emcllem2 - Metamath Proof Explorer

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Theorem emcllem2 27192

Description: Lemma for emcl 27198. 𝐹 is increasing, and 𝐺 is decreasing. (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))))

**Assertion**
Ref	Expression
emcllem2	⊢ (𝑁 ∈ ℕ → ((𝐹‘(𝑁 + 1)) ≤ (𝐹‘𝑁) ∧ (𝐺‘𝑁) ≤ (𝐺‘(𝑁 + 1))))

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

**Proof of Theorem emcllem2**
Step	Hyp	Ref	Expression
1		peano2nn 12256	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
2	1	nnrecred 12298	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ∈ ℝ)
3	1	nnrpd 13070	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ⁺)
4	3	relogcld 26819	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘(𝑁 + 1)) ∈ ℝ)
5		nnrp 13040	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
6	5	relogcld 26819	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℝ)
7	4, 6	resubcld 11653	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((log‘(𝑁 + 1)) − (log‘𝑁)) ∈ ℝ)
8		fzfid 14023	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
9		elfznn 13594	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...𝑁) → 𝑚 ∈ ℕ)
10	9	adantl 487	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ)
11	10	nnrecred 12298	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → (1 / 𝑚) ∈ ℝ)
12	8, 11	fsumrecl 15804	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) ∈ ℝ)
13	3	rpreccld 13082	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ∈ ℝ⁺)
14	13	rpge0d 13076	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 0 ≤ (1 / (𝑁 + 1)))
15		1div1e1 11916	. . . . . . . . . . . 12 ⊢ (1 / 1) = 1
16		1re 11219	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
17		ltaddrp 13067	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → 1 < (1 + 𝑁))
18	16, 5, 17	sylancr 599	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 1 < (1 + 𝑁))
19		ax-1cn 11169	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
20		nncn 12252	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
21		addcom 11407	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (1 + 𝑁) = (𝑁 + 1))
22	19, 20, 21	sylancr 599	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (1 + 𝑁) = (𝑁 + 1))
23	18, 22	breqtrd 5139	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 1 < (𝑁 + 1))
24	15, 23	eqbrtrid 5148	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (1 / 1) < (𝑁 + 1))
25	1	nnred 12259	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ)
26	1	nngt0d 12296	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1))
27		0lt1 11747	. . . . . . . . . . . . 13 ⊢ 0 < 1
28		ltrec1 12113	. . . . . . . . . . . . 13 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ ((𝑁 + 1) ∈ ℝ ∧ 0 < (𝑁 + 1))) → ((1 / 1) < (𝑁 + 1) ↔ (1 / (𝑁 + 1)) < 1))
29	16, 27, 28	mpanl12 715	. . . . . . . . . . . 12 ⊢ (((𝑁 + 1) ∈ ℝ ∧ 0 < (𝑁 + 1)) → ((1 / 1) < (𝑁 + 1) ↔ (1 / (𝑁 + 1)) < 1))
30	25, 26, 29	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((1 / 1) < (𝑁 + 1) ↔ (1 / (𝑁 + 1)) < 1))
31	24, 30	mpbid 235	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) < 1)
32	2, 14, 31	eflegeo 16195	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (exp‘(1 / (𝑁 + 1))) ≤ (1 / (1 − (1 / (𝑁 + 1)))))
33	25	recnd 11248	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℂ)
34		nnne0 12281	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
35	1	nnne0d 12297	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ≠ 0)
36	20, 33, 34, 35	recdivd 12019	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 / (𝑁 + 1))) = ((𝑁 + 1) / 𝑁))
37		1cnd 11213	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
38	33, 37, 33, 35	divsubdird 12041	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) − 1) / (𝑁 + 1)) = (((𝑁 + 1) / (𝑁 + 1)) − (1 / (𝑁 + 1))))
39		pncan 11474	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
40	20, 19, 39	sylancl 598	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
41	40	oveq1d 7431	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) − 1) / (𝑁 + 1)) = (𝑁 / (𝑁 + 1)))
42	33, 35	dividd 12000	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) / (𝑁 + 1)) = 1)
43	42	oveq1d 7431	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / (𝑁 + 1)) − (1 / (𝑁 + 1))) = (1 − (1 / (𝑁 + 1))))
44	38, 41, 43	3eqtr3rd 2809	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (1 − (1 / (𝑁 + 1))) = (𝑁 / (𝑁 + 1)))
45	44	oveq2d 7432	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / (1 − (1 / (𝑁 + 1)))) = (1 / (𝑁 / (𝑁 + 1))))
46	3, 5	rpdivcld 13089	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) / 𝑁) ∈ ℝ⁺)
47	46	reeflogd 26820	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (exp‘(log‘((𝑁 + 1) / 𝑁))) = ((𝑁 + 1) / 𝑁))
48	36, 45, 47	3eqtr4d 2810	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (1 / (1 − (1 / (𝑁 + 1)))) = (exp‘(log‘((𝑁 + 1) / 𝑁))))
49	32, 48	breqtrd 5139	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (exp‘(1 / (𝑁 + 1))) ≤ (exp‘(log‘((𝑁 + 1) / 𝑁))))
50	3, 5	relogdivd 26822	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) / 𝑁)) = ((log‘(𝑁 + 1)) − (log‘𝑁)))
51	50, 7	eqeltrd 2865	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) / 𝑁)) ∈ ℝ)
52		efle 16192	. . . . . . . . 9 ⊢ (((1 / (𝑁 + 1)) ∈ ℝ ∧ (log‘((𝑁 + 1) / 𝑁)) ∈ ℝ) → ((1 / (𝑁 + 1)) ≤ (log‘((𝑁 + 1) / 𝑁)) ↔ (exp‘(1 / (𝑁 + 1))) ≤ (exp‘(log‘((𝑁 + 1) / 𝑁)))))
