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Theorem isermono 9906
Description: The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
Hypotheses
Ref Expression
isermono.1  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
isermono.2  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
isermono.3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  RR )
isermono.4  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  0  <_  ( F `  x
) )
Assertion
Ref Expression
isermono  |-  ( ph  ->  (  seq M (  +  ,  F ,  RR ) `  K )  <_  (  seq M
(  +  ,  F ,  RR ) `  N
) )
Distinct variable groups:    x, F    x, K    x, M    x, N    ph, x

Proof of Theorem isermono
Dummy variables  k  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isermono.2 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
2 elfzuz 9436 . . . 4  |-  ( k  e.  ( K ... N )  ->  k  e.  ( ZZ>= `  K )
)
3 isermono.1 . . . 4  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
4 uztrn 9035 . . . 4  |-  ( ( k  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
52, 3, 4syl2anr 284 . . 3  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  k  e.  ( ZZ>= `  M )
)
6 isermono.3 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  RR )
76adantlr 461 . . 3  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  RR )
8 readdcl 7468 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
98adantl 271 . . 3  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  ->  ( x  +  y )  e.  RR )
105, 7, 9iseqcl 9881 . 2  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  (  seq M (  +  ,  F ,  RR ) `  k )  e.  RR )
11 fveq2 5305 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
1211breq2d 3857 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
0  <_  ( F `  x )  <->  0  <_  ( F `  ( k  +  1 ) ) ) )
13 isermono.4 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  0  <_  ( F `  x
) )
1413ralrimiva 2446 . . . . . 6  |-  ( ph  ->  A. x  e.  ( ( K  +  1 ) ... N ) 0  <_  ( F `  x ) )
1514adantr 270 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  A. x  e.  ( ( K  + 
1 ) ... N
) 0  <_  ( F `  x )
)
16 simpr 108 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... ( N  -  1 ) ) )
173adantr 270 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  K  e.  ( ZZ>= `  M )
)
18 eluzelz 9028 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
1917, 18syl 14 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  K  e.  ZZ )
201adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  N  e.  ( ZZ>= `  K )
)
21 eluzelz 9028 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ZZ )
2220, 21syl 14 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  N  e.  ZZ )
23 peano2zm 8788 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
2422, 23syl 14 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( N  -  1 )  e.  ZZ )
25 elfzelz 9440 . . . . . . . . 9  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  k  e.  ZZ )
2625adantl 271 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ZZ )
27 1zzd 8777 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  1  e.  ZZ )
28 fzaddel 9473 . . . . . . . 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 ) ) ) )
2919, 24, 26, 27, 28syl22anc 1175 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  e.  ( K ... ( N  -  1 ) )  <->  ( k  +  1 )  e.  ( ( K  +  1 ) ... ( ( N  -  1 )  +  1 ) ) ) )
3016, 29mpbid 145 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( ( K  + 
1 ) ... (
( N  -  1 )  +  1 ) ) )
31 zcn 8755 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
32 ax-1cn 7438 . . . . . . . . 9  |-  1  e.  CC
33 npcan 7691 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
3431, 32, 33sylancl 404 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
( N  -  1 )  +  1 )  =  N )
3522, 34syl 14 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( ( N  -  1 )  +  1 )  =  N )
3635oveq2d 5668 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( ( K  +  1 ) ... ( ( N  -  1 )  +  1 ) )  =  ( ( K  + 
1 ) ... N
) )
3730, 36eleqtrd 2166 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
3812, 15, 37rspcdva 2727 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  0  <_  ( F `  ( k  +  1 ) ) )
39 fzelp1 9488 . . . . . . . 8  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  k  e.  ( K ... (
( N  -  1 )  +  1 ) ) )
4039adantl 271 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... ( ( N  -  1 )  +  1 ) ) )
4135oveq2d 5668 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( K ... ( ( N  - 
1 )  +  1 ) )  =  ( K ... N ) )
4240, 41eleqtrd 2166 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... N ) )
4342, 10syldan 276 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq M (  +  ,  F ,  RR ) `  k )  e.  RR )
4411eleq1d 2156 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
( F `  x
)  e.  RR  <->  ( F `  ( k  +  1 ) )  e.  RR ) )
456ralrimiva 2446 . . . . . . 7  |-  ( ph  ->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  RR )
4645adantr 270 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  A. x  e.  ( ZZ>= `  M )
( F `  x
)  e.  RR )
47 fzss1 9477 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( K ... N )  C_  ( M ... N ) )
4817, 47syl 14 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( K ... N )  C_  ( M ... N ) )
49 fzp1elp1 9489 . . . . . . . . . 10  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  (
k  +  1 )  e.  ( K ... ( ( N  - 
1 )  +  1 ) ) )
5049adantl 271 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( K ... (
( N  -  1 )  +  1 ) ) )
5150, 41eleqtrd 2166 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( K ... N
) )
5248, 51sseldd 3026 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( M ... N
) )
53 elfzuz 9436 . . . . . . 7  |-  ( ( k  +  1 )  e.  ( M ... N )  ->  (
k  +  1 )  e.  ( ZZ>= `  M
) )
5452, 53syl 14 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( ZZ>= `  M )
)
5544, 46, 54rspcdva 2727 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( F `  ( k  +  1 ) )  e.  RR )
5643, 55addge01d 8010 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( 0  <_  ( F `  ( k  +  1 ) )  <->  (  seq M (  +  ,  F ,  RR ) `  k )  <_  (
(  seq M (  +  ,  F ,  RR ) `  k )  +  ( F `  ( k  +  1 ) ) ) ) )
5738, 56mpbid 145 . . 3  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq M (  +  ,  F ,  RR ) `  k )  <_  (
(  seq M (  +  ,  F ,  RR ) `  k )  +  ( F `  ( k  +  1 ) ) ) )
5842, 5syldan 276 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( ZZ>= `  M )
)
596adantlr 461 . . . 4  |-  ( ( ( ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  RR )
608adantl 271 . . . 4  |-  ( ( ( ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  ->  ( x  +  y )  e.  RR )
6158, 59, 60iseqp1 9882 . . 3  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq M (  +  ,  F ,  RR ) `  ( k  +  1 ) )  =  ( (  seq M (  +  ,  F ,  RR ) `  k )  +  ( F `  ( k  +  1 ) ) ) )
6257, 61breqtrrd 3871 . 2  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq M (  +  ,  F ,  RR ) `  k )  <_  (  seq M (  +  ,  F ,  RR ) `  ( k  +  1 ) ) )
631, 10, 62monoord 9904 1  |-  ( ph  ->  (  seq M (  +  ,  F ,  RR ) `  K )  <_  (  seq M
(  +  ,  F ,  RR ) `  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359    C_ wss 2999   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   CCcc 7348   RRcr 7349   0cc0 7350   1c1 7351    + caddc 7353    <_ cle 7523    - cmin 7653   ZZcz 8750   ZZ>=cuz 9019   ...cfz 9424    seqcseq4 9851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-ltadd 7461
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-inn 8423  df-n0 8674  df-z 8751  df-uz 9020  df-fz 9425  df-iseq 9853
This theorem is referenced by: (None)
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