MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  emcllem5 Unicode version

Theorem emcllem5 20293
Description: Lemma for emcl 20296. The partial sums of the series  T, which is used in the definition df-em 20287, is in fact the same as  G. (Contributed by Mario Carneiro, 11-Jul-2014.)
Hypotheses
Ref Expression
emcl.1  |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m
)  -  ( log `  n ) ) )
emcl.2  |-  G  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m
)  -  ( log `  ( n  +  1 ) ) ) )
emcl.3  |-  H  =  ( n  e.  NN  |->  ( log `  ( 1  +  ( 1  /  n ) ) ) )
emcl.4  |-  T  =  ( n  e.  NN  |->  ( ( 1  /  n )  -  ( log `  ( 1  +  ( 1  /  n
) ) ) ) )
Assertion
Ref Expression
emcllem5  |-  G  =  seq  1 (  +  ,  T )
Distinct variable groups:    m, H    m, n, T
Allowed substitution hints:    F( m, n)    G( m, n)    H( n)

Proof of Theorem emcllem5
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elfznn 10819 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  m  e.  NN )
21adantl 452 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  NN )
32nncnd 9762 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  CC )
4 ax-1cn 8795 . . . . . . . . . . . 12  |-  1  e.  CC
54a1i 10 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  1  e.  CC )
62nnne0d 9790 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  =/=  0
)
73, 5, 3, 6divdird 9574 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( m  +  1 )  /  m )  =  ( ( m  /  m
)  +  ( 1  /  m ) ) )
83, 6dividd 9534 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  /  m )  =  1 )
98oveq1d 5873 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( m  /  m )  +  ( 1  /  m
) )  =  ( 1  +  ( 1  /  m ) ) )
107, 9eqtrd 2315 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( m  +  1 )  /  m )  =  ( 1  +  ( 1  /  m ) ) )
1110fveq2d 5529 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
( m  +  1 )  /  m ) )  =  ( log `  ( 1  +  ( 1  /  m ) ) ) )
12 peano2nn 9758 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  (
m  +  1 )  e.  NN )
132, 12syl 15 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  + 
1 )  e.  NN )
1413nnrpd 10389 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  + 
1 )  e.  RR+ )
152nnrpd 10389 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  RR+ )
1614, 15relogdivd 19977 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
( m  +  1 )  /  m ) )  =  ( ( log `  ( m  +  1 ) )  -  ( log `  m
) ) )
1711, 16eqtr3d 2317 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
1  +  ( 1  /  m ) ) )  =  ( ( log `  ( m  +  1 ) )  -  ( log `  m
) ) )
1817sumeq2dv 12176 . . . . . 6  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( log `  (
1  +  ( 1  /  m ) ) )  =  sum_ m  e.  ( 1 ... n
) ( ( log `  ( m  +  1 ) )  -  ( log `  m ) ) )
19 fveq2 5525 . . . . . . 7  |-  ( x  =  m  ->  ( log `  x )  =  ( log `  m
) )
20 fveq2 5525 . . . . . . 7  |-  ( x  =  ( m  + 
1 )  ->  ( log `  x )  =  ( log `  (
m  +  1 ) ) )
21 fveq2 5525 . . . . . . 7  |-  ( x  =  1  ->  ( log `  x )  =  ( log `  1
) )
22 fveq2 5525 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  ( log `  x )  =  ( log `  (
n  +  1 ) ) )
23 nnz 10045 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  ZZ )
24 peano2nn 9758 . . . . . . . 8  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  NN )
25 nnuz 10263 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
2624, 25syl6eleq 2373 . . . . . . 7  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  ( ZZ>= `  1
) )
27 elfznn 10819 . . . . . . . . . . 11  |-  ( x  e.  ( 1 ... ( n  +  1 ) )  ->  x  e.  NN )
2827adantl 452 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  x  e.  NN )
2928nnrpd 10389 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  x  e.  RR+ )
3029relogcld 19974 . . . . . . . 8  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  ( log `  x
)  e.  RR )
3130recnd 8861 . . . . . . 7  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  ( log `  x
)  e.  CC )
3219, 20, 21, 22, 23, 26, 31fsumtscop2 12263 . . . . . 6  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( ( log `  ( m  +  1 ) )  -  ( log `  m ) )  =  ( ( log `  ( n  +  1 ) )  -  ( log `  1 ) ) )
33 log1 19939 . . . . . . . 8  |-  ( log `  1 )  =  0
3433oveq2i 5869 . . . . . . 7  |-  ( ( log `  ( n  +  1 ) )  -  ( log `  1
) )  =  ( ( log `  (
n  +  1 ) )  -  0 )
3524nnrpd 10389 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  RR+ )
3635relogcld 19974 . . . . . . . . 9  |-  ( n  e.  NN  ->  ( log `  ( n  + 
1 ) )  e.  RR )
3736recnd 8861 . . . . . . . 8  |-  ( n  e.  NN  ->  ( log `  ( n  + 
1 ) )  e.  CC )
3837subid1d 9146 . . . . . . 7  |-  ( n  e.  NN  ->  (
( log `  (
n  +  1 ) )  -  0 )  =  ( log `  (
n  +  1 ) ) )
3934, 38syl5eq 2327 . . . . . 6  |-  ( n  e.  NN  ->  (
( log `  (
n  +  1 ) )  -  ( log `  1 ) )  =  ( log `  (
n  +  1 ) ) )
4018, 32, 393eqtrd 2319 . . . . 5  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( log `  (
1  +  ( 1  /  m ) ) )  =  ( log `  ( n  +  1 ) ) )
4140oveq2d 5874 . . . 4  |-  ( n  e.  NN  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  sum_ m  e.  ( 1 ... n ) ( log `  (
1  +  ( 1  /  m ) ) ) )  =  (
sum_ m  e.  (
1 ... n ) ( 1  /  m )  -  ( log `  (
n  +  1 ) ) ) )
42 fzfid 11035 . . . . . 6  |-  ( n  e.  NN  ->  (
1 ... n )  e. 
