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Theorem sermono 11283
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 10988 . . . 4  |-  ( k  e.  ( K ... N )  ->  k  e.  ( ZZ>= `  K )
)
3 sermono.1 . . . 4  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
4 uztrn 10435 . . . 4  |-  ( ( k  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
52, 3, 4syl2anr 465 . . 3  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  k  e.  ( ZZ>= `  M )
)
6 elfzuz3 10989 . . . . . . 7  |-  ( k  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  k )
)
76adantl 453 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  N  e.  ( ZZ>= `  k )
)
8 fzss2 11025 . . . . . 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 3292 . . . 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 696 . . . 4  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  x  e.  ( M ... N
) )  ->  ( F `  x )  e.  RR )
1310, 12syldan 457 . . 3  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  x  e.  ( M ... k
) )  ->  ( F `  x )  e.  RR )
14 readdcl 9007 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
1514adantl 453 . . 3  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  ->  ( x  +  y )  e.  RR )
165, 13, 15seqcl 11271 . 2  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  (  seq  M (  +  ,  F
) `  k )  e.  RR )
17 simpr 448 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... ( N  -  1 ) ) )
183adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  K  e.  ( ZZ>= `  M )
)
19 eluzelz 10429 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
2018, 19syl 16 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  K  e.  ZZ )
211adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  N  e.  ( ZZ>= `  K )
)
22 eluzelz 10429 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ZZ )
2321, 22syl 16 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  N  e.  ZZ )
24 peano2zm 10253 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
2523, 24syl 16 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( N  -  1 )  e.  ZZ )
26 elfzelz 10992 . . . . . . . . 9  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  k  e.  ZZ )
2726adantl 453 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ZZ )
28 1z 10244 . . . . . . . . 9  |-  1  e.  ZZ
2928a1i 11 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  1  e.  ZZ )
30 fzaddel 11020 . . . . . . . 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 ) ) ) )
3120, 25, 27, 29, 30syl22anc 1185 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  e.  ( K ... ( N  -  1 ) )  <->  ( k  +  1 )  e.  ( ( K  +  1 ) ... ( ( N  -  1 )  +  1 ) ) ) )
3217, 31mpbid 202 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( ( K  + 
1 ) ... (
( N  -  1 )  +  1 ) ) )
33 zcn 10220 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
34 ax-1cn 8982 . . . . . . . . 9  |-  1  e.  CC
35 npcan 9247 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
3633, 34, 35sylancl 644 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
( N  -  1 )  +  1 )  =  N )
3723, 36syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( ( N  -  1 )  +  1 )  =  N )
3837oveq2d 6037 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( ( K  +  1 ) ... ( ( N  -  1 )  +  1 ) )  =  ( ( K  + 
1 ) ... N
) )
3932, 38eleqtrd 2464 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
40 sermono.4 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  0  <_  ( F `  x
) )
4140ralrimiva 2733 . . . . . 6  |-  ( ph  ->  A. x  e.  ( ( K  +  1 ) ... N ) 0  <_  ( F `  x ) )
4241adantr 452 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  A. x  e.  ( ( K  + 
1 ) ... N
) 0  <_  ( F `  x )
)
43 fveq2 5669 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
4443breq2d 4166 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
0  <_  ( F `  x )  <->  0  <_  ( F `  ( k  +  1 ) ) ) )
4544rspcv 2992 . . . . 5  |-  ( ( k  +  1 )  e.  ( ( K  +  1 ) ... N )  ->  ( A. x  e.  (
( K  +  1 ) ... N ) 0  <_  ( F `  x )  ->  0  <_  ( F `  (
k  +  1 ) ) ) )
4639, 42, 45sylc 58 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  0  <_  ( F `  ( k  +  1 ) ) )
47 fzelp1 11032 . . . . . . . 8  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  k  e.  ( K ... (
( N  -  1 )  +  1 ) ) )
4847adantl 453 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... ( ( N  -  1 )  +  1 ) ) )
4937oveq2d 6037 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( K ... ( ( N  - 
1 )  +  1 ) )  =  ( K ... N ) )
5048, 49eleqtrd 2464 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... N ) )
5150, 16syldan 457 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  k )  e.  RR )
52 fzss1 11024 . . . . . . . 8  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( K ... N )  C_  ( M ... N ) )
5318, 52syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( K ... N )  C_  ( M ... N ) )
54 fzp1elp1 11033 . . . . . . . . 9  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  (
k  +  1 )  e.  ( K ... ( ( N  - 
1 )  +  1 ) ) )
5554adantl 453 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( K ... (
( N  -  1 )  +  1 ) ) )
5655, 49eleqtrd 2464 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( K ... N
) )
5753, 56sseldd 3293 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( M ... N
) )
5811ralrimiva 2733 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( M ... N ) ( F `  x
)  e.  RR )
5958adantr 452 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  A. x  e.  ( M ... N
) ( F `  x )  e.  RR )
6043eleq1d 2454 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  (
( F `  x
)  e.  RR  <->  ( F `  ( k  +  1 ) )  e.  RR ) )
6160rspcv 2992 . . . . . 6  |-  ( ( k  +  1 )  e.  ( M ... N )  ->  ( A. x  e.  ( M ... N ) ( F `  x )  e.  RR  ->  ( F `  ( k  +  1 ) )  e.  RR ) )
6257, 59, 61sylc 58 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( F `  ( k  +  1 ) )  e.  RR )
6351, 62addge01d 9547 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( 0  <_  ( F `  ( k  +  1 ) )  <->  (  seq  M (  +  ,  F
) `  k )  <_  ( (  seq  M
(  +  ,  F
) `  k )  +  ( F `  ( k  +  1 ) ) ) ) )
6446, 63mpbid 202 . . 3  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  k )  <_  ( (  seq  M
(  +  ,  F
) `  k )  +  ( F `  ( k  +  1 ) ) ) )
6550, 5syldan 457 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( ZZ>= `  M )
)
66 seqp1 11266 . . . 4  |-  ( k  e.  ( ZZ>= `  M
)  ->  (  seq  M (  +  ,  F
) `  ( k  +  1 ) )  =  ( (  seq 
M (  +  ,  F ) `  k
)  +  ( F `
 ( k  +  1 ) ) ) )
6765, 66syl 16 . . 3  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  ( k  +  1 ) )  =  ( (  seq 
M (  +  ,  F ) `  k
)  +  ( F `
 ( k  +  1 ) ) ) )
6864, 67breqtrrd 4180 . 2  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  k )  <_  (  seq  M (  +  ,  F ) `
 ( k  +  1 ) ) )
691, 16, 68monoord 11281 1  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 K )  <_ 
(  seq  M (  +  ,  F ) `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650    C_ wss 3264   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   CCcc 8922   RRcr 8923   0cc0 8924   1c1 8925    + caddc 8927    <_ cle 9055    - cmin 9224   ZZcz 10215   ZZ>=cuz 10421   ...cfz 10976    seq cseq 11251
This theorem is referenced by:  cvgcmp  12523  isumsup2  12554  climcnds  12559  ovolunlem1a  19260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-seq 11252
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