emcllem3 - Metamath Proof Explorer

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Theorem emcllem3 26052

Description: Lemma for emcl 26057. 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 11915	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
2	1	nnrpd 12699	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ⁺)
3		nnrp 12670	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
4	2, 3	relogdivd 25686	. . 3 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) / 𝑁)) = ((log‘(𝑁 + 1)) − (log‘𝑁)))
5		nncn 11911	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
6		1cnd 10901	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
7		nnne0 11937	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
8	5, 6, 5, 7	divdird 11719	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) / 𝑁) = ((𝑁 / 𝑁) + (1 / 𝑁)))
9	5, 7	dividd 11679	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 / 𝑁) = 1)
10	9	oveq1d 7270	. . . . 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 6760	. . 3 ⊢ (𝑁 ∈ ℕ → (log‘(1 + (1 / 𝑁))) = (log‘((𝑁 + 1) / 𝑁)))
13		fzfid 13621	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
14		elfznn 13214	. . . . . . . 8 ⊢ (𝑚 ∈ (1...𝑁) → 𝑚 ∈ ℕ)
15	14	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ)
16	15	nnrecred 11954	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → (1 / 𝑚) ∈ ℝ)
17	13, 16	fsumrecl 15374	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) ∈ ℝ)
18	17	recnd 10934	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) ∈ ℂ)
19	3	relogcld 25683	. . . . 5 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℝ)
20	19	recnd 10934	. . . 4 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℂ)
21	2	relogcld 25683	. . . . 5 ⊢ (𝑁 ∈ ℕ → (log‘(𝑁 + 1)) ∈ ℝ)
22	21	recnd 10934	. . . 4 ⊢ (𝑁 ∈ ℕ → (log‘(𝑁 + 1)) ∈ ℂ)
23	18, 20, 22	nnncan1d 11296	. . 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 7263	. . . . 5 ⊢ (𝑛 = 𝑁 → (1 / 𝑛) = (1 / 𝑁))
26	25	oveq2d 7271	. . . 4 ⊢ (𝑛 = 𝑁 → (1 + (1 / 𝑛)) = (1 + (1 / 𝑁)))
27	26	fveq2d 6760	. . 3 ⊢ (𝑛 = 𝑁 → (log‘(1 + (1 / 𝑛))) = (log‘(1 + (1 / 𝑁))))
28		emcl.3	. . 3 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛))))
29		fvex 6769	. . 3 ⊢ (log‘(1 + (1 / 𝑁))) ∈ V
30	27, 28, 29	fvmpt 6857	. 2 ⊢ (𝑁 ∈ ℕ → (𝐻‘𝑁) = (log‘(1 + (1 / 𝑁))))
31		oveq2 7263	. . . . . 6 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
32	31	sumeq1d 15341	. . . . 5 ⊢ (𝑛 = 𝑁 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...𝑁)(1 / 𝑚))
33		fveq2 6756	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘𝑛) = (log‘𝑁))
34	32, 33	oveq12d 7273	. . . 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 7288	. . . 4 ⊢ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ∈ V
37	34, 35, 36	fvmpt 6857	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
38		fvoveq1 7278	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘(𝑛 + 1)) = (log‘(𝑁 + 1)))
39	32, 38	oveq12d 7273	. . . 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 7288	. . . 4 ⊢ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ V
42	39, 40, 41	fvmpt 6857	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐺‘𝑁) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))))
43	37, 42	oveq12d 7273	. 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 395 = wceq 1539 ∈ wcel 2108 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 1c1 10803 + caddc 10805 − cmin 11135 / cdiv 11562 ℕcn 11903 ...cfz 13168 Σcsu 15325 logclog 25615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617

This theorem is referenced by: emcllem6 26055

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