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Theorem emcllem5 23912
Description: Lemma for emcl 23915. The partial sums of the series  T, which is used in the definition df-em 23905, 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 11829 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  m  e.  NN )
21adantl 467 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  NN )
32nncnd 10626 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  CC )
4 1cnd 9660 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  1  e.  CC )
52nnne0d 10655 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  =/=  0
)
63, 4, 3, 5divdird 10422 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( m  +  1 )  /  m )  =  ( ( m  /  m
)  +  ( 1  /  m ) ) )
73, 5dividd 10382 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  /  m )  =  1 )
87oveq1d 6317 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( m  /  m )  +  ( 1  /  m
) )  =  ( 1  +  ( 1  /  m ) ) )
96, 8eqtrd 2463 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( m  +  1 )  /  m )  =  ( 1  +  ( 1  /  m ) ) )
109fveq2d 5882 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
( m  +  1 )  /  m ) )  =  ( log `  ( 1  +  ( 1  /  m ) ) ) )
11 peano2nn 10622 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  (
m  +  1 )  e.  NN )
122, 11syl 17 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  + 
1 )  e.  NN )
1312nnrpd 11340 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  + 
1 )  e.  RR+ )
142nnrpd 11340 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  RR+ )
1513, 14relogdivd 23562 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
( m  +  1 )  /  m ) )  =  ( ( log `  ( m  +  1 ) )  -  ( log `  m
) ) )
1610, 15eqtr3d 2465 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
1  +  ( 1  /  m ) ) )  =  ( ( log `  ( m  +  1 ) )  -  ( log `  m
) ) )
1716sumeq2dv 13757 . . . . . 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 5878 . . . . . . 7  |-  ( x  =  m  ->  ( log `  x )  =  ( log `  m
) )
19 fveq2 5878 . . . . . . 7  |-  ( x  =  ( m  + 
1 )  ->  ( log `  x )  =  ( log `  (
m  +  1 ) ) )
20 fveq2 5878 . . . . . . 7  |-  ( x  =  1  ->  ( log `  x )  =  ( log `  1
) )
21 fveq2 5878 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  ( log `  x )  =  ( log `  (
n  +  1 ) ) )
22 nnz 10960 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  ZZ )
23 peano2nn 10622 . . . . . . . 8  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  NN )
24 nnuz 11195 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
2523, 24syl6eleq 2520 . . . . . . 7  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  ( ZZ>= `  1
) )
26 elfznn 11829 . . . . . . . . . . 11  |-  ( x  e.  ( 1 ... ( n  +  1 ) )  ->  x  e.  NN )
2726adantl 467 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  x  e.  NN )
2827nnrpd 11340 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  x  e.  RR+ )
2928relogcld 23559 . . . . . . . 8  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  ( log `  x
)  e.  RR )
3029recnd 9670 . . . . . . 7  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  ( log `  x
)  e.  CC )
3118, 19, 20, 21, 22, 25, 30telfsum2 13853 . . . . . 6  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( ( log `  ( m  +  1 ) )  -  ( log `  m ) )  =  ( ( log `  ( n  +  1 ) )  -  ( log `  1 ) ) )
32 log1 23522 . . . . . . . 8  |-  ( log `  1 )  =  0
3332oveq2i 6313 . . . . . . 7  |-  ( ( log `  ( n  +  1 ) )  -  ( log `  1
) )  =  ( ( log `  (
n  +  1 ) )  -  0 )
3423nnrpd 11340 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  RR+ )
3534relogcld 23559 . . . . . . . . 9  |-  ( n  e.  NN  ->  ( log `  ( n  + 
1 ) )  e.  RR )
3635recnd 9670 . . . . . . . 8  |-  ( n  e.  NN  ->  ( log `  ( n  + 
1 ) )  e.  CC )
3736subid1d 9976 . . . . . . 7  |-  ( n  e.  NN  ->  (
( log `  (
n  +  1 ) )  -  0 )  =  ( log `  (
n  +  1 ) ) )
3833, 37syl5eq 2475 . . . . . 6  |-  ( n  e.  NN  ->  (
( log `  (
n  +  1 ) )  -  ( log `  1 ) )  =  ( log `  (
n  +  1 ) ) )
3917, 31, 383eqtrd 2467 . . . . 5  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( log `  (
1  +  ( 1  /  m ) ) )  =  ( log `  ( n  +  1 ) ) )
4039oveq2d 6318 . . . 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 12186 . . . . . 6  |-  ( n  e.  NN  ->  (
1 ... n )  e. 