Fin )
432nnrecred 9791 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  RR )
4443recnd 8861 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  CC )
45 1rp 10358 . . . . . . . . 9  |-  1  e.  RR+
4615rpreccld 10400 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  RR+ )
47 rpaddcl 10374 . . . . . . . . 9  |-  ( ( 1  e.  RR+  /\  (
1  /  m )  e.  RR+ )  ->  (
1  +  ( 1  /  m ) )  e.  RR+ )
4845, 46, 47sylancr 644 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  +  ( 1  /  m
) )  e.  RR+ )
4948relogcld 19974 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
1  +  ( 1  /  m ) ) )  e.  RR )
5049recnd 8861 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
1  +  ( 1  /  m ) ) )  e.  CC )
5142, 44, 50fsumsub 12250 . . . . 5  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  =  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  sum_ m  e.  ( 1 ... n ) ( log `  ( 1  +  ( 1  /  m ) ) ) ) )
52 oveq2 5866 . . . . . . . . 9  |-  ( n  =  m  ->  (
1  /  n )  =  ( 1  /  m ) )
5352oveq2d 5874 . . . . . . . . . 10  |-  ( n  =  m  ->  (
1  +  ( 1  /  n ) )  =  ( 1  +  ( 1  /  m
) ) )
5453fveq2d 5529 . . . . . . . . 9  |-  ( n  =  m  ->  ( log `  ( 1  +  ( 1  /  n
) ) )  =  ( log `  (
1  +  ( 1  /  m ) ) ) )
5552, 54oveq12d 5876 . . . . . . . 8  |-  ( n  =  m  ->  (
( 1  /  n
)  -  ( log `  ( 1  +  ( 1  /  n ) ) ) )  =  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m
) ) ) ) )
56 emcl.4 . . . . . . . 8  |-  T  =  ( n  e.  NN  |->  ( ( 1  /  n )  -  ( log `  ( 1  +  ( 1  /  n
) ) ) ) )
57 ovex 5883 . . . . . . . 8  |-  ( ( 1  /  m )  -  ( log `  (
1  +  ( 1  /  m ) ) ) )  e.  _V
5855, 56, 57fvmpt 5602 . . . . . . 7  |-  ( m  e.  NN  ->  ( T `  m )  =  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) ) )
592, 58syl 15 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( T `  m )  =  ( ( 1  /  m
)  -  ( log `  ( 1  +  ( 1  /  m ) ) ) ) )
60 id 19 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  NN )
6160, 25syl6eleq 2373 . . . . . 6  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
6243, 49resubcld 9211 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  e.  RR )
6362recnd 8861 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  e.  CC )
6459, 61, 63fsumser 12203 . . . . 5  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  =  (  seq  1 (  +  ,  T ) `  n
) )
6551, 64eqtr3d 2317 . . . 4  |-  ( n  e.  NN  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  sum_ m  e.  ( 1 ... n ) ( log `  (
1  +  ( 1  /  m ) ) ) )  =  (  seq  1 (  +  ,  T ) `  n ) )
6641, 65eqtr3d 2317 . . 3  |-  ( n  e.  NN  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  (
n  +  1 ) ) )  =  (  seq  1 (  +  ,  T ) `  n ) )
6766mpteq2ia 4102 . 2  |-  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 ) ) ) )  =  ( n  e.  NN  |->  (  seq  1 (  +  ,  T ) `  n ) )
68 emcl.2 . 2  |-  G  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m
)  -  ( log `  ( n  +  1 ) ) ) )
69 1z 10053 . . . . 5  |-  1  e.  ZZ
70 seqfn 11058 . . . . 5  |-  ( 1  e.  ZZ  ->  seq  1 (  +  ,  T )  Fn  ( ZZ>=
`  1 ) )
7169, 70ax-mp 8 . . . 4  |-  seq  1
(  +  ,  T
)  Fn  ( ZZ>= ` 
1 )
7225fneq2i 5339 . . . 4  |-  (  seq  1 (  +  ,  T )  Fn  NN  <->  seq  1 (  +  ,  T )  Fn  ( ZZ>=
`  1 ) )
7371, 72mpbir 200 . . 3  |-  seq  1
(  +  ,  T
)  Fn  NN
74 dffn5 5568 . . 3  |-  (  seq  1 (  +  ,  T )  Fn  NN  <->  seq  1 (  +  ,  T )  =  ( n  e.  NN  |->  (  seq  1 (  +  ,  T ) `  n ) ) )
7573, 74mpbi 199 . 2  |-  seq  1
(  +  ,  T
)  =  ( n  e.  NN  |->  (  seq  1 (  +  ,  T ) `  n
) )
7667, 68, 753eqtr4i 2313 1  |-  G  =  seq  1 (  +  ,  T )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    - cmin 9037    / cdiv 9423   NNcn 9746   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   ...cfz 10782    seq cseq 11046   sum_csu 12158   logclog 19912
This theorem is referenced by:  emcllem6  20294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914
  Copyright terms: Public domain W3C validator