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Theorem ser3mono 10282
Description: The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
Hypotheses
Ref Expression
sermono.1  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
sermono.2  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
ser3mono.3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  RR )
sermono.4  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  0  <_  ( F `  x
) )
Assertion
Ref Expression
ser3mono  |-  ( 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 ser3mono
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 eqid 2140 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
3 sermono.1 . . . . . 6  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
4 eluzel2 9355 . . . . . 6  |-  ( K  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
53, 4syl 14 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
65adantr 274 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  M  e.  ZZ )
7 ser3mono.3 . . . . 5  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  RR )
87adantlr 469 . . . 4  |-  ( ( ( ph  /\  k  e.  ( K ... N
) )  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  RR )
92, 6, 8serfre 10279 . . 3  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  seq M (  +  ,  F ) : ( ZZ>= `  M
) --> RR )
10 elfzuz 9833 . . . 4  |-  ( k  e.  ( K ... N )  ->  k  e.  ( ZZ>= `  K )
)
11 uztrn 9366 . . . 4  |-  ( ( k  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
1210, 3, 11syl2anr 288 . . 3  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  k  e.  ( ZZ>= `  M )
)
139, 12ffvelrnd 5564 . 2  |-  ( (
ph  /\  k  e.  ( K ... N ) )  ->  (  seq M (  +  ,  F ) `  k
)  e.  RR )
14 fveq2 5429 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
1514breq2d 3949 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
0  <_  ( F `  x )  <->  0  <_  ( F `  ( k  +  1 ) ) ) )
16 sermono.4 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  0  <_  ( F `  x
) )
1716ralrimiva 2508 . . . . . 6  |-  ( ph  ->  A. x  e.  ( ( K  +  1 ) ... N ) 0  <_  ( F `  x ) )
1817adantr 274 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  A. x  e.  ( ( K  + 
1 ) ... N
) 0  <_  ( F `  x )
)
19 simpr 109 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... ( N  -  1 ) ) )
203adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  K  e.  ( ZZ>= `  M )
)
21 eluzelz 9359 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
2220, 21syl 14 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  K  e.  ZZ )
231adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  N  e.  ( ZZ>= `  K )
)
24 eluzelz 9359 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ZZ )
2523, 24syl 14 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  N  e.  ZZ )
26 peano2zm 9116 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
2725, 26syl 14 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( N  -  1 )  e.  ZZ )
28 elfzelz 9837 . . . . . . . . 9  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  k  e.  ZZ )
2928adantl 275 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ZZ )
30 1zzd 9105 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  1  e.  ZZ )
31 fzaddel 9870 . . . . . . . 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 ) ) ) )
3222, 27, 29, 30, 31syl22anc 1218 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  e.  ( K ... ( N  -  1 ) )  <->  ( k  +  1 )  e.  ( ( K  +  1 ) ... ( ( N  -  1 )  +  1 ) ) ) )
3319, 32mpbid 146 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( ( K  + 
1 ) ... (
( N  -  1 )  +  1 ) ) )
34 zcn 9083 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
35 ax-1cn 7737 . . . . . . . . 9  |-  1  e.  CC
36 npcan 7995 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
3734, 35, 36sylancl 410 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
( N  -  1 )  +  1 )  =  N )
3825, 37syl 14 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( ( N  -  1 )  +  1 )  =  N )
3938oveq2d 5798 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( ( K  +  1 ) ... ( ( N  -  1 )  +  1 ) )  =  ( ( K  + 
1 ) ... N
) )
4033, 39eleqtrd 2219 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
4115, 18, 40rspcdva 2798 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  0  <_  ( F `  ( k  +  1 ) ) )
42 fzelp1 9885 . . . . . . . 8  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  k  e.  ( K ... (
( N  -  1 )  +  1 ) ) )
4342adantl 275 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... ( ( N  -  1 )  +  1 ) ) )
4438oveq2d 5798 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( K ... ( ( N  - 
1 )  +  1 ) )  =  ( K ... N ) )
4543, 44eleqtrd 2219 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( K ... N ) )
4645, 13syldan 280 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq M (  +  ,  F ) `  k
)  e.  RR )
4714eleq1d 2209 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
( F `  x
)  e.  RR  <->  ( F `  ( k  +  1 ) )  e.  RR ) )
487ralrimiva 2508 . . . . . . 7  |-  ( ph  ->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  RR )
4948adantr 274 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  A. x  e.  ( ZZ>= `  M )
( F `  x
)  e.  RR )
50 fzss1 9874 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( K ... N )  C_  ( M ... N ) )
5120, 50syl 14 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( K ... N )  C_  ( M ... N ) )
52 fzp1elp1 9886 . . . . . . . . . 10  |-  ( k  e.  ( K ... ( N  -  1
) )  ->  (
k  +  1 )  e.  ( K ... ( ( N  - 
1 )  +  1 ) ) )
5352adantl 275 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( K ... (
( N  -  1 )  +  1 ) ) )
5453, 44eleqtrd 2219 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( K ... N
) )
5551, 54sseldd 3103 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( M ... N
) )
56 elfzuz 9833 . . . . . . 7  |-  ( ( k  +  1 )  e.  ( M ... N )  ->  (
k  +  1 )  e.  ( ZZ>= `  M
) )
5755, 56syl 14 . . . . . 6  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( k  +  1 )  e.  ( ZZ>= `  M )
)
5847, 49, 57rspcdva 2798 . . . . 5  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( F `  ( k  +  1 ) )  e.  RR )
5946, 58addge01d 8319 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  ( 0  <_  ( F `  ( k  +  1 ) )  <->  (  seq M (  +  ,  F ) `  k
)  <_  ( (  seq M (  +  ,  F ) `  k
)  +  ( F `
 ( k  +  1 ) ) ) ) )
6041, 59mpbid 146 . . 3  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq M (  +  ,  F ) `  k
)  <_  ( (  seq M (  +  ,  F ) `  k
)  +  ( F `
 ( k  +  1 ) ) ) )
6145, 12syldan 280 . . . 4  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  k  e.  ( ZZ>= `  M )
)
627adantlr 469 . . . 4  |-  ( ( ( ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  RR )
63 readdcl 7770 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
6463adantl 275 . . . 4  |-  ( ( ( ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  ->  ( x  +  y )  e.  RR )
6561, 62, 64seq3p1 10266 . . 3  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq M (  +  ,  F ) `  (
k  +  1 ) )  =  ( (  seq M (  +  ,  F ) `  k )  +  ( F `  ( k  +  1 ) ) ) )
6660, 65breqtrrd 3964 . 2  |-  ( (
ph  /\  k  e.  ( K ... ( N  -  1 ) ) )  ->  (  seq M (  +  ,  F ) `  k
)  <_  (  seq M (  +  ,  F ) `  (
k  +  1 ) ) )
671, 13, 66monoord 10280 1  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 K )  <_ 
(  seq M (  +  ,  F ) `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   A.wral 2417    C_ wss 3076   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   CCcc 7642   RRcr 7643   0cc0 7644   1c1 7645    + caddc 7647    <_ cle 7825    - cmin 7957   ZZcz 9078   ZZ>=cuz 9350   ...cfz 9821    seqcseq 10249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-ltadd 7760
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-fz 9822  df-seqfrec 10250
This theorem is referenced by: (None)
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