Theorem emcllem5 24726

Description: Lemma for emcl 24729. The partial sums of the series T, which is used in the definition df-em 24719, 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.

Step	Hyp	Ref	Expression
1		elfznn 12370	. . . . . . . . . . . . 13 |- ( m e. ( 1 ... n ) -> m e. NN )
2	1	adantl 482	. . . . . . . . . . . 12 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> m e. NN )
3	2	nncnd 11036	. . . . . . . . . . 11 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> m e. CC )
4		1cnd 10056	. . . . . . . . . . 11 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> 1 e. CC )
5	2	nnne0d 11065	. . . . . . . . . . 11 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> m =/= 0 )
6	3, 4, 3, 5	divdird 10839	. . . . . . . . . 10 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( ( m + 1 ) / m ) = ( ( m / m ) + ( 1 / m ) ) )
7	3, 5	dividd 10799	. . . . . . . . . . 11 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( m / m ) = 1 )
8	7	oveq1d 6665	. . . . . . . . . 10 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( ( m / m ) + ( 1 / m ) ) = ( 1 + ( 1 / m ) ) )
9	6, 8	eqtrd 2656	. . . . . . . . 9 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( ( m + 1 ) / m ) = ( 1 + ( 1 / m ) ) )
10	9	fveq2d 6195	. . . . . . . 8 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( log ` ( ( m + 1 ) / m ) ) = ( log ` ( 1 + ( 1 / m ) ) ) )
11		peano2nn 11032	. . . . . . . . . . 11 |- ( m e. NN -> ( m + 1 ) e. NN )
12	2, 11	syl 17	. . . . . . . . . 10 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( m + 1 ) e. NN )
13	12	nnrpd 11870	. . . . . . . . 9 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( m + 1 ) e. RR+ )
14	2	nnrpd 11870	. . . . . . . . 9 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> m e. RR+ )
15	13, 14	relogdivd 24372	. . . . . . . 8 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( log ` ( ( m + 1 ) / m ) ) = ( ( log ` ( m + 1 ) ) - ( log ` m ) ) )
16	10, 15	eqtr3d 2658	. . . . . . 7 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( log ` ( 1 + ( 1 / m ) ) ) = ( ( log ` ( m + 1 ) ) - ( log ` m ) ) )
17	16	sumeq2dv 14433	. . . . . 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 6191	. . . . . . 7 |- ( x = m -> ( log ` x ) = ( log ` m ) )
19		fveq2 6191	. . . . . . 7 |- ( x = ( m + 1 ) -> ( log ` x ) = ( log ` ( m + 1 ) ) )
20		fveq2 6191	. . . . . . 7 |- ( x = 1 -> ( log ` x ) = ( log ` 1 ) )
21		fveq2 6191	. . . . . . 7 |- ( x = ( n + 1 ) -> ( log ` x ) = ( log ` ( n + 1 ) ) )
22		nnz 11399	. . . . . . 7 |- ( n e. NN -> n e. ZZ )
23		peano2nn 11032	. . . . . . . 8 |- ( n e. NN -> ( n + 1 ) e. NN )
24		nnuz 11723	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
25	23, 24	syl6eleq 2711	. . . . . . 7 |- ( n e. NN -> ( n + 1 ) e. ( ZZ>= ` 1 ) )
26		elfznn 12370	. . . . . . . . . . 11 |- ( x e. ( 1 ... ( n + 1 ) ) -> x e. NN )
27	26	adantl 482	. . . . . . . . . 10 |- ( ( n e. NN /\ x e. ( 1 ... ( n + 1 ) ) ) -> x e. NN )
28	27	nnrpd 11870	. . . . . . . . 9 |- ( ( n e. NN /\ x e. ( 1 ... ( n + 1 ) ) ) -> x e. RR+ )
29	28	relogcld 24369	. . . . . . . 8 |- ( ( n e. NN /\ x e. ( 1 ... ( n + 1 ) ) ) -> ( log ` x ) e. RR )
30	29	recnd 10068	. . . . . . 7 |- ( ( n e. NN /\ x e. ( 1 ... ( n + 1 ) ) ) -> ( log ` x ) e. CC )
31	18, 19, 20, 21, 22, 25, 30	telfsum2 14537	. . . . . 6 |- ( n e. NN -> sum_ m e. ( 1 ... n ) ( ( log ` ( m + 1 ) ) - ( log ` m ) ) = ( ( log ` ( n + 1 ) ) - ( log ` 1 ) ) )
32		log1 24332	. . . . . . . 8 |- ( log ` 1 ) = 0
33	32	oveq2i 6661	. . . . . . 7 |- ( ( log ` ( n + 1 ) ) - ( log ` 1 ) ) = ( ( log ` ( n + 1 ) ) - 0 )
34	23	nnrpd 11870	. . . . . . . . . 10 |- ( n e. NN -> ( n + 1 ) e. RR+ )
35	34	relogcld 24369	. . . . . . . . 9 |- ( n e. NN -> ( log ` ( n + 1 ) ) e. RR )
36	35	recnd 10068	. . . . . . . 8 |- ( n e. NN -> ( log ` ( n + 1 ) ) e. CC )
