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Theorem emcllem5 20791
Description: Lemma for emcl 20794. The partial sums of the series  T, which is used in the definition df-em 20784, 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 11036 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  m  e.  NN )
21adantl 453 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  NN )
32nncnd 9972 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  CC )
4 ax-1cn 9004 . . . . . . . . . . . 12  |-  1  e.  CC
54a1i 11 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  1  e.  CC )
62nnne0d 10000 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  =/=  0
)
73, 5, 3, 6divdird 9784 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( m  +  1 )  /  m )  =  ( ( m  /  m
)  +  ( 1  /  m ) ) )
83, 6dividd 9744 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  /  m )  =  1 )
98oveq1d 6055 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( m  /  m )  +  ( 1  /  m
) )  =  ( 1  +  ( 1  /  m ) ) )
107, 9eqtrd 2436 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( m  +  1 )  /  m )  =  ( 1  +  ( 1  /  m ) ) )
1110fveq2d 5691 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
( m  +  1 )  /  m ) )  =  ( log `  ( 1  +  ( 1  /  m ) ) ) )
12 peano2nn 9968 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  (
m  +  1 )  e.  NN )
132, 12syl 16 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  + 
1 )  e.  NN )
1413nnrpd 10603 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  + 
1 )  e.  RR+ )
152nnrpd 10603 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  RR+ )
1614, 15relogdivd 20474 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
( m  +  1 )  /  m ) )  =  ( ( log `  ( m  +  1 ) )  -  ( log `  m
) ) )
1711, 16eqtr3d 2438 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
1  +  ( 1  /  m ) ) )  =  ( ( log `  ( m  +  1 ) )  -  ( log `  m
) ) )
1817sumeq2dv 12452 . . . . . 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 5687 . . . . . . 7  |-  ( x  =  m  ->  ( log `  x )  =  ( log `  m
) )
20 fveq2 5687 . . . . . . 7  |-  ( x  =  ( m  + 
1 )  ->  ( log `  x )  =  ( log `  (
m  +  1 ) ) )
21 fveq2 5687 . . . . . . 7  |-  ( x  =  1  ->  ( log `  x )  =  ( log `  1
) )
22 fveq2 5687 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  ( log `  x )  =  ( log `  (
n  +  1 ) ) )
23 nnz 10259 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  ZZ )
24 peano2nn 9968 . . . . . . . 8  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  NN )
25 nnuz 10477 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
2624, 25syl6eleq 2494 . . . . . . 7  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  ( ZZ>= `  1
) )
27 elfznn 11036 . . . . . . . . . . 11  |-  ( x  e.  ( 1 ... ( n  +  1 ) )  ->  x  e.  NN )
2827adantl 453 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  x  e.  NN )
2928nnrpd 10603 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  x  e.  RR+ )
3029relogcld 20471 . . . . . . . 8  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  ( log `  x
)  e.  RR )
3130recnd 9070 . . . . . . 7  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  ( log `  x
)  e.  CC )
3219, 20, 21, 22, 23, 26, 31fsumtscop2 12539 . . . . . 6  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( ( log `  ( m  +  1 ) )  -  ( log `  m ) )  =  ( ( log `  ( n  +  1 ) )  -  ( log `  1 ) ) )
33 log1 20433 . . . . . . . 8  |-  ( log `  1 )  =  0
3433oveq2i 6051 . . . . . . 7  |-  ( ( log `  ( n  +  1 ) )  -  ( log `  1
) )  =  ( ( log `  (
n  +  1 ) )  -  0 )
3524nnrpd 10603 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  RR+ )
3635relogcld 20471 . . . . . . . . 9  |-  ( n  e.  NN  ->  ( log `  ( n  + 
1 ) )  e.  RR )
3736recnd 9070 . . . . . . . 8  |-  ( n  e.  NN  ->  ( log `  ( n  + 
1 ) )  e.  CC )
3837subid1d 9356 . . . . . . 7  |-  ( n  e.  NN  ->  (
( log `  (
n  +  1 ) )  -  0 )  =  ( log `  (
n  +  1 ) ) )
3934, 38syl5eq 2448 . . . . . 6  |-  ( n  e.  NN  ->  (
( log `  (
n  +  1 ) )  -  ( log `  1 ) )  =  ( log `  (
n  +  1 ) ) )
4018, 32, 393eqtrd 2440 . . . . 5  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( log `  (
1  +  ( 1  /  m ) ) )  =  ( log `  ( n  +  1 ) ) )
4140oveq2d 6056 . . . 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 11267 . . . . . 6  |-  ( n  e.  NN  ->  (
1 ... n )  e. 
