emcllem3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > emcllem3		Structured version Visualization version GIF version

Theorem emcllem3 26147

Description: Lemma for emcl 26152. The function 𝐻 is the difference between 𝐹 and 𝐺. (Contributed by Mario Carneiro, 11-Jul-2014.)

**Hypotheses**
Ref	Expression
emcl.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)))
emcl.2	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))))
emcl.3	⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛))))

**Assertion**
Ref	Expression
emcllem3	⊢ (𝑁 ∈ ℕ → (𝐻‘𝑁) = ((𝐹‘𝑁) − (𝐺‘𝑁)))

Distinct variable groups: 𝑚,𝐻 𝑚,𝑛,𝑁 Allowed substitution hints: 𝐹(𝑚,𝑛) 𝐺(𝑚,𝑛) 𝐻(𝑛)

**Proof of Theorem emcllem3**
Step	Hyp	Ref	Expression
1		peano2nn 11985	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
2	1	nnrpd 12770	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ⁺)
3		nnrp 12741	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
4	2, 3	relogdivd 25781	. . 3 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) / 𝑁)) = ((log‘(𝑁 + 1)) − (log‘𝑁)))
5		nncn 11981	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
6		1cnd 10970	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
7		nnne0 12007	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
8	5, 6, 5, 7	divdird 11789	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) / 𝑁) = ((𝑁 / 𝑁) + (1 / 𝑁)))
9	5, 7	dividd 11749	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 / 𝑁) = 1)
10	9	oveq1d 7290	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 𝑁) + (1 / 𝑁)) = (1 + (1 / 𝑁)))
11	8, 10	eqtr2d 2779	. . . 4 ⊢ (𝑁 ∈ ℕ → (1 + (1 / 𝑁)) = ((𝑁 + 1) / 𝑁))
12	11	fveq2d 6778	. . 3 ⊢ (𝑁 ∈ ℕ → (log‘(1 + (1 / 𝑁))) = (log‘((𝑁 + 1) / 𝑁)))
13		fzfid 13693	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
14		elfznn 13285	. . . . . . . 8 ⊢ (𝑚 ∈ (1...𝑁) → 𝑚 ∈ ℕ)
15	14	adantl 482	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ)
16	15	nnrecred 12024	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → (1 / 𝑚) ∈ ℝ)
17	13, 16	fsumrecl 15446	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) ∈ ℝ)
18	17	recnd 11003	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) ∈ ℂ)
19	3	relogcld 25778	. . . . 5 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℝ)
20	19	recnd 11003	. . . 4 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℂ)
21	2	relogcld 25778	. . . . 5 ⊢ (𝑁 ∈ ℕ → (log‘(𝑁 + 1)) ∈ ℝ)
22	21	recnd 11003	. . . 4 ⊢ (𝑁 ∈ ℕ → (log‘(𝑁 + 1)) ∈ ℂ)
23	18, 20, 22	nnncan1d 11366	. . 3 ⊢ (𝑁 ∈ ℕ → ((Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) − (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1)))) = ((log‘(𝑁 + 1)) − (log‘𝑁)))
24	4, 12, 23	3eqtr4d 2788	. 2 ⊢ (𝑁 ∈ ℕ → (log‘(1 + (1 / 𝑁))) = ((Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) − (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1)))))
25		oveq2 7283	. . . . 5 ⊢ (𝑛 = 𝑁 → (1 / 𝑛) = (1 / 𝑁))
26	25	oveq2d 7291	. . . 4 ⊢ (𝑛 = 𝑁 → (1 + (1 / 𝑛)) = (1 + (1 / 𝑁)))
27	26	fveq2d 6778	. . 3 ⊢ (𝑛 = 𝑁 → (log‘(1 + (1 / 𝑛))) = (log‘(1 + (1 / 𝑁))))
28		emcl.3	. . 3 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛))))
29		fvex 6787	. . 3 ⊢ (log‘(1 + (1 / 𝑁))) ∈ V
30	27, 28, 29	fvmpt 6875	. 2 ⊢ (𝑁 ∈ ℕ → (𝐻‘𝑁) = (log‘(1 + (1 / 𝑁))))
31		oveq2 7283	. . . . . 6 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
32	31	sumeq1d 15413	. . . . 5 ⊢ (𝑛 = 𝑁 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...𝑁)(1 / 𝑚))
33		fveq2 6774	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘𝑛) = (log‘𝑁))
34	32, 33	oveq12d 7293	. . . 4 ⊢ (𝑛 = 𝑁 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
35		emcl.1	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)))
36		ovex 7308	. . . 4 ⊢ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ∈ V
37	34, 35, 36	fvmpt 6875	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
38		fvoveq1 7298	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘(𝑛 + 1)) = (log‘(𝑁 + 1)))
39	32, 38	oveq12d 7293	. . . 4 ⊢ (𝑛 = 𝑁 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))))
40		emcl.2	. . . 4 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))))
41		ovex 7308	. . . 4 ⊢ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ V
42	39, 40, 41	fvmpt 6875	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐺‘𝑁) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))))
43	37, 42	oveq12d 7293	. 2 ⊢ (𝑁 ∈ ℕ → ((𝐹‘𝑁) − (𝐺‘𝑁)) = ((Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) − (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1)))))
44	24, 30, 43	3eqtr4d 2788	1 ⊢ (𝑁 ∈ ℕ → (𝐻‘𝑁) = ((𝐹‘𝑁) − (𝐺‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 1c1 10872 + caddc 10874 − cmin 11205 / cdiv 11632 ℕcn 11973 ...cfz 13239 Σcsu 15397 logclog 25710

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712

This theorem is referenced by: emcllem6 26150

W3C validator