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Theorem isumltss 12587
Description: A partial sum of a series with positive terms is less than the infinite sum. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)
Hypotheses
Ref Expression
isumltss.1  |-  Z  =  ( ZZ>= `  M )
isumltss.2  |-  ( ph  ->  M  e.  ZZ )
isumltss.3  |-  ( ph  ->  A  e.  Fin )
isumltss.4  |-  ( ph  ->  A  C_  Z )
isumltss.5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
isumltss.6  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  RR+ )
isumltss.7  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
Assertion
Ref Expression
isumltss  |-  ( ph  -> 
sum_ k  e.  A  B  <  sum_ k  e.  Z  B )
Distinct variable groups:    A, k    k, F    k, M    ph, k    k, Z
Allowed substitution hint:    B( k)

Proof of Theorem isumltss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isumltss.2 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
2 isumltss.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
32uzinf 11264 . . . . 5  |-  ( M  e.  ZZ  ->  -.  Z  e.  Fin )
41, 3syl 16 . . . 4  |-  ( ph  ->  -.  Z  e.  Fin )
5 ssdif0 3650 . . . . 5  |-  ( Z 
C_  A  <->  ( Z  \  A )  =  (/) )
6 isumltss.4 . . . . . 6  |-  ( ph  ->  A  C_  Z )
7 eqss 3327 . . . . . . 7  |-  ( A  =  Z  <->  ( A  C_  Z  /\  Z  C_  A ) )
8 isumltss.3 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
9 eleq1 2468 . . . . . . . 8  |-  ( A  =  Z  ->  ( A  e.  Fin  <->  Z  e.  Fin ) )
108, 9syl5ibcom 212 . . . . . . 7  |-  ( ph  ->  ( A  =  Z  ->  Z  e.  Fin ) )
117, 10syl5bir 210 . . . . . 6  |-  ( ph  ->  ( ( A  C_  Z  /\  Z  C_  A
)  ->  Z  e.  Fin ) )
126, 11mpand 657 . . . . 5  |-  ( ph  ->  ( Z  C_  A  ->  Z  e.  Fin )
)
135, 12syl5bir 210 . . . 4  |-  ( ph  ->  ( ( Z  \  A )  =  (/)  ->  Z  e.  Fin )
)
144, 13mtod 170 . . 3  |-  ( ph  ->  -.  ( Z  \  A )  =  (/) )
15 neq0 3602 . . 3  |-  ( -.  ( Z  \  A
)  =  (/)  <->  E. x  x  e.  ( Z  \  A ) )
1614, 15sylib 189 . 2  |-  ( ph  ->  E. x  x  e.  ( Z  \  A
) )
178adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  A  e.  Fin )
186adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  A  C_  Z
)
1918sselda 3312 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  A )  ->  k  e.  Z )
20 isumltss.6 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  RR+ )
2120adantlr 696 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  B  e.  RR+ )
2221rpred 10608 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  B  e.  RR )
2319, 22syldan 457 . . . 4  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  A )  ->  B  e.  RR )
2417, 23fsumrecl 12487 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  e.  RR )
25 snfi 7150 . . . . 5  |-  { x }  e.  Fin
26 unfi 7337 . . . . 5  |-  ( ( A  e.  Fin  /\  { x }  e.  Fin )  ->  ( A  u.  { x } )  e. 
Fin )
2717, 25, 26sylancl 644 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  u.  { x } )  e.  Fin )
28 eldifi 3433 . . . . . . . . 9  |-  ( x  e.  ( Z  \  A )  ->  x  e.  Z )
2928snssd 3907 . . . . . . . 8  |-  ( x  e.  ( Z  \  A )  ->  { x }  C_  Z )
306, 29anim12i 550 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  C_  Z  /\  { x }  C_  Z ) )
31 unss 3485 . . . . . . 7  |-  ( ( A  C_  Z  /\  { x }  C_  Z
)  <->  ( A  u.  { x } )  C_  Z )
3230, 31sylib 189 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  u.  { x } ) 
C_  Z )
3332sselda 3312 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  ( A  u.  {
x } ) )  ->  k  e.  Z
)
3433, 22syldan 457 . . . 4  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  ( A  u.  {
x } ) )  ->  B  e.  RR )
3527, 34fsumrecl 12487 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  ( A  u.  {
x } ) B  e.  RR )
361adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  M  e.  ZZ )
37 isumltss.5 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
3837adantlr 696 . . . 4  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  ( F `  k )  =  B )
39 isumltss.7 . . . . 5  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
4039adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )
412, 36, 38, 22, 40isumrecl 12508 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  Z  B  e.  RR )
4225a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  { x }  e.  Fin )
43 vex 2923 . . . . . . . 8  |-  x  e. 
_V
4443snnz 3886 . . . . . . 7  |-  { x }  =/=  (/)
4544a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  { x }  =/=  (/) )
4629adantl 453 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  { x }  C_  Z )
4746sselda 3312 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  { x } )  ->  k  e.  Z
)
4847, 21syldan 457 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  { x } )  ->  B  e.  RR+ )
4942, 45, 48fsumrpcl 12490 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e. 
{ x } B  e.  RR+ )
5024, 49ltaddrpd 10637 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  ( sum_ k  e.  A  B  +  sum_ k  e.  {
x } B ) )
51 eldifn 3434 . . . . . . 7  |-  ( x  e.  ( Z  \  A )  ->  -.  x  e.  A )
5251adantl 453 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  -.  x  e.  A )
53 disjsn 3832 . . . . . 6  |-  ( ( A  i^i  { x } )  =  (/)  <->  -.  x  e.  A )
5452, 53sylibr 204 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  i^i  { x } )  =  (/) )
55 eqidd 2409 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  u.  { x } )  =  ( A  u.  { x } ) )
5621rpcnd 10610 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  B  e.  CC )
5733, 56syldan 457 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  ( A  u.  {
x } ) )  ->  B  e.  CC )
5854, 55, 27, 57fsumsplit 12492 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  ( A  u.  {
x } ) B  =  ( sum_ k  e.  A  B  +  sum_ k  e.  { x } B ) )
5950, 58breqtrrd 4202 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  sum_ k  e.  ( A  u.  { x } ) B )
6021rpge0d 10612 . . . 4  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  0  <_  B )
612, 36, 27, 32, 38, 22, 60, 40isumless 12584 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  ( A  u.  {
x } ) B  <_  sum_ k  e.  Z  B )
6224, 35, 41, 59, 61ltletrd 9190 . 2  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  sum_ k  e.  Z  B
)
6316, 62exlimddv 1645 1  |-  ( ph  -> 
sum_ k  e.  A  B  <  sum_ k  e.  Z  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2571    \ cdif 3281    u. cun 3282    i^i cin 3283    C_ wss 3284   (/)c0 3592   {csn 3778   class class class wbr 4176   dom cdm 4841   ` cfv 5417  (class class class)co 6044   Fincfn 7072   CCcc 8948   RRcr 8949    + caddc 8953    < clt 9080   ZZcz 10242   ZZ>=cuz 10448   RR+crp 10572    seq cseq 11282    ~~> cli 12237   sum_csu 12438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-fz 11004  df-fzo 11095  df-fl 11161  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-rlim 12242  df-sum 12439
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