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Theorem sermono 12113
Description: The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-Jun-2013.)
Hypotheses
Ref Expression
sermono.1  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
sermono.2  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
sermono.3  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  e.  RR )
sermono.4  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  0  <_  ( F `  x
) )
Assertion
Ref Expression
sermono  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 K )  <_ 
(  seq M (  +  ,  F ) `  N ) )
Distinct variable groups:    x, F    x, K    x, M    x, N    ph, x

Proof of Theorem sermono
Dummy variables  k 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sermono.2 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
2 elfzuz 11688 . . . 4  |-  ( k  e.  ( K ... N )  ->  k  e.  ( ZZ>= `  K )
)
3 sermono.1 . . . 4  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
4 uztrn 11101 . . . 4  |-  ( ( k  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
52, 3, 4syl2anr 478 . . 3  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  k  e.  ( ZZ>= `  M )
)
6 elfzuz3 11689 . . . . . . 7  |-  ( k  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  k )
)
76adantl 466 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  N  e.  ( ZZ>= `  k )
)
8 fzss2 11727 . . . . . 6  |-  ( N  e.  ( ZZ>= `  k
)  ->  ( M ... k )  C_  ( M ... N ) )
97, 8syl 16 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  ( M ... k )  C_  ( M ... N ) )
109sselda 3486 . . . 4  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  x  e.  ( M ... k
) )  ->  x  e.  ( M ... N
) )
11 sermono.3 . . . . 5  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  e.  RR )
1211adantlr 714 . . . 4  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  x  e.  ( M ... N
) )  ->  ( F `  x )  e.  RR )
1310, 12syldan 470 . . 3  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  x  e.  ( M ... k
) )  ->  ( F `  x )  e.  RR )
14 readdcl 9573 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
1514adantl 466 . . 3  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  ->  ( x  +  y )  e.  RR )
165, 13, 15seqcl 12101 . 2  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  (  seq M (  +  ,  F ) `  k
)  e.  RR )
17 simpr 461 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... ( N  -  1 ) ) )
183adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  K  e.  ( ZZ>= `  M )
)
19 eluzelz 11094 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
2018, 19syl 16 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  K  e.  ZZ )
211adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  N  e.  ( ZZ>= `  K )
)
22 eluzelz 11094 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ZZ )
2321, 22syl 16 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  N  e.  ZZ )
24 peano2zm 10908 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
2523, 24syl 16 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( N  -  1 )  e.  ZZ )
26 elfzelz 11692 . . . . . . . . 9  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  k  e.  ZZ )
2726adantl 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ZZ )
28 1zzd 10896 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  1  e.  ZZ )
29 fzaddel 11722 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  ( N  -  1 )  e.  ZZ )  /\  ( k  e.  ZZ  /\  1  e.  ZZ ) )  -> 
( k  e.  ( K ... ( N  -  1 ) )  <-> 
( k  +  1 )  e.  ( ( K  +  1 ) ... ( ( N  -  1 )  +  1 ) ) ) )
3020, 25, 27, 28, 29syl22anc 1228 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  e.  ( K ... ( N  -  1 ) )  <->  ( k  +  1 )  e.  ( ( K  +  1 ) ... ( ( N  -  1 )  +  1 ) ) ) )
3117, 30mpbid 210 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( ( K  + 
1 ) ... (
( N  -  1 )  +  1 ) ) )
32 zcn 10870 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
33 ax-1cn 9548 . . . . . . . . 9  |-  1  e.  CC
34 npcan 9829 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
3532, 33, 34sylancl 662 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
( N  -  1 )  +  1 )  =  N )
3623, 35syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( ( N  -  1 )  +  1 )  =  N )
