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Theorem isumltss 13672
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 12079 . . . . 5  |-  ( M  e.  ZZ  ->  -.  Z  e.  Fin )
41, 3syl 16 . . . 4  |-  ( ph  ->  -.  Z  e.  Fin )
5 ssdif0 3888 . . . . 5  |-  ( Z 
C_  A  <->  ( Z  \  A )  =  (/) )
6 isumltss.4 . . . . . 6  |-  ( ph  ->  A  C_  Z )
7 eqss 3514 . . . . . . 7  |-  ( A  =  Z  <->  ( A  C_  Z  /\  Z  C_  A ) )
8 isumltss.3 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
9 eleq1 2529 . . . . . . . 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 3804 . . 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 3499 . . . . 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 11281 . . . . 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 13568 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  e.  RR )
25 snfi 7615 . . . . 5  |-  { x }  e.  Fin
26 unfi 7805 . . . . 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 3622 . . . . . . . . 9  |-  ( x  e.  ( Z  \  A )  ->  x  e.  Z )
2928snssd 4177 . . . . . . . 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 3674 . . . . . . 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 3499 . . . . 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 13568 . . 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 13592 . . 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 3112 . . . . . . . 8  |-  x  e. 
_V
4443snnz 4150 . . . . . . 7  |-  { x }  =/=  (/)
4544a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  { x }  =/=  (/) )
4629adantl 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  { x }  C_  Z )
4746sselda 3499 . . . . . . 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 13571 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e. 
{ x } B  e.  RR+ )
5024, 49ltaddrpd 11310 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  ( sum_ k  e.  A  B  +  sum_ k  e.  {
x } B ) )
51 eldifn 3623 . . . . . . 7  |-  ( x  e.  ( Z  \  A )  ->  -.  x  e.  A )
5251adantl 466 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  -.  x  e.  A )
53 disjsn 4092 . . . . . 6  |-  ( ( A  i^i  { x } )  =  (/)  <->  -.  x  e.  A )
5452, 53sylibr 212 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  i^i  { x } )  =  (/) )
55 eqidd 2458 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  u.  { x } )  =  ( A  u.  { x } ) )
5621rpcnd 11283 . . . . . 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 13574 . . . 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 4482 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  sum_ k  e.  ( A  u.  { x } ) B )
6021rpge0d 11285 . . . 4  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  0  <_  B )
612, 36, 27, 32, 38, 22, 60, 40isumless 13669 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  ( A  u.  {
x } ) B  <_  sum_ k  e.  Z  B )
6224, 35, 41, 59, 61ltletrd 9759 . 2  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  sum_ k  e.  Z  B
)
6316, 62exlimddv 1727 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 1395   E.wex 1613    e. wcel 1819    =/= wne 2652    \ cdif 3468    u. cun 3469    i^i cin 3470    C_ wss 3471   (/)c0 3793   {csn 4032   class class class wbr 4456   dom cdm 5008   ` cfv 5594  (class class class)co 6296   Fincfn 7535   CCcc 9507   RRcr 9508    + caddc 9512    < clt 9645   ZZcz 10885   ZZ>=cuz 11106   RR+crp 11245    seqcseq 12110    ~~> cli 13319   sum_csu 13520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521
This theorem is referenced by: (None)
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