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Theorem isumltss 12398
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 11117 . . . . 5  |-  ( M  e.  ZZ  ->  -.  Z  e.  Fin )
41, 3syl 15 . . . 4  |-  ( ph  ->  -.  Z  e.  Fin )
5 ssdif0 3589 . . . . 5  |-  ( Z 
C_  A  <->  ( Z  \  A )  =  (/) )
6 isumltss.4 . . . . . 6  |-  ( ph  ->  A  C_  Z )
7 eqss 3270 . . . . . . 7  |-  ( A  =  Z  <->  ( A  C_  Z  /\  Z  C_  A ) )
8 isumltss.3 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
9 eleq1 2418 . . . . . . . 8  |-  ( A  =  Z  ->  ( A  e.  Fin  <->  Z  e.  Fin ) )
108, 9syl5ibcom 211 . . . . . . 7  |-  ( ph  ->  ( A  =  Z  ->  Z  e.  Fin ) )
117, 10syl5bir 209 . . . . . 6  |-  ( ph  ->  ( ( A  C_  Z  /\  Z  C_  A
)  ->  Z  e.  Fin ) )
126, 11mpand 656 . . . . 5  |-  ( ph  ->  ( Z  C_  A  ->  Z  e.  Fin )
)
135, 12syl5bir 209 . . . 4  |-  ( ph  ->  ( ( Z  \  A )  =  (/)  ->  Z  e.  Fin )
)
144, 13mtod 168 . . 3  |-  ( ph  ->  -.  ( Z  \  A )  =  (/) )
15 neq0 3541 . . 3  |-  ( -.  ( Z  \  A
)  =  (/)  <->  E. x  x  e.  ( Z  \  A ) )
1614, 15sylib 188 . 2  |-  ( ph  ->  E. x  x  e.  ( Z  \  A
) )
178adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  A  e.  Fin )
186adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  A  C_  Z
)
1918sselda 3256 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  A )  ->  k  e.  Z )
20 isumltss.6 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  RR+ )
2120adantlr 695 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  B  e.  RR+ )
2221rpred 10479 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  B  e.  RR )
2319, 22syldan 456 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  A )  ->  B  e.  RR )
2417, 23fsumrecl 12298 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  e.  RR )
25 snfi 7026 . . . . . . 7  |-  { x }  e.  Fin
26 unfi 7211 . . . . . . 7  |-  ( ( A  e.  Fin  /\  { x }  e.  Fin )  ->  ( A  u.  { x } )  e. 
Fin )
2717, 25, 26sylancl 643 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  u.  { x } )  e.  Fin )
28 eldifi 3374 . . . . . . . . . . 11  |-  ( x  e.  ( Z  \  A )  ->  x  e.  Z )
2928snssd 3839 . . . . . . . . . 10  |-  ( x  e.  ( Z  \  A )  ->  { x }  C_  Z )
306, 29anim12i 549 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  C_  Z  /\  { x }  C_  Z ) )
31 unss 3425 . . . . . . . . 9  |-  ( ( A  C_  Z  /\  { x }  C_  Z
)  <->  ( A  u.  { x } )  C_  Z )
3230, 31sylib 188 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  u.  { x } ) 
C_  Z )
3332sselda 3256 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  ( A  u.  {
x } ) )  ->  k  e.  Z
)
3433, 22syldan 456 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  ( A  u.  {
x } ) )  ->  B  e.  RR )
3527, 34fsumrecl 12298 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  ( A  u.  {
x } ) B  e.  RR )
361adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  M  e.  ZZ )
37 isumltss.5 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
3837adantlr 695 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  ( F `  k )  =  B )
39 isumltss.7 . . . . . . 7  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
4039adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )
412, 36, 38, 22, 40isumrecl 12319 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  Z  B  e.  RR )
4225a1i 10 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  { x }  e.  Fin )
43 vex 2867 . . . . . . . . . 10  |-  x  e. 
_V
4443snnz 3820 . . . . . . . . 9  |-  { x }  =/=  (/)
4544a1i 10 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  { x }  =/=  (/) )
4629adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  { x }  C_  Z )
4746sselda 3256 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  { x } )  ->  k  e.  Z
)
4847, 21syldan 456 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  { x } )  ->  B  e.  RR+ )
4942, 45, 48fsumrpcl 12301 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e. 
{ x } B  e.  RR+ )
5024, 49ltaddrpd 10508 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  ( sum_ k  e.  A  B  +  sum_ k  e.  {
x } B ) )
51 eldifn 3375 . . . . . . . . 9  |-  ( x  e.  ( Z  \  A )  ->  -.  x  e.  A )
5251adantl 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  -.  x  e.  A )
53 disjsn 3769 . . . . . . . 8  |-  ( ( A  i^i  { x } )  =  (/)  <->  -.  x  e.  A )
5452, 53sylibr 203 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  i^i  { x } )  =  (/) )
55 eqidd 2359 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  u.  { x } )  =  ( A  u.  { x } ) )
5621rpcnd 10481 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  B  e.  CC )
5733, 56syldan 456 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  ( A  u.  {
x } ) )  ->  B  e.  CC )
5854, 55, 27, 57fsumsplit 12303 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  ( A  u.  {
x } ) B  =  ( sum_ k  e.  A  B  +  sum_ k  e.  { x } B ) )
5950, 58breqtrrd 4128 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  sum_ k  e.  ( A  u.  { x } ) B )
6021rpge0d 10483 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  0  <_  B )
612, 36, 27, 32, 38, 22, 60, 40isumless 12395 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  ( A  u.  {
x } ) B  <_  sum_ k  e.  Z  B )
6224, 35, 41, 59, 61ltletrd 9063 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  sum_ k  e.  Z  B
)
6362ex 423 . . 3  |-  ( ph  ->  ( x  e.  ( Z  \  A )  ->  sum_ k  e.  A  B  <  sum_ k  e.  Z  B ) )
6463exlimdv 1636 . 2  |-  ( ph  ->  ( E. x  x  e.  ( Z  \  A )  ->  sum_ k  e.  A  B  <  sum_ k  e.  Z  B
) )
6516, 64mpd 14 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 358   E.wex 1541    = wceq 1642    e. wcel 1710    =/= wne 2521    \ cdif 3225    u. cun 3226    i^i cin 3227    C_ wss 3228   (/)c0 3531   {csn 3716   class class class wbr 4102   dom cdm 4768   ` cfv 5334  (class class class)co 5942   Fincfn 6948   CCcc 8822   RRcr 8823    + caddc 8827    < clt 8954   ZZcz 10113   ZZ>=cuz 10319   RR+crp 10443    seq cseq 11135    ~~> cli 12048   sum_csu 12249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-oi 7312  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-fz 10872  df-fzo 10960  df-fl 11014  df-seq 11136  df-exp 11195  df-hash 11428  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-clim 12052  df-rlim 12053  df-sum 12250
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