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Theorem isumltss 12633
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 11310 . . . . 5  |-  ( M  e.  ZZ  ->  -.  Z  e.  Fin )
41, 3syl 16 . . . 4  |-  ( ph  ->  -.  Z  e.  Fin )
5 ssdif0 3688 . . . . 5  |-  ( Z 
C_  A  <->  ( Z  \  A )  =  (/) )
6 isumltss.4 . . . . . 6  |-  ( ph  ->  A  C_  Z )
7 eqss 3365 . . . . . . 7  |-  ( A  =  Z  <->  ( A  C_  Z  /\  Z  C_  A ) )
8 isumltss.3 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
9 eleq1 2498 . . . . . . . 8  |-  ( A  =  Z  ->  ( A  e.  Fin  <->  Z  e.  Fin ) )
108, 9syl5ibcom 213 . . . . . . 7  |-  ( ph  ->  ( A  =  Z  ->  Z  e.  Fin ) )
117, 10syl5bir 211 . . . . . 6  |-  ( ph  ->  ( ( A  C_  Z  /\  Z  C_  A
)  ->  Z  e.  Fin ) )
126, 11mpand 658 . . . . 5  |-  ( ph  ->  ( Z  C_  A  ->  Z  e.  Fin )
)
135, 12syl5bir 211 . . . 4  |-  ( ph  ->  ( ( Z  \  A )  =  (/)  ->  Z  e.  Fin )
)
144, 13mtod 171 . . 3  |-  ( ph  ->  -.  ( Z  \  A )  =  (/) )
15 neq0 3640 . . 3  |-  ( -.  ( Z  \  A
)  =  (/)  <->  E. x  x  e.  ( Z  \  A ) )
1614, 15sylib 190 . 2  |-  ( ph  ->  E. x  x  e.  ( Z  \  A
) )
178adantr 453 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  A  e.  Fin )
186adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  A  C_  Z
)
1918sselda 3350 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  A )  ->  k  e.  Z )
20 isumltss.6 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  RR+ )
2120adantlr 697 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  B  e.  RR+ )
2221rpred 10653 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  B  e.  RR )
2319, 22syldan 458 . . . 4  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  A )  ->  B  e.  RR )
2417, 23fsumrecl 12533 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  e.  RR )
25 snfi 7190 . . . . 5  |-  { x }  e.  Fin
26 unfi 7377 . . . . 5  |-  ( ( A  e.  Fin  /\  { x }  e.  Fin )  ->  ( A  u.  { x } )  e. 
Fin )
2717, 25, 26sylancl 645 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  u.  { x } )  e.  Fin )
28 eldifi 3471 . . . . . . . . 9  |-  ( x  e.  ( Z  \  A )  ->  x  e.  Z )
2928snssd 3945 . . . . . . . 8  |-  ( x  e.  ( Z  \  A )  ->  { x }  C_  Z )
306, 29anim12i 551 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  C_  Z  /\  { x }  C_  Z ) )
31 unss 3523 . . . . . . 7  |-  ( ( A  C_  Z  /\  { x }  C_  Z
)  <->  ( A  u.  { x } )  C_  Z )
3230, 31sylib 190 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  u.  { x } ) 
C_  Z )
3332sselda 3350 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  ( A  u.  {
x } ) )  ->  k  e.  Z
)
3433, 22syldan 458 . . . 4  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  ( A  u.  {
x } ) )  ->  B  e.  RR )
3527, 34fsumrecl 12533 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  ( A  u.  {
x } ) B  e.  RR )
361adantr 453 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  M  e.  ZZ )
37 isumltss.5 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
3837adantlr 697 . . . 4  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  ( F `  k )  =  B )
39 isumltss.7 . . . . 5  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
4039adantr 453 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )
412, 36, 38, 22, 40isumrecl 12554 . . 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 2961 . . . . . . . 8  |-  x  e. 
_V
4443snnz 3924 . . . . . . 7  |-  { x }  =/=  (/)
4544a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  { x }  =/=  (/) )
4629adantl 454 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  { x }  C_  Z )
4746sselda 3350 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  { x } )  ->  k  e.  Z
)
4847, 21syldan 458 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  { x } )  ->  B  e.  RR+ )
4942, 45, 48fsumrpcl 12536 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e. 
{ x } B  e.  RR+ )
5024, 49ltaddrpd 10682 . . . 4  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  ( sum_ k  e.  A  B  +  sum_ k  e.  {
x } B ) )
51 eldifn 3472 . . . . . . 7  |-  ( x  e.  ( Z  \  A )  ->  -.  x  e.  A )
5251adantl 454 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  -.  x  e.  A )
53 disjsn 3870 . . . . . 6  |-  ( ( A  i^i  { x } )  =  (/)  <->  -.  x  e.  A )
5452, 53sylibr 205 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  i^i  { x } )  =  (/) )
55 eqidd 2439 . . . . 5  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  ( A  u.  { x } )  =  ( A  u.  { x } ) )
5621rpcnd 10655 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  B  e.  CC )
5733, 56syldan 458 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  ( A  u.  {
x } ) )  ->  B  e.  CC )
5854, 55, 27, 57fsumsplit 12538 . . . 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 4241 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  sum_ k  e.  ( A  u.  { x } ) B )
6021rpge0d 10657 . . . 4  |-  ( ( ( ph  /\  x  e.  ( Z  \  A
) )  /\  k  e.  Z )  ->  0  <_  B )
612, 36, 27, 32, 38, 22, 60, 40isumless 12630 . . 3  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  ( A  u.  {
x } ) B  <_  sum_ k  e.  Z  B )
6224, 35, 41, 59, 61ltletrd 9235 . 2  |-  ( (
ph  /\  x  e.  ( Z  \  A ) )  ->  sum_ k  e.  A  B  <  sum_ k  e.  Z  B
)
6316, 62exlimddv 1649 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 360   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601    \ cdif 3319    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630   {csn 3816   class class class wbr 4215   dom cdm 4881   ` cfv 5457  (class class class)co 6084   Fincfn 7112   CCcc 8993   RRcr 8994    + caddc 8998    < clt 9125   ZZcz 10287   ZZ>=cuz 10493   RR+crp 10617    seq cseq 11328    ~~> cli 12283   sum_csu 12484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485
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