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Theorem isumrpcl 11515
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, 3eleqtrdi 2280 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 9550 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 14 . . 3  |-  ( ph  ->  N  e.  ZZ )
7 uzss 9561 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  M ) )
84, 7syl 14 . . . . . 6  |-  ( ph  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
)
98, 1, 33sstr4g 3210 . . . . 5  |-  ( ph  ->  W  C_  Z )
109sselda 3167 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  k  e.  Z )
11 isumrpcl.4 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
1210, 11syldan 282 . . 3  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  =  A )
13 isumrpcl.5 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR+ )
1413rpred 9709 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR )
1510, 14syldan 282 . . 3  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  RR )
16 isumrpcl.6 . . . 4  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
1711, 13eqeltrd 2264 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR+ )
1817rpcnd 9711 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
193, 2, 18iserex 11360 . . . 4  |-  ( ph  ->  (  seq M (  +  ,  F )  e.  dom  ~~>  <->  seq N (  +  ,  F )  e.  dom  ~~>  ) )
2016, 19mpbid 147 . . 3  |-  ( ph  ->  seq N (  +  ,  F )  e. 
dom 
~~>  )
211, 6, 12, 15, 20isumrecl 11450 . 2  |-  ( ph  -> 
sum_ k  e.  W  A  e.  RR )
22 fveq2 5527 . . . 4  |-  ( k  =  N  ->  ( F `  k )  =  ( F `  N ) )
2322eleq1d 2256 . . 3  |-  ( k  =  N  ->  (
( F `  k
)  e.  RR+  <->  ( F `  N )  e.  RR+ ) )
2417ralrimiva 2560 . . 3  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  RR+ )
2523, 24, 2rspcdva 2858 . 2  |-  ( ph  ->  ( F `  N
)  e.  RR+ )
268sselda 3167 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  k  e.  ( ZZ>= `  M )
)
2726, 3eleqtrrdi 2281 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  k  e.  Z )
2827, 17syldan 282 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  ->  ( F `  k )  e.  RR+ )
29 rpaddcl 9690 . . . . 5  |-  ( ( k  e.  RR+  /\  y  e.  RR+ )  ->  (
k  +  y )  e.  RR+ )
3029adantl 277 . . . 4  |-  ( (
ph  /\  ( k  e.  RR+  /\  y  e.  RR+ ) )  ->  (
k  +  y )  e.  RR+ )
316, 28, 30seq3-1 10473 . . 3  |-  ( ph  ->  (  seq N (  +  ,  F ) `
 N )  =  ( F `  N
) )
32 uzid 9555 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
336, 32syl 14 . . . . 5  |-  ( ph  ->  N  e.  ( ZZ>= `  N ) )
3433, 1eleqtrrdi 2281 . . . 4  |-  ( ph  ->  N  e.  W )
3515recnd 7999 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  CC )
361, 6, 12, 35, 20isumclim2 11443 . . . 4  |-  ( ph  ->  seq N (  +  ,  F )  ~~>  sum_ k  e.  W  A )
379sseld 3166 . . . . . . 7  |-  ( ph  ->  ( m  e.  W  ->  m  e.  Z ) )
38 fveq2 5527 . . . . . . . . 9  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
3938eleq1d 2256 . . . . . . . 8  |-  ( k  =  m  ->  (
( F `  k
)  e.  RR+  <->  ( F `  m )  e.  RR+ ) )
4039rspcv 2849 . . . . . . 7  |-  ( m  e.  Z  ->  ( A. k  e.  Z  ( F `  k )  e.  RR+  ->  ( F `
 m )  e.  RR+ ) )
4137, 24, 40syl6ci 1455 . . . . . 6  |-  ( ph  ->  ( m  e.  W  ->  ( F `  m
)  e.  RR+ )
)
4241imp 124 . . . . 5  |-  ( (
ph  /\  m  e.  W )  ->  ( F `  m )  e.  RR+ )
4342rpred 9709 . . . 4  |-  ( (
ph  /\  m  e.  W )  ->  ( F `  m )  e.  RR )
4442rpge0d 9713 . . . 4  |-  ( (
ph  /\  m  e.  W )  ->  0  <_  ( F `  m
) )
451, 34, 36, 43, 44climserle 11366 . . 3  |-  ( ph  ->  (  seq N (  +  ,  F ) `
 N )  <_  sum_ k  e.  W  A
)
4631, 45eqbrtrrd 4039 . 2  |-  ( ph  ->  ( F `  N
)  <_  sum_ k  e.  W  A )
4721, 25, 46rpgecld 9749 1  |-  ( ph  -> 
sum_ k  e.  W  A  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1363    e. wcel 2158   A.wral 2465    C_ wss 3141   dom cdm 4638   ` cfv 5228  (class class class)co 5888   RRcr 7823    + caddc 7827    <_ cle 8006   ZZcz 9266   ZZ>=cuz 9541   RR+crp 9666    seqcseq 10458    ~~> cli 11299   sum_csu 11374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-frec 6405  df-1o 6430  df-oadd 6434  df-er 6548  df-en 6754  df-dom 6755  df-fin 6756  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-q 9633  df-rp 9667  df-fz 10022  df-fzo 10156  df-seqfrec 10459  df-exp 10533  df-ihash 10769  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021  df-clim 11300  df-sumdc 11375
This theorem is referenced by:  effsumlt  11713  eirraplem  11797
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