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Theorem sermono 11834
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 11445 . . . 4  |-  ( k  e.  ( K ... N )  ->  k  e.  ( ZZ>= `  K )
)
3 sermono.1 . . . 4  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
4 uztrn 10873 . . . 4  |-  ( ( k  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
52, 3, 4syl2anr 475 . . 3  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  k  e.  ( ZZ>= `  M )
)
6 elfzuz3 11446 . . . . . . 7  |-  ( k  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  k )
)
76adantl 463 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  N  e.  ( ZZ>= `  k )
)
8 fzss2 11494 . . . . . 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 3353 . . . 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 709 . . . 4  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  x  e.  ( M ... N
) )  ->  ( F `  x )  e.  RR )
1310, 12syldan 467 . . 3  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  x  e.  ( M ... k
) )  ->  ( F `  x )  e.  RR )
14 readdcl 9361 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
1514adantl 463 . . 3  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  ->  ( x  +  y )  e.  RR )
165, 13, 15seqcl 11822 . 2  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  (  seq M (  +  ,  F ) `  k
)  e.  RR )
17 simpr 458 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... ( N  -  1 ) ) )
183adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  K  e.  ( ZZ>= `  M )
)
19 eluzelz 10866 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
2018, 19syl 16 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  K  e.  ZZ )
211adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  N  e.  ( ZZ>= `  K )
)
22 eluzelz 10866 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ZZ )
2321, 22syl 16 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  N  e.  ZZ )
24 peano2zm 10684 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
2523, 24syl 16 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( N  -  1 )  e.  ZZ )
26 elfzelz 11449 . . . . . . . . 9  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  k  e.  ZZ )
2726adantl 463 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ZZ )
28 1zzd 10673 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  1  e.  ZZ )
29 fzaddel 11489 . . . . . . . 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 1214 . . . . . . 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 10647 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
33 ax-1cn 9336 . . . . . . . . 9  |-  1  e.  CC
34 npcan 9615 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
3532, 33, 34sylancl 657 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
( N  -  1 )  +  1 )  =  N )
3623, 35syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( ( N  -  1 )  +  1 )  =  N )
3736oveq2d 6106 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( ( K  +  1 ) ... ( ( N  -  1 )  +  1 ) )  =  ( ( K  + 
1 ) ... N
) )
3831, 37eleqtrd 2517 . . . . 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 2797 . . . . . 6  |-  ( ph  ->  A. x  e.  ( ( K  +  1 ) ... N ) 0  <_  ( F `  x ) )
4140adantr 462 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  A. x  e.  ( ( K  + 
1 ) ... N
) 0  <_  ( F `  x )
)
42 fveq2 5688 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
4342breq2d 4301 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
0  <_  ( F `  x )  <->  0  <_  ( F `  ( k  +  1 ) ) ) )
4443rspcv 3066 . . . . 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 11503 . . . . . . . 8  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  k  e.  ( K ... (
( N  -  1 )  +  1 ) ) )
4746adantl 463 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... ( ( N  -  1 )  +  1 ) ) )
4836oveq2d 6106 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( K ... ( ( N  - 
1 )  +  1 ) )  =  ( K ... N ) )
4947, 48eleqtrd 2517 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... N ) )
5049, 16syldan 467 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq M (  +  ,  F ) `  k
)  e.  RR )
51 fzss1 11493 . . . . . . . 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 11505 . . . . . . . . 9  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  (
k  +  1 )  e.  ( K ... ( ( N  - 
1 )  +  1 ) ) )
5453adantl 463 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( K ... (
( N  -  1 )  +  1 ) ) )
5554, 48eleqtrd 2517 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( K ... N
) )
5652, 55sseldd 3354 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( M ... N
) )
5711ralrimiva 2797 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( M ... N ) ( F `  x
)  e.  RR )
5857adantr 462 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  A. x  e.  ( M ... N
) ( F `  x )  e.  RR )
5942eleq1d 2507 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  (
( F `  x
)  e.  RR  <->  ( F `  ( k  +  1 ) )  e.  RR ) )
6059rspcv 3066 . . . . . 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 9923 . . . 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 467 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( ZZ>= `  M )
)
65 seqp1 11817 . . . 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 4315 . 2  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq M (  +  ,  F ) `  k
)  <_  (  seq M (  +  ,  F ) `  (
k  +  1 ) ) )
681, 16, 67monoord 11832 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 1364    e. wcel 1761   A.wral 2713    C_ wss 3325   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    <_ cle 9415    - cmin 9591   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433    seqcseq 11802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-seq 11803
This theorem is referenced by:  cvgcmp  13275  isumsup2  13305  climcnds  13310  ovolunlem1a  20938  mblfinlem2  28354
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