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Theorem emcllem5 22536
Description: Lemma for emcl 22539. The partial sums of the series  T, which is used in the definition df-em 22529, 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 11599 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  m  e.  NN )
21adantl 466 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  NN )
32nncnd 10453 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  CC )
4 1cnd 9517 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  1  e.  CC )
52nnne0d 10481 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  =/=  0
)
63, 4, 3, 5divdird 10260 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( m  +  1 )  /  m )  =  ( ( m  /  m
)  +  ( 1  /  m ) ) )
73, 5dividd 10220 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  /  m )  =  1 )
87oveq1d 6218 . . . . . . . . . 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 5806 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
( m  +  1 )  /  m ) )  =  ( log `  ( 1  +  ( 1  /  m ) ) ) )
11 peano2nn 10449 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  (
m  +  1 )  e.  NN )
122, 11syl 16 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  + 
1 )  e.  NN )
1312nnrpd 11141 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( m  + 
1 )  e.  RR+ )
142nnrpd 11141 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  m  e.  RR+ )
1513, 14relogdivd 22218 . . . . . . . 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 13302 . . . . . 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 5802 . . . . . . 7  |-  ( x  =  m  ->  ( log `  x )  =  ( log `  m
) )
19 fveq2 5802 . . . . . . 7  |-  ( x  =  ( m  + 
1 )  ->  ( log `  x )  =  ( log `  (
m  +  1 ) ) )
20 fveq2 5802 . . . . . . 7  |-  ( x  =  1  ->  ( log `  x )  =  ( log `  1
) )
21 fveq2 5802 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  ( log `  x )  =  ( log `  (
n  +  1 ) ) )
22 nnz 10783 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  ZZ )
23 peano2nn 10449 . . . . . . . 8  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  NN )
24 nnuz 11011 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
2523, 24syl6eleq 2552 . . . . . . 7  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  ( ZZ>= `  1
) )
26 elfznn 11599 . . . . . . . . . . 11  |-  ( x  e.  ( 1 ... ( n  +  1 ) )  ->  x  e.  NN )
2726adantl 466 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  x  e.  NN )
2827nnrpd 11141 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  x  e.  RR+ )
2928relogcld 22215 . . . . . . . 8  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  ( log `  x
)  e.  RR )
3029recnd 9527 . . . . . . 7  |-  ( ( n  e.  NN  /\  x  e.  ( 1 ... ( n  + 
1 ) ) )  ->  ( log `  x
)  e.  CC )
3118, 19, 20, 21, 22, 25, 30fsumtscop2 13390 . . . . . 6  |-  ( n  e.  NN  ->  sum_ m  e.  ( 1 ... n
) ( ( log `  ( m  +  1 ) )  -  ( log `  m ) )  =  ( ( log `  ( n  +  1 ) )  -  ( log `  1 ) ) )
32 log1 22177 . . . . . . . 8  |-  ( log `  1 )  =  0
3332oveq2i 6214 . . . . . . 7  |-  ( ( log `  ( n  +  1 ) )  -  ( log `  1
) )  =  ( ( log `  (
n  +  1 ) )  -  0 )
3423nnrpd 11141 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  RR+ )
3534relogcld 22215 . . . . . . . . 9  |-  ( n  e.  NN  ->  ( log `  ( n  + 
1 ) )  e.  RR )
3635recnd 9527 . . . . . . . 8  |-  ( n  e.  NN  ->  ( log `  ( n  + 
1 ) )  e.  CC )
3736subid1d 9823 . . . . . . 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 6219 . . . 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 11916 . . . . . 6  |-  ( n  e.  NN  ->  (
1 ... n )  e. 