Fin )
432nnrecred 10001 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  RR )
4443recnd 9070 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  CC )
45 1rp 10572 . . . . . . . . 9  |-  1  e.  RR+
4615rpreccld 10614 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  RR+ )
47 rpaddcl 10588 . . . . . . . . 9  |-  ( ( 1  e.  RR+  /\  (
1  /  m )  e.  RR+ )  ->  (
1  +  ( 1  /  m ) )  e.  RR+ )
4845, 46, 47sylancr 645 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  +  ( 1  /  m
) )  e.  RR+ )
4948relogcld 20471 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
1  +  ( 1  /  m ) ) )  e.  RR )
5049recnd 9070 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
1  +  ( 1  /  m ) ) )  e.  CC )
5142, 44, 50fsumsub 12526 . . . . 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 6048 . . . . . . . . 9  |-  ( n  =  m  ->  (
1  /  n )  =  ( 1  /  m ) )
5352oveq2d 6056 . . . . . . . . . 10  |-  ( n  =  m  ->  (
1  +  ( 1  /  n ) )  =  ( 1  +  ( 1  /  m
) ) )
5453fveq2d 5691 . . . . . . . . 9  |-  ( n  =  m  ->  ( log `  ( 1  +  ( 1  /  n
) ) )  =  ( log `  (
1  +  ( 1  /  m ) ) ) )
5552, 54oveq12d 6058 . . . . . . . 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 6065 . . . . . . . 8  |-  ( ( 1  /  m )  -  ( log `  (
1  +  ( 1  /  m ) ) ) )  e.  _V
5855, 56, 57fvmpt 5765 . . . . . . 7  |-  ( m  e.  NN  ->  ( T `  m )  =  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) ) )
592, 58syl 16 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( T `  m )  =  ( ( 1  /  m
)  -  ( log `  ( 1  +  ( 1  /  m ) ) ) ) )
60 id 20 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  NN )
6160, 25syl6eleq 2494 . . . . . 6  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
6243, 49resubcld 9421 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  e.  RR )
6362recnd 9070 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  e.  CC )
6459, 61, 63fsumser 12479 . . . . 5  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  =  (  seq  1 (  +  ,  T ) `  n
) )
6551, 64eqtr3d 2438 . . . 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 2438 . . 3  |-  ( n  e.  NN  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  (
n  +  1 ) ) )  =  (  seq  1 (  +  ,  T ) `  n ) )
6766mpteq2ia 4251 . 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 10267 . . . . 5  |-  1  e.  ZZ
70 seqfn 11290 . . . . 5  |-  ( 1  e.  ZZ  ->  seq  1 (  +  ,  T )  Fn  ( ZZ>=
`  1 ) )
7169, 70ax-mp 8 . . . 4  |-  seq  1
(  +  ,  T
)  Fn  ( ZZ>= ` 
1 )
7225fneq2i 5499 . . . 4  |-  (  seq  1 (  +  ,  T )  Fn  NN  <->  seq  1 (  +  ,  T )  Fn  ( ZZ>=
`  1 ) )
7371, 72mpbir 201 . . 3  |-  seq  1
(  +  ,  T
)  Fn  NN
74 dffn5 5731 . . 3  |-  (  seq  1 (  +  ,  T )  Fn  NN  <->  seq  1 (  +  ,  T )  =  ( n  e.  NN  |->  (  seq  1 (  +  ,  T ) `  n ) ) )
7573, 74mpbi 200 . 2  |-  seq  1
(  +  ,  T
)  =  ( n  e.  NN  |->  (  seq  1 (  +  ,  T ) `  n
) )
7667, 68, 753eqtr4i 2434 1  |-  G  =  seq  1 (  +  ,  T )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1721    e. cmpt 4226    Fn wfn 5408   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    - cmin 9247    / cdiv 9633   NNcn 9956   ZZcz 10238   ZZ>=cuz 10444   RR+crp 10568   ...cfz 10999    seq cseq 11278   sum_csu 12434   logclog 20405
This theorem is referenced by:  emcllem6  20792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407
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