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Theorem emcllem5 23530
Description: Lemma for emcl 23533. The partial sums of the series  T, which is used in the definition df-em 23523, 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 11717 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  m  e.  NN )
21adantl 464 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  NN )
32nncnd 10547 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  CC )
4 1cnd 9601 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  1  e.  CC )
52nnne0d 10576 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  =/=  0
)
63, 4, 3, 5divdird 10354 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( m  +  1 )  /  m )  =  ( ( m  /  m
)  +  ( 1  /  m ) ) )
73, 5dividd 10314 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  /  m )  =  1 )
87oveq1d 6285 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( m  /  m )  +  ( 1  /  m
) )  =  ( 1  +  ( 1  /  m ) ) )
96, 8eqtrd 2495 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( m  +  1 )  /  m )  =  ( 1  +  ( 1  /  m ) ) )
109fveq2d 5852 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
( m  +  1 )  /  m ) )  =  ( log `  ( 1  +  ( 1  /  m ) ) ) )
11 peano2nn 10543 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  (
m  +  1 )  e.  NN )
122, 11syl 16 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  + 
1 )  e.  NN )
1312nnrpd 11257 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  + 
1 )  e.  RR+ )
142nnrpd 11257 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  RR+ )
1513, 14relogdivd 23182 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
( m  +  1 )  /  m ) )  =  ( ( log `  ( m  +  1 ) )  -  ( log `  m
) ) )
1610, 15eqtr3d 2497 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
1  +  ( 1  /  m ) ) )  =  ( ( log `  ( m  +  1 ) )  -  ( log `  m
) ) )
1716sumeq2dv 13610 . . . . . 6  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( log `  (
1  +  ( 1  /  m ) ) )  =  sum_ m  e.  ( 1 ... n
) ( ( log `  ( m  +  1 ) )  -  ( log `  m ) ) )
18 fveq2 5848 . . . . . . 7  |-  ( x  =  m  ->  ( log `  x )  =  ( log `  m
) )
19 fveq2 5848 . . . . . . 7  |-  ( x  =  ( m  + 
1 )  ->  ( log `  x )  =  ( log `  (
m  +  1 ) ) )
20 fveq2 5848 . . . . . . 7  |-  ( x  =  1  ->  ( log `  x )  =  ( log `  1
) )
21 fveq2 5848 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  ( log `  x )  =  ( log `  (
n  +  1 ) ) )
22 nnz 10882 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  ZZ )
23 peano2nn 10543 . . . . . . . 8  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  NN )
24 nnuz 11117 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
2523, 24syl6eleq 2552 . . . . . . 7  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  ( ZZ>= `  1
) )
26 elfznn 11717 . . . . . . . . . . 11  |-  ( x  e.  ( 1 ... ( n  +  1 ) )  ->  x  e.  NN )
2726adantl 464 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  x  e.  NN )
2827nnrpd 11257 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  x  e.  RR+ )
2928relogcld 23179 . . . . . . . 8  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  ( log `  x
)  e.  RR )
3029recnd 9611 . . . . . . 7  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  ( log `  x
)  e.  CC )
3118, 19, 20, 21, 22, 25, 30telfsum2 13704 . . . . . 6  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( ( log `  ( m  +  1 ) )  -  ( log `  m ) )  =  ( ( log `  ( n  +  1 ) )  -  ( log `  1 ) ) )
32 log1 23142 . . . . . . . 8  |-  ( log `  1 )  =  0
3332oveq2i 6281 . . . . . . 7  |-  ( ( log `  ( n  +  1 ) )  -  ( log `  1
) )  =  ( ( log `  (
n  +  1 ) )  -  0 )
3423nnrpd 11257 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  RR+ )
3534relogcld 23179 . . . . . . . . 9  |-  ( n  e.  NN  ->  ( log `  ( n  + 
1 ) )  e.  RR )
3635recnd 9611 . . . . . . . 8  |-  ( n  e.  NN  ->  ( log `  ( n  + 
1 ) )  e.  CC )
3736subid1d 9911 . . . . . . 7  |-  ( n  e.  NN  ->  (
( log `  (
n  +  1 ) )  -  0 )  =  ( log `  (
n  +  1 ) ) )
3833, 37syl5eq 2507 . . . . . 6  |-  ( n  e.  NN  ->  (
( log `  (
n  +  1 ) )  -  ( log `  1 ) )  =  ( log `  (
n  +  1 ) ) )
3917, 31, 383eqtrd 2499 . . . . 5  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( log `  (
1  +  ( 1  /  m ) ) )  =  ( log `  ( n  +  1 ) ) )
4039oveq2d 6286 . . . 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 ) ) ) )
41 fzfid 12068 . . . . . 6  |-  ( n  e.  NN  ->  (
1 ... n )  e. 
