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Theorem sermono 11094
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 10810 . . . 4  |-  ( k  e.  ( K ... N )  ->  k  e.  ( ZZ>= `  K )
)
3 sermono.1 . . . 4  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
4 uztrn 10260 . . . 4  |-  ( ( k  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
52, 3, 4syl2anr 464 . . 3  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  k  e.  ( ZZ>= `  M )
)
6 elfzuz3 10811 . . . . . . 7  |-  ( k  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  k )
)
76adantl 452 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  N  e.  ( ZZ>= `  k )
)
8 fzss2 10847 . . . . . 6  |-  ( N  e.  ( ZZ>= `  k
)  ->  ( M ... k )  C_  ( M ... N ) )
97, 8syl 15 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  ( M ... k )  C_  ( M ... N ) )
109sselda 3193 . . . 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 695 . . . 4  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  x  e.  ( M ... N
) )  ->  ( F `  x )  e.  RR )
1310, 12syldan 456 . . 3  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  x  e.  ( M ... k
) )  ->  ( F `  x )  e.  RR )
14 readdcl 8836 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
1514adantl 452 . . 3  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  ->  ( x  +  y )  e.  RR )
165, 13, 15seqcl 11082 . 2  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  (  seq  M (  +  ,  F
) `  k )  e.  RR )
17 simpr 447 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... ( N  -  1 ) ) )
183adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  K  e.  ( ZZ>= `  M )
)
19 eluzelz 10254 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
2018, 19syl 15 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  K  e.  ZZ )
211adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  N  e.  ( ZZ>= `  K )
)
22 eluzelz 10254 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ZZ )
2321, 22syl 15 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  N  e.  ZZ )
24 peano2zm 10078 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
2523, 24syl 15 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( N  -  1 )  e.  ZZ )
26 elfzelz 10814 . . . . . . . . 9  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  k  e.  ZZ )
2726adantl 452 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ZZ )
28 1z 10069 . . . . . . . . 9  |-  1  e.  ZZ
2928a1i 10 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  1  e.  ZZ )
30 fzaddel 10842 . . . . . . . 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 1183 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  e.  ( K ... ( N  -  1 ) )  <->  ( k  +  1 )  e.  ( ( K  +  1 ) ... ( ( N  -  1 )  +  1 ) ) ) )
3217, 31mpbid 201 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( ( K  + 
1 ) ... (
( N  -  1 )  +  1 ) ) )
33 zcn 10045 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
34 ax-1cn 8811 . . . . . . . . 9  |-  1  e.  CC
35 npcan 9076 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
3633, 34, 35sylancl 643 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
( N  -  1 )  +  1 )  =  N )
3723, 36syl 15 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( ( N  -  1 )  +  1 )  =  N )
3837oveq2d 5890 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( ( K  +  1 ) ... ( ( N  -  1 )  +  1 ) )  =  ( ( K  + 
1 ) ... N
) )
3932, 38eleqtrd 2372 . . . . 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 2639 . . . . . 6  |-  ( ph  ->  A. x  e.  ( ( K  +  1 ) ... N ) 0  <_  ( F `  x ) )
4241adantr 451 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  A. x  e.  ( ( K  + 
1 ) ... N
) 0  <_  ( F `  x )
)
43 fveq2 5541 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
4443breq2d 4051 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
0  <_  ( F `  x )  <->  0  <_  ( F `  ( k  +  1 ) ) ) )
4544rspcv 2893 . . . . 5  |-  ( ( k  +  1 )  e.  ( ( K  +  1 ) ... N )  ->  ( A. x  e.  (
( K  +  1 ) ... N ) 0  <_  ( F `  x )  ->  0  <_  ( F `  (
k  +  1 ) ) ) )
4639, 42, 45sylc 56 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  0  <_  ( F `  ( k  +  1 ) ) )
47 fzelp1 10854 . . . . . . . 8  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  k  e.  ( K ... (
( N  -  1 )  +  1 ) ) )
4847adantl 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... ( ( N  -  1 )  +  1 ) ) )
4937oveq2d 5890 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( K ... ( ( N  - 
1 )  +  1 ) )  =  ( K ... N ) )
5048, 49eleqtrd 2372 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... N ) )
5150, 16syldan 456 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  k )  e.  RR )
52 fzss1 10846 . . . . . . . 8  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( K ... N )  C_  ( M ... N ) )
5318, 52syl 15 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( K ... N )  C_  ( M ... N ) )
54 fzp1elp1 10855 . . . . . . . . 9  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  (
k  +  1 )  e.  ( K ... ( ( N  - 
1 )  +  1 ) ) )
5554adantl 452 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( K ... (
( N  -  1 )  +  1 ) ) )
5655, 49eleqtrd 2372 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( K ... N
) )
5753, 56sseldd 3194 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( M ... N
) )
5811ralrimiva 2639 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( M ... N ) ( F `  x
)  e.  RR )
5958adantr 451 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  A. x  e.  ( M ... N
) ( F `  x )  e.  RR )
6043eleq1d 2362 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  (
( F `  x
)  e.  RR  <->  ( F `  ( k  +  1 ) )  e.  RR ) )
6160rspcv 2893 . . . . . 6  |-  ( ( k  +  1 )  e.  ( M ... N )  ->  ( A. x  e.  ( M ... N ) ( F `  x )  e.  RR  ->  ( F `  ( k  +  1 ) )  e.  RR ) )
6257, 59, 61sylc 56 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( F `  ( k  +  1 ) )  e.  RR )
6351, 62addge01d 9376 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( 0  <_  ( F `  ( k  +  1 ) )  <->  (  seq  M (  +  ,  F
) `  k )  <_  ( (  seq  M
(  +  ,  F
) `  k )  +  ( F `  ( k  +  1 ) ) ) ) )
6446, 63mpbid 201 . . 3  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  k )  <_  ( (  seq  M
(  +  ,  F
) `  k )  +  ( F `  ( k  +  1 ) ) ) )
6550, 5syldan 456 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( ZZ>= `  M )
)
66 seqp1 11077 . . . 4  |-  ( k  e.  ( ZZ>= `  M
)  ->  (  seq  M (  +  ,  F
) `  ( k  +  1 ) )  =  ( (  seq 
M (  +  ,  F ) `  k
)  +  ( F `
 ( k  +  1 ) ) ) )
6765, 66syl 15 . . 3  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  ( k  +  1 ) )  =  ( (  seq 
M (  +  ,  F ) `  k
)  +  ( F `
 ( k  +  1 ) ) ) )
6864, 67breqtrrd 4065 . 2  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  k )  <_  (  seq  M (  +  ,  F ) `
 ( k  +  1 ) ) )
691, 16, 68monoord 11092 1  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 K )  <_ 
(  seq  M (  +  ,  F ) `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    <_ cle 8884    - cmin 9053   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062
This theorem is referenced by:  cvgcmp  12290  isumsup2  12321  climcnds  12326  ovolunlem1a  18871
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063
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