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Theorem isumrpcl 13629
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 2539 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 11094 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 16 . . 3  |-  ( ph  ->  N  e.  ZZ )
7 uzss 11105 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  M ) )
84, 7syl 16 . . . . . 6  |-  ( ph  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
)
98, 1, 33sstr4g 3527 . . . . 5  |-  ( ph  ->  W  C_  Z )
109sselda 3486 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  k  e.  Z )
11 isumrpcl.4 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
1210, 11syldan 470 . . 3  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  =  A )
13 isumrpcl.5 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR+ )
1413rpred 11260 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR )
1510, 14syldan 470 . . 3  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  RR )
16 isumrpcl.6 . . . 4  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
1711, 13eqeltrd 2529 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR+ )
1817rpcnd 11262 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
193, 2, 18iserex 13453 . . . 4  |-  ( ph  ->  (  seq M (  +  ,  F )  e.  dom  ~~>  <->  seq N (  +  ,  F )  e.  dom  ~~>  ) )
2016, 19mpbid 210 . . 3  |-  ( ph  ->  seq N (  +  ,  F )  e. 
dom 
~~>  )
211, 6, 12, 15, 20isumrecl 13554 . 2  |-  ( ph  -> 
sum_ k  e.  W  A  e.  RR )
2217ralrimiva 2855 . . 3  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  RR+ )
23 fveq2 5852 . . . . 5  |-  ( k  =  N  ->  ( F `  k )  =  ( F `  N ) )
2423eleq1d 2510 . . . 4  |-  ( k  =  N  ->  (
( F `  k
)  e.  RR+  <->  ( F `  N )  e.  RR+ ) )
2524rspcv 3190 . . 3  |-  ( N  e.  Z  ->  ( A. k  e.  Z  ( F `  k )  e.  RR+  ->  ( F `
 N )  e.  RR+ ) )
262, 22, 25sylc 60 . 2  |-  ( ph  ->  ( F `  N
)  e.  RR+ )
27 seq1 12094 . . . 4  |-  ( N  e.  ZZ  ->  (  seq N (  +  ,  F ) `  N
)  =  ( F `
 N ) )
286, 27syl 16 . . 3  |-  ( ph  ->  (  seq N (  +  ,  F ) `
 N )  =  ( F `  N
) )
29 uzid 11099 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
306, 29syl 16 . . . . 5  |-  ( ph  ->  N  e.  ( ZZ>= `  N ) )
3130, 1syl6eleqr 2540 . . . 4  |-  ( ph  ->  N  e.  W )
3215recnd 9620 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  CC )
331, 6, 12, 32, 20isumclim2 13547 . . . 4  |-  ( ph  ->  seq N (  +  ,  F )  ~~>  sum_ k  e.  W  A )
349sseld 3485 . . . . . . 7  |-  ( ph  ->  ( m  e.  W  ->  m  e.  Z ) )
35 fveq2 5852 . . . . . . . . 9  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
3635eleq1d 2510 . . . . . . . 8  |-  ( k  =  m  ->  (
( F `  k
)  e.  RR+  <->  ( F `  m )  e.  RR+ ) )
3736rspcv 3190 . . . . . . 7  |-  ( m  e.  Z  ->  ( A. k  e.  Z  ( F `  k )  e.  RR+  ->  ( F `
 m )  e.  RR+ ) )
3834, 22, 37syl6ci 65 . . . . . 6  |-  ( ph  ->  ( m  e.  W  ->  ( F `  m
)  e.  RR+ )
)
3938imp 429 . . . . 5  |-  ( (
ph  /\  m  e.  W )  ->  ( F `  m )  e.  RR+ )
4039rpred 11260 . . . 4  |-  ( (
ph  /\  m  e.  W )  ->  ( F `  m )  e.  RR )
4139rpge0d 11264 . . . 4  |-  ( (
ph  /\  m  e.  W )  ->  0  <_  ( F `  m
) )
421, 31, 33, 40, 41climserle 13459 . . 3  |-  ( ph  ->  (  seq N (  +  ,  F ) `
 N )  <_  sum_ k  e.  W  A
)
4328, 42eqbrtrrd 4455 . 2  |-  ( ph  ->  ( F `  N
)  <_  sum_ k  e.  W  A )
4421, 26, 43rpgecld 11295 1  |-  ( ph  -> 
sum_ k  e.  W  A  e.  RR+ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   A.wral 2791    C_ wss 3458   dom cdm 4985   ` cfv 5574   RRcr 9489    + caddc 9493    <_ cle 9627   ZZcz 10865   ZZ>=cuz 11085   RR+crp 11224    seqcseq 12081    ~~> cli 13281   sum_csu 13482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-fz 11677  df-fzo 11799  df-fl 11903  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-rlim 13286  df-sum 13483
This theorem is referenced by:  effsumlt  13718  eirrlem  13809  aaliou3lem3  22605
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