Fin )
422nnrecred 10577 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  RR )
4342recnd 9611 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  CC )
44 1rp 11225 . . . . . . . . 9  |-  1  e.  RR+
4514rpreccld 11269 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  RR+ )
46 rpaddcl 11242 . . . . . . . . 9  |-  ( ( 1  e.  RR+  /\  (
1  /  m )  e.  RR+ )  ->  (
1  +  ( 1  /  m ) )  e.  RR+ )
4744, 45, 46sylancr 661 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  +  ( 1  /  m
) )  e.  RR+ )
4847relogcld 23179 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
1  +  ( 1  /  m ) ) )  e.  RR )
4948recnd 9611 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
1  +  ( 1  /  m ) ) )  e.  CC )
5041, 43, 49fsumsub 13688 . . . . 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 ) ) ) ) )
51 oveq2 6278 . . . . . . . . 9  |-  ( n  =  m  ->  (
1  /  n )  =  ( 1  /  m ) )
5251oveq2d 6286 . . . . . . . . . 10  |-  ( n  =  m  ->  (
1  +  ( 1  /  n ) )  =  ( 1  +  ( 1  /  m
) ) )
5352fveq2d 5852 . . . . . . . . 9  |-  ( n  =  m  ->  ( log `  ( 1  +  ( 1  /  n
) ) )  =  ( log `  (
1  +  ( 1  /  m ) ) ) )
5451, 53oveq12d 6288 . . . . . . . 8  |-  ( n  =  m  ->  (
( 1  /  n
)  -  ( log `  ( 1  +  ( 1  /  n ) ) ) )  =  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m
) ) ) ) )
55 emcl.4 . . . . . . . 8  |-  T  =  ( n  e.  NN  |->  ( ( 1  /  n )  -  ( log `  ( 1  +  ( 1  /  n
) ) ) ) )
56 ovex 6298 . . . . . . . 8  |-  ( ( 1  /  m )  -  ( log `  (
1  +  ( 1  /  m ) ) ) )  e.  _V
5754, 55, 56fvmpt 5931 . . . . . . 7  |-  ( m  e.  NN  ->  ( T `  m )  =  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) ) )
582, 57syl 16 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( T `  m )  =  ( ( 1  /  m
)  -  ( log `  ( 1  +  ( 1  /  m ) ) ) ) )
59 id 22 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  NN )
6059, 24syl6eleq 2552 . . . . . 6  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
6142, 48resubcld 9983 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  e.  RR )
6261recnd 9611 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  e.  CC )
6358, 60, 62fsumser 13637 . . . . 5  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  =  (  seq 1 (  +  ,  T ) `  n
) )
6450, 63eqtr3d 2497 . . . 4  |-  ( n  e.  NN  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  sum_ m  e.  ( 1 ... n ) ( log `  (
1  +  ( 1  /  m ) ) ) )  =  (  seq 1 (  +  ,  T ) `  n ) )
6540, 64eqtr3d 2497 . . 3  |-  ( n  e.  NN  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  (
n  +  1 ) ) )  =  (  seq 1 (  +  ,  T ) `  n ) )
6665mpteq2ia 4521 . 2  |-  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 ) ) ) )  =  ( n  e.  NN  |->  (  seq 1 (  +  ,  T ) `  n ) )
67 emcl.2 . 2  |-  G  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m
)  -  ( log `  ( n  +  1 ) ) ) )
68 1z 10890 . . . . 5  |-  1  e.  ZZ
69 seqfn 12104 . . . . 5  |-  ( 1  e.  ZZ  ->  seq 1 (  +  ,  T )  Fn  ( ZZ>=
`  1 ) )
7068, 69ax-mp 5 . . . 4  |-  seq 1
(  +  ,  T
)  Fn  ( ZZ>= ` 
1 )
7124fneq2i 5658 . . . 4  |-  (  seq 1 (  +  ,  T )  Fn  NN  <->  seq 1 (  +  ,  T )  Fn  ( ZZ>=
`  1 ) )
7270, 71mpbir 209 . . 3  |-  seq 1
(  +  ,  T
)  Fn  NN
73 dffn5 5893 . . 3  |-  (  seq 1 (  +  ,  T )  Fn  NN  <->  seq 1 (  +  ,  T )  =  ( n  e.  NN  |->  (  seq 1 (  +  ,  T ) `  n ) ) )
7472, 73mpbi 208 . 2  |-  seq 1
(  +  ,  T
)  =  ( n  e.  NN  |->  (  seq 1 (  +  ,  T ) `  n
) )
7566, 67, 743eqtr4i 2493 1  |-  G  =  seq 1 (  +  ,  T )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398    e. wcel 1823    |-> cmpt 4497    Fn wfn 5565   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482    + caddc 9484    - cmin 9796    / cdiv 10202   NNcn 10531   ZZcz 10860   ZZ>=cuz 11082   RR+crp 11221   ...cfz 11675    seqcseq 12092   sum_csu 13593   logclog 23111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12093  df-exp 12152  df-fac 12339  df-bc 12366  df-hash 12391  df-shft 12985  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-limsup 13379  df-clim 13396  df-rlim 13397  df-sum 13594  df-ef 13888  df-sin 13890  df-cos 13891  df-pi 13893  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-hom 14811  df-cco 14812  df-rest 14915  df-topn 14916  df-0g 14934  df-gsum 14935  df-topgen 14936  df-pt 14937  df-prds 14940  df-xrs 14994  df-qtop 14999  df-imas 15000  df-xps 15002  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-mulg 16262  df-cntz 16557  df-cmn 17002  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-fbas 18614  df-fg 18615  df-cnfld 18619  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-cld 19690  df-ntr 19691  df-cls 19692  df-nei 19769  df-lp 19807  df-perf 19808  df-cn 19898  df-cnp 19899  df-haus 19986  df-tx 20232  df-hmeo 20425  df-fil 20516  df-fm 20608  df-flim 20609  df-flf 20610  df-xms 20992  df-ms 20993  df-tms 20994  df-cncf 21551  df-limc 22439  df-dv 22440  df-log 23113
This theorem is referenced by:  emcllem6  23531
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