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Theorem isumltss 13424
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 11900 . . . . 5  |-  ( M  e.  ZZ  ->  -.  Z  e.  Fin )
41, 3syl 16 . . . 4  |-  ( ph  ->  -.  Z  e.  Fin )
5 ssdif0 3840 . . . . 5  |-  ( Z 
C_  A  <->  ( Z  \  A )  =  (/) )
6 isumltss.4 . . . . . 6  |-  ( ph  ->  A  C_  Z )
7 eqss 3474 . . . . . . 7  |-  ( A  =  Z  <->  ( A  C_  Z  /\  Z  C_  A ) )
8 isumltss.3 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
9 eleq1 2524 . . . . . . . 8  |-  ( A  =  Z  ->  ( A  e.  Fin  <->  Z  e.  Fin ) )
108, 9syl5ibcom 220 . . . . . . 7  |-  ( ph  ->  ( A  =  Z  ->  Z  e.  Fin ) )
117, 10syl5bir 218 . . . . . 6  |-  ( ph  ->  ( ( A  C_  Z  /\  Z  C_  A
)  ->  Z  e.  Fin ) )
126, 11mpand 675 . . . . 5  |-  ( ph  ->  ( Z  C_  A  ->  Z  e.  Fin )
)
135, 12syl5bir 218 . . . 4  |-  ( ph  ->  ( ( Z  \  A )  =  (/)  ->  Z  e.  Fin )
)
144, 13mtod 177 . . 3  |-  ( ph  ->  -.  ( Z  \  A )  =  (/) )
15 neq0 3750 . . 3  |-  ( -.  ( Z  \  A
)  =  (/)  <->  E. x  x  e.  ( Z  \  A ) )
1614, 15sylib 196 . 2  |-  ( ph  ->  E. x  x  e.  ( Z  \  A
) )
178adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  A  e.  Fin )
186adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  A  C_  Z
)
1918sselda 3459 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  A )  ->  k  e.  Z )
20 isumltss.6 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  RR+ )
2120adantlr 714 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  B  e.  RR+ )
2221rpred 11133 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  B  e.  RR )
2319, 22syldan 470 . . . 4  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  A )  ->  B  e.  RR )
2417, 23fsumrecl 13324 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  e.  RR )
25 snfi 7495 . . . . 5  |-  { x }  e.  Fin
26 unfi 7685 . . . . 5  |-  ( ( A  e.  Fin  /\  { x }  e.  Fin )  ->  ( A  u.  { x } )  e. 
Fin )
2717, 25, 26sylancl 662 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  u.  { x } )  e.  Fin )
28 eldifi 3581 . . . . . . . . 9  |-  ( x  e.  ( Z  \  A )  ->  x  e.  Z )
2928snssd 4121 . . . . . . . 8  |-  ( x  e.  ( Z  \  A )  ->  { x }  C_  Z )
306, 29anim12i 566 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  C_  Z  /\  { x }  C_  Z ) )
31 unss 3633 . . . . . . 7  |-  ( ( A  C_  Z  /\  { x }  C_  Z
)  <->  ( A  u.  { x } )  C_  Z )
3230, 31sylib 196 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  u.  { x } ) 
C_  Z )
3332sselda 3459 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  ( A  u.  {
x } ) )  ->  k  e.  Z
)
3433, 22syldan 470 . . . 4  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  ( A  u.  {
x } ) )  ->  B  e.  RR )
3527, 34fsumrecl 13324 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  ( A  u.  {
x } ) B  e.  RR )
361adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  M  e.  ZZ )
37 isumltss.5 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
3837adantlr 714 . . . 4  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  ( F `  k )  =  B )
39 isumltss.7 . . . . 5  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
4039adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  seq M (  +  ,  F )  e.  dom  ~~>  )
412, 36, 38, 22, 40isumrecl 13345 . . 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 3075 . . . . . . . 8  |-  x  e. 
_V
4443snnz 4096 . . . . . . 7  |-  { x }  =/=  (/)
4544a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  { x }  =/=  (/) )
4629adantl 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  { x }  C_  Z )
4746sselda 3459 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  { x } )  ->  k  e.  Z
)
4847, 21syldan 470 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  { x } )  ->  B  e.  RR+ )
4942, 45, 48fsumrpcl 13327 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e. 
{ x } B  e.  RR+ )
5024, 49ltaddrpd 11162 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  ( sum_ k  e.  A  B  +  sum_ k  e.  {
x } B ) )
51 eldifn 3582 . . . . . . 7  |-  ( x  e.  ( Z  \  A )  ->  -.  x  e.  A )
5251adantl 466 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  -.  x  e.  A )
53 disjsn 4039 . . . . . 6  |-  ( ( A  i^i  { x } )  =  (/)  <->  -.  x  e.  A )
5452, 53sylibr 212 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  i^i  { x } )  =  (/) )
55 eqidd 2453 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  u.  { x } )  =  ( A  u.  { x } ) )
5621rpcnd 11135 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  B  e.  CC )
5733, 56syldan 470 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  ( A  u.  {
x } ) )  ->  B  e.  CC )
5854, 55, 27, 57fsumsplit 13329 . . . 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 4421 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  sum_ k  e.  ( A  u.  { x } ) B )
6021rpge0d 11137 . . . 4  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  0  <_  B )
612, 36, 27, 32, 38, 22, 60, 40isumless 13421 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  ( A  u.  {
x } ) B  <_  sum_ k  e.  Z  B )
6224, 35, 41, 59, 61ltletrd 9637 . 2  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  sum_ k  e.  Z  B
)
6316, 62exlimddv 1693 1  |-  ( ph  -> 
sum_ k  e.  A  B  <  sum_ k  e.  Z  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2645    \ cdif 3428    u. cun 3429    i^i cin 3430    C_ wss 3431   (/)c0 3740   {csn 3980   class class class wbr 4395   dom cdm 4943   ` cfv 5521  (class class class)co 6195   Fincfn 7415   CCcc 9386   RRcr 9387    + caddc 9391    < clt 9524   ZZcz 10752   ZZ>=cuz 10967   RR+crp 11097    seqcseq 11918    ~~> cli 13075   sum_csu 13276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-pm 7322  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-fz 11550  df-fzo 11661  df-fl 11754  df-seq 11919  df-exp 11978  df-hash 12216  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-clim 13079  df-rlim 13080  df-sum 13277
This theorem is referenced by: (None)
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