37	36	subid1d 10381	. . . . . . 7 |- ( n e. NN -> ( ( log ` ( n + 1 ) ) - 0 ) = ( log ` ( n + 1 ) ) )
38	33, 37	syl5eq 2668	. . . . . 6 |- ( n e. NN -> ( ( log ` ( n + 1 ) ) - ( log ` 1 ) ) = ( log ` ( n + 1 ) ) )
39	17, 31, 38	3eqtrd 2660	. . . . 5 |- ( n e. NN -> sum_ m e. ( 1 ... n ) ( log ` ( 1 + ( 1 / m ) ) ) = ( log ` ( n + 1 ) ) )
40	39	oveq2d 6666	. . . 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 12772	. . . . . 6 |- ( n e. NN -> ( 1 ... n ) e. Fin )
42	2	nnrecred 11066	. . . . . . 7 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( 1 / m ) e. RR )
43	42	recnd 10068	. . . . . 6 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( 1 / m ) e. CC )
44		1rp 11836	. . . . . . . . 9 |- 1 e. RR+
45	14	rpreccld 11882	. . . . . . . . 9 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( 1 / m ) e. RR+ )
46		rpaddcl 11854	. . . . . . . . 9 |- ( ( 1 e. RR+ /\ ( 1 / m ) e. RR+ ) -> ( 1 + ( 1 / m ) ) e. RR+ )
47	44, 45, 46	sylancr 695	. . . . . . . 8 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( 1 + ( 1 / m ) ) e. RR+ )
48	47	relogcld 24369	. . . . . . 7 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( log ` ( 1 + ( 1 / m ) ) ) e. RR )
49	48	recnd 10068	. . . . . 6 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( log ` ( 1 + ( 1 / m ) ) ) e. CC )
50	41, 43, 49	fsumsub 14520	. . . . 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 6658	. . . . . . . . 9 |- ( n = m -> ( 1 / n ) = ( 1 / m ) )
52	51	oveq2d 6666	. . . . . . . . . 10 |- ( n = m -> ( 1 + ( 1 / n ) ) = ( 1 + ( 1 / m ) ) )
53	52	fveq2d 6195	. . . . . . . . 9 |- ( n = m -> ( log ` ( 1 + ( 1 / n ) ) ) = ( log ` ( 1 + ( 1 / m ) ) ) )
54	51, 53	oveq12d 6668	. . . . . . . 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 6678	. . . . . . . 8 |- ( ( 1 / m ) - ( log ` ( 1 + ( 1 / m ) ) ) ) e. _V
57	54, 55, 56	fvmpt 6282	. . . . . . 7 |- ( m e. NN -> ( T ` m ) = ( ( 1 / m ) - ( log ` ( 1 + ( 1 / m ) ) ) ) )
58	2, 57	syl 17	. . . . . 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 )
60	59, 24	syl6eleq 2711	. . . . . 6 |- ( n e. NN -> n e. ( ZZ>= ` 1 ) )
61	42, 48	resubcld 10458	. . . . . . 7 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( ( 1 / m ) - ( log ` ( 1 + ( 1 / m ) ) ) ) e. RR )
62	61	recnd 10068	. . . . . 6 |- ( ( n e. NN /\ m e. ( 1 ... n ) ) -> ( ( 1 / m ) - ( log ` ( 1 + ( 1 / m ) ) ) ) e. CC )
63	58, 60, 62	fsumser 14461	. . . . 5 |- ( n e. NN -> sum_ m e. ( 1 ... n ) ( ( 1 / m ) - ( log ` ( 1 + ( 1 / m ) ) ) ) = ( seq 1 ( + , T ) ` n ) )
64	50, 63	eqtr3d 2658	. . . 4 |- ( n e. NN -> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - sum_ m e. ( 1 ... n ) ( log ` ( 1 + ( 1 / m ) ) ) ) = ( seq 1 ( + , T ) ` n ) )
65	40, 64	eqtr3d 2658	. . 3 |- ( n e. NN -> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` ( n + 1 ) ) ) = ( seq 1 ( + , T ) ` n ) )
66	65	mpteq2ia 4740	. 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 11407	. . . . 5 |- 1 e. ZZ
69		seqfn 12813	. . . . 5 |- ( 1 e. ZZ -> seq 1 ( + , T ) Fn ( ZZ>= ` 1 ) )
70	68, 69	ax-mp 5	. . . 4 |- seq 1 ( + , T ) Fn ( ZZ>= ` 1 )
71	24	fneq2i 5986	. . . 4 |- ( seq 1 ( + , T ) Fn NN <-> seq 1 ( + , T ) Fn ( ZZ>= ` 1 ) )
72	70, 71	mpbir 221	. . 3 |- seq 1 ( + , T ) Fn NN
73		dffn5 6241	. . 3 |- ( seq 1 ( + , T ) Fn NN <-> seq 1 ( + , T ) = ( n e. NN |-> ( seq 1 ( + , T ) ` n ) ) )
74	72, 73	mpbi 220	. 2 |- seq 1 ( + , T ) = ( n e. NN |-> ( seq 1 ( + , T ) ` n ) )
75	66, 67, 74	3eqtr4i 2654	1 |- G = seq 1 ( + , T )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   / cdiv 10684   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   ...cfz 12326   seqcseq 12801   sum_csu 14416   logclog 24301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303

This theorem is referenced by:  emcllem6  24727