Fin )
422nnrecred 10482 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  RR )
4342recnd 9527 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  CC )
44 1rp 11110 . . . . . . . . 9  |-  1  e.  RR+
4514rpreccld 11152 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  /  m )  e.  RR+ )
46 rpaddcl 11126 . . . . . . . . 9  |-  ( ( 1  e.  RR+  /\  (
1  /  m )  e.  RR+ )  ->  (
1  +  ( 1  /  m ) )  e.  RR+ )
4744, 45, 46sylancr 663 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( 1  +  ( 1  /  m
) )  e.  RR+ )
4847relogcld 22215 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
1  +  ( 1  /  m ) ) )  e.  RR )
4948recnd 9527 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( log `  (
1  +  ( 1  /  m ) ) )  e.  CC )
5041, 43, 49fsumsub 13377 . . . . 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 6211 . . . . . . . . 9  |-  ( n  =  m  ->  (
1  /  n )  =  ( 1  /  m ) )
5251oveq2d 6219 . . . . . . . . . 10  |-  ( n  =  m  ->  (
1  +  ( 1  /  n ) )  =  ( 1  +  ( 1  /  m
) ) )
5352fveq2d 5806 . . . . . . . . 9  |-  ( n  =  m  ->  ( log `  ( 1  +  ( 1  /  n
) ) )  =  ( log `  (
1  +  ( 1  /  m ) ) ) )
5451, 53oveq12d 6221 . . . . . . . 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 6228 . . . . . . . 8  |-  ( ( 1  /  m )  -  ( log `  (
1  +  ( 1  /  m ) ) ) )  e.  _V
5754, 55, 56fvmpt 5886 . . . . . . 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 9891 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  e.  RR )
6261recnd 9527 . . . . . 6  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( 1  /  m )  -  ( log `  ( 1  +  ( 1  /  m ) ) ) )  e.  CC )
6358, 60, 62fsumser 13329 . . . . 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 4485 . 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 10791 . . . . 5  |-  1  e.  ZZ
69 seqfn 11939 . . . . 5  |-  ( 1  e.  ZZ  ->  seq 1 (  +  ,  T )  Fn  ( ZZ>=
`  1 ) )
7068, 69ax-mp 5 . . . 4  |-  seq 1
(  +  ,  T
)  Fn  ( ZZ>= ` 
1 )
7124fneq2i 5617 . . . 4  |-  (  seq 1 (  +  ,  T )  Fn  NN  <->  seq 1 (  +  ,  T )  Fn  ( ZZ>=
`  1 ) )
7270, 71mpbir 209 . . 3  |-  seq 1
(  +  ,  T
)  Fn  NN
73 dffn5 5849 . . 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 369    = wceq 1370    e. wcel 1758    |-> cmpt 4461    Fn wfn 5524   ` cfv 5529  (class class class)co 6203   0cc0 9397   1c1 9398    + caddc 9400    - cmin 9710    / cdiv 10108   NNcn 10437   ZZcz 10761   ZZ>=cuz 10976   RR+crp 11106   ...cfz 11558    seqcseq 11927   sum_csu 13285   logclog 22149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475  ax-addf 9476  ax-mulf 9477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-fi 7776  df-sup 7806  df-oi 7839  df-card 8224  df-cda 8452  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-q 11069  df-rp 11107  df-xneg 11204  df-xadd 11205  df-xmul 11206  df-ioo 11419  df-ioc 11420  df-ico 11421  df-icc 11422  df-fz 11559  df-fzo 11670  df-fl 11763  df-mod 11830  df-seq 11928  df-exp 11987  df-fac 12173  df-bc 12200  df-hash 12225  df-shft 12678  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-limsup 13071  df-clim 13088  df-rlim 13089  df-sum 13286  df-ef 13475  df-sin 13477  df-cos 13478  df-pi 13480  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-mulr 14375  df-starv 14376  df-sca 14377  df-vsca 14378  df-ip 14379  df-tset 14380  df-ple 14381  df-ds 14383  df-unif 14384  df-hom 14385  df-cco 14386  df-rest 14484  df-topn 14485  df-0g 14503  df-gsum 14504  df-topgen 14505  df-pt 14506  df-prds 14509  df-xrs 14563  df-qtop 14568  df-imas 14569  df-xps 14571  df-mre 14647  df-mrc 14648  df-acs 14650  df-mnd 15538  df-submnd 15588  df-mulg 15671  df-cntz 15958  df-cmn 16404  df-psmet 17944  df-xmet 17945  df-met 17946  df-bl 17947  df-mopn 17948  df-fbas 17949  df-fg 17950  df-cnfld 17954  df-top 18645  df-bases 18647  df-topon 18648  df-topsp 18649  df-cld 18765  df-ntr 18766  df-cls 18767  df-nei 18844  df-lp 18882  df-perf 18883  df-cn 18973  df-cnp 18974  df-haus 19061  df-tx 19277  df-hmeo 19470  df-fil 19561  df-fm 19653  df-flim 19654  df-flf 19655  df-xms 20037  df-ms 20038  df-tms 20039  df-cncf 20596  df-limc 21484  df-dv 21485  df-log 22151
This theorem is referenced by:  emcllem6  22537
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