ph  ->  ( m  e.  W  ->  m  e.  Z ) )
35 fveq2 5525 . . . . . . . . . 10  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
3635eleq1d 2349 . . . . . . . . 9  |-  ( k  =  m  ->  (
( F `  k
)  e.  RR+  <->  ( F `  m )  e.  RR+ ) )
3736rspcv 2880 . . . . . . . 8  |-  ( m  e.  Z  ->  ( A. k  e.  Z  ( F `  k )  e.  RR+  ->  ( F `
 m )  e.  RR+ ) )
3822, 37syl5com 26 . . . . . . 7  |-  ( ph  ->  ( m  e.  Z  ->  ( F `  m
)  e.  RR+ )
)
3934, 38syld 40 . . . . . 6  |-  ( ph  ->  ( m  e.  W  ->  ( F `  m
)  e.  RR+ )
)
4039imp 418 . . . . 5  |-  ( (
ph  /\  m  e.  W )  ->  ( F `  m )  e.  RR+ )
4140rpred 10390 . . . 4  |-  ( (
ph  /\  m  e.  W )  ->  ( F `  m )  e.  RR )
4240rpge0d 10394 . . . 4  |-  ( (
ph  /\  m  e.  W )  ->  0  <_  ( F `  m
) )
431, 31, 33, 41, 42climserle 12136 . . 3  |-  ( ph  ->  (  seq  N (  +  ,  F ) `
 N )  <_  sum_ k  e.  W  A
)
4428, 43eqbrtrrd 4045 . 2  |-  ( ph  ->  ( F `  N
)  <_  sum_ k  e.  W  A )
4521, 26, 44rpgecld 10425 1  |-  ( ph  -> 
sum_ k  e.  W  A  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   dom cdm 4689   ` cfv 5255   RRcr 8736    + caddc 8740    <_ cle 8868   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354    seq cseq 11046    ~~> cli 11958   sum_csu 12158
This theorem is referenced by:  effsumlt  12391  eirrlem  12482  aaliou3lem3  19724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159
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