3736oveq2d 6293 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( ( K  +  1 ) ... ( ( N  -  1 )  +  1 ) )  =  ( ( K  + 
1 ) ... N
) )
3831, 37eleqtrd 2531 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
39 sermono.4 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  0  <_  ( F `  x
) )
4039ralrimiva 2855 . . . . . 6  |-  ( ph  ->  A. x  e.  ( ( K  +  1 ) ... N ) 0  <_  ( F `  x ) )
4140adantr 465 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  A. x  e.  ( ( K  + 
1 ) ... N
) 0  <_  ( F `  x )
)
42 fveq2 5852 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
4342breq2d 4445 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
0  <_  ( F `  x )  <->  0  <_  ( F `  ( k  +  1 ) ) ) )
4443rspcv 3190 . . . . 5  |-  ( ( k  +  1 )  e.  ( ( K  +  1 ) ... N )  ->  ( A. x  e.  (
( K  +  1 ) ... N ) 0  <_  ( F `  x )  ->  0  <_  ( F `  (
k  +  1 ) ) ) )
4538, 41, 44sylc 60 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  0  <_  ( F `  ( k  +  1 ) ) )
46 fzelp1 11736 . . . . . . . 8  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  k  e.  ( K ... (
( N  -  1 )  +  1 ) ) )
4746adantl 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... ( ( N  -  1 )  +  1 ) ) )
4836oveq2d 6293 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( K ... ( ( N  - 
1 )  +  1 ) )  =  ( K ... N ) )
4947, 48eleqtrd 2531 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... N ) )
5049, 16syldan 470 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq M (  +  ,  F ) `  k
)  e.  RR )
51 fzss1 11726 . . . . . . . 8  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( K ... N )  C_  ( M ... N ) )
5218, 51syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( K ... N )  C_  ( M ... N ) )
53 fzp1elp1 11737 . . . . . . . . 9  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  (
k  +  1 )  e.  ( K ... ( ( N  - 
1 )  +  1 ) ) )
5453adantl 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( K ... (
( N  -  1 )  +  1 ) ) )
5554, 48eleqtrd 2531 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( K ... N
) )
5652, 55sseldd 3487 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( M ... N
) )
5711ralrimiva 2855 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( M ... N ) ( F `  x
)  e.  RR )
5857adantr 465 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  A. x  e.  ( M ... N
) ( F `  x )  e.  RR )
5942eleq1d 2510 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  (
( F `  x
)  e.  RR  <->  ( F `  ( k  +  1 ) )  e.  RR ) )
6059rspcv 3190 . . . . . 6  |-  ( ( k  +  1 )  e.  ( M ... N )  ->  ( A. x  e.  ( M ... N ) ( F `  x )  e.  RR  ->  ( F `  ( k  +  1 ) )  e.  RR ) )
6156, 58, 60sylc 60 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( F `  ( k  +  1 ) )  e.  RR )
6250, 61addge01d 10141 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( 0  <_  ( F `  ( k  +  1 ) )  <->  (  seq M (  +  ,  F ) `  k
)  <_  ( (  seq M (  +  ,  F ) `  k
)  +  ( F `
 ( k  +  1 ) ) ) ) )
6345, 62mpbid 210 . . 3  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq M (  +  ,  F ) `  k
)  <_  ( (  seq M (  +  ,  F ) `  k
)  +  ( F `
 ( k  +  1 ) ) ) )
6449, 5syldan 470 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( ZZ>= `  M )
)
65 seqp1 12096 . . . 4  |-  ( k  e.  ( ZZ>= `  M
)  ->  (  seq M (  +  ,  F ) `  (
k  +  1 ) )  =  ( (  seq M (  +  ,  F ) `  k )  +  ( F `  ( k  +  1 ) ) ) )
6664, 65syl 16 . . 3  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq M (  +  ,  F ) `  (
k  +  1 ) )  =  ( (  seq M (  +  ,  F ) `  k )  +  ( F `  ( k  +  1 ) ) ) )
6763, 66breqtrrd 4459 . 2  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq M (  +  ,  F ) `  k
)  <_  (  seq M (  +  ,  F ) `  (
k  +  1 ) ) )
681, 16, 67monoord 12111 1  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 K )  <_ 
(  seq M (  +  ,  F ) `  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   A.wral 2791    C_ wss 3458   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493    <_ cle 9627    - cmin 9805   ZZcz 10865   ZZ>=cuz 11085   ...cfz 11676    seqcseq 12081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-seq 12082
This theorem is referenced by:  cvgcmp  13604  isumsup2  13632  climcnds  13637  ovolunlem1a  21773  mblfinlem2  30020
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