Fin )
422nnrecred 10656 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  RR )
4342recnd 9670 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  CC )
44 1rp 11307 . . . . . . . . 9  |-  1  e.  RR+
4514rpreccld 11352 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  RR+ )
46 rpaddcl 11324 . . . . . . . . 9  |-  ( ( 1  e.  RR+  /\  (
1  /  m )  e.  RR+ )  ->  (
1  +  ( 1  /  m ) )  e.  RR+ )
4744, 45, 46sylancr 667 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  +  ( 1  /  m
) )  e.  RR+ )
4847relogcld 23559 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
1  +  ( 1  /  m ) ) )  e.  RR )
4948recnd 9670 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
1  +  ( 1  /  m ) ) )  e.  CC )
5041, 43, 49fsumsub 13837 . . . . 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 6310 . . . . . . . . 9  |-  ( n  =  m  ->  (
1  /  n )  =  ( 1  /  m ) )
5251oveq2d 6318 . . . . . . . . . 10  |-  ( n  =  m  ->  (
1  +  ( 1  /  n ) )  =  ( 1  +  ( 1  /  m
) ) )
5352fveq2d 5882 . . . . . . . . 9  |-  ( n  =  m  ->  ( log `  ( 1  +  ( 1  /  n
) ) )  =  ( log `  (
1  +  ( 1  /  m ) ) ) )
5451, 53oveq12d 6320 . . . . . . . 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 6330 . . . . . . . 8  |-  ( ( 1  /  m )  -  ( log `  (
1  +  ( 1  /  m ) ) ) )  e.  _V
5754, 55, 56fvmpt 5961 . . . . . . 7  |-  ( m  e.  NN  ->  ( T `  m )  =  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) ) )
582, 57syl 17 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( T `  m )  =  ( ( 1  /  m
)  -  ( log `  ( 1  +  ( 1  /  m ) ) ) ) )
59 id 23 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  NN )
6059, 24syl6eleq 2520 . . . . . 6  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
6142, 48resubcld 10048 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  e.  RR )
6261recnd 9670 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  e.  CC )
6358, 60, 62fsumser 13784 . . . . 5  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  =  (  seq 1 (  +  ,  T ) `  n
) )
6450, 63eqtr3d 2465 . . . 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 2465 . . 3  |-  ( n  e.  NN  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  (
n  +  1 ) ) )  =  (  seq 1 (  +  ,  T ) `  n ) )
6665mpteq2ia 4503 . 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 10968 . . . . 5  |-  1  e.  ZZ
69 seqfn 12225 . . . . 5  |-  ( 1  e.  ZZ  ->  seq 1 (  +  ,  T )  Fn  ( ZZ>=
`  1 ) )
7068, 69ax-mp 5 . . . 4  |-  seq 1
(  +  ,  T
)  Fn  ( ZZ>= ` 
1 )
7124fneq2i 5686 . . . 4  |-  (  seq 1 (  +  ,  T )  Fn  NN  <->  seq 1 (  +  ,  T )  Fn  ( ZZ>=
`  1 ) )
7270, 71mpbir 212 . . 3  |-  seq 1
(  +  ,  T
)  Fn  NN
73 dffn5 5923 . . 3  |-  (  seq 1 (  +  ,  T )  Fn  NN  <->  seq 1 (  +  ,  T )  =  ( n  e.  NN  |->  (  seq 1 (  +  ,  T ) `  n ) ) )
7472, 73mpbi 211 . 2  |-  seq 1
(  +  ,  T
)  =  ( n  e.  NN  |->  (  seq 1 (  +  ,  T ) `  n
) )
7566, 67, 743eqtr4i 2461 1  |-  G  =  seq 1 (  +  ,  T )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437    e. wcel 1868    |-> cmpt 4479    Fn wfn 5593   ` cfv 5598  (class class class)co 6302   0cc0 9540   1c1 9541    + caddc 9543    - cmin 9861    / cdiv 10270   NNcn 10610   ZZcz 10938   ZZ>=cuz 11160   RR+crp 11303   ...cfz 11785    seqcseq 12213   sum_csu 13740   logclog 23491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619  ax-mulf 9620
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-fi 7928  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xneg 11410  df-xadd 11411  df-xmul 11412  df-ioo 11640  df-ioc 11641  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13119  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-limsup 13514  df-clim 13540  df-rlim 13541  df-sum 13741  df-ef 14109  df-sin 14111  df-cos 14112  df-pi 14114  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-hom 15202  df-cco 15203  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-pt 15331  df-prds 15334  df-xrs 15388  df-qtop 15394  df-imas 15395  df-xps 15398  df-mre 15480  df-mrc 15481  df-acs 15483  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-mulg 16664  df-cntz 16959  df-cmn 17420  df-psmet 18950  df-xmet 18951  df-met 18952  df-bl 18953  df-mopn 18954  df-fbas 18955  df-fg 18956  df-cnfld 18959  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-cld 20021  df-ntr 20022  df-cls 20023  df-nei 20101  df-lp 20139  df-perf 20140  df-cn 20230  df-cnp 20231  df-haus 20318  df-tx 20564  df-hmeo 20757  df-fil 20848  df-fm 20940  df-flim 20941  df-flf 20942  df-xms 21322  df-ms 21323  df-tms 21324  df-cncf 21897  df-limc 22808  df-dv 22809  df-log 23493
This theorem is referenced by:  emcllem6  23913
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