53	2, 51, 52	syl2anc 596	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((1 / (𝑁 + 1)) ≤ (log‘((𝑁 + 1) / 𝑁)) ↔ (exp‘(1 / (𝑁 + 1))) ≤ (exp‘(log‘((𝑁 + 1) / 𝑁)))))
54	49, 53	mpbird 260	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ≤ (log‘((𝑁 + 1) / 𝑁)))
55	54, 50	breqtrd 5139	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ≤ ((log‘(𝑁 + 1)) − (log‘𝑁)))
56	2, 7, 12, 55	leadd2dd 11840	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) + (1 / (𝑁 + 1))) ≤ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) + ((log‘(𝑁 + 1)) − (log‘𝑁))))
57		id 23	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ)
58		nnuz 12913	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
59	57, 58	eleqtrdi 2875	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘1))
60		elfznn 13594	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(𝑁 + 1)) → 𝑚 ∈ ℕ)
61	60	adantl 487	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → 𝑚 ∈ ℕ)
62	61	nnrecred 12298	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → (1 / 𝑚) ∈ ℝ)
63	62	recnd 11248	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → (1 / 𝑚) ∈ ℂ)
64		oveq2 7424	. . . . . 6 ⊢ (𝑚 = (𝑁 + 1) → (1 / 𝑚) = (1 / (𝑁 + 1)))
65	59, 63, 64	fsump1 15826	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) + (1 / (𝑁 + 1))))
66	4	recnd 11248	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘(𝑁 + 1)) ∈ ℂ)
67	12	recnd 11248	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) ∈ ℂ)
68	6	recnd 11248	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℂ)
69	66, 67, 68	addsub12d 11603	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((log‘(𝑁 + 1)) + (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁))) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) + ((log‘(𝑁 + 1)) − (log‘𝑁))))
70	56, 65, 69	3brtr4d 5145	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) ≤ ((log‘(𝑁 + 1)) + (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁))))
71		fzfid 14023	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 + 1)) ∈ Fin)
72	71, 62	fsumrecl 15804	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) ∈ ℝ)
73	12, 6	resubcld 11653	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ∈ ℝ)
74	72, 4, 73	lesubadd2d 11824	. . . 4 ⊢ (𝑁 ∈ ℕ → ((Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘(𝑁 + 1))) ≤ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ↔ Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) ≤ ((log‘(𝑁 + 1)) + (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))))
75	70, 74	mpbird 260	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘(𝑁 + 1))) ≤ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
76		oveq2 7424	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → (1...𝑛) = (1...(𝑁 + 1)))
77	76	sumeq1d 15771	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚))
78		fveq2 6885	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → (log‘𝑛) = (log‘(𝑁 + 1)))
79	77, 78	oveq12d 7434	. . . . 5 ⊢ (𝑛 = (𝑁 + 1) → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘(𝑁 + 1))))
80		emcl.1	. . . . 5 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)))
81		ovex 7449	. . . . 5 ⊢ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ V
82	79, 80, 81	fvmpt 6993	. . . 4 ⊢ ((𝑁 + 1) ∈ ℕ → (𝐹‘(𝑁 + 1)) = (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘(𝑁 + 1))))
83	1, 82	syl 18	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) = (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘(𝑁 + 1))))
84		oveq2 7424	. . . . . 6 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
85	84	sumeq1d 15771	. . . . 5 ⊢ (𝑛 = 𝑁 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...𝑁)(1 / 𝑚))
86		fveq2 6885	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘𝑛) = (log‘𝑁))
87	85, 86	oveq12d 7434	. . . 4 ⊢ (𝑛 = 𝑁 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
88		ovex 7449	. . . 4 ⊢ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ∈ V
89	87, 80, 88	fvmpt 6993	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
90	75, 83, 89	3brtr4d 5145	. 2 ⊢ (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) ≤ (𝐹‘𝑁))
91		peano2nn 12256	. . . . . . . . . 10 ⊢ ((𝑁 + 1) ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℕ)
92	1, 91	syl 18	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℕ)
93	92	nnrpd 13070	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℝ⁺)
94	93	relogcld 26819	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) + 1)) ∈ ℝ)
95	94, 4	resubcld 11653	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((log‘((𝑁 + 1) + 1)) − (log‘(𝑁 + 1))) ∈ ℝ)
96		logdifbnd 27189	. . . . . . 7 ⊢ ((𝑁 + 1) ∈ ℝ⁺ → ((log‘((𝑁 + 1) + 1)) − (log‘(𝑁 + 1))) ≤ (1 / (𝑁 + 1)))
97	3, 96	syl 18	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((log‘((𝑁 + 1) + 1)) − (log‘(𝑁 + 1))) ≤ (1 / (𝑁 + 1)))
98	95, 2, 12, 97	leadd2dd 11840	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) + ((log‘((𝑁 + 1) + 1)) − (log‘(𝑁 + 1)))) ≤ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) + (1 / (𝑁 + 1))))
99	94	recnd 11248	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) + 1)) ∈ ℂ)
100	67, 66, 99	subadd23d 11602	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) + (log‘((𝑁 + 1) + 1))) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) + ((log‘((𝑁 + 1) + 1)) − (log‘(𝑁 + 1)))))
101	98, 100, 65	3brtr4d 5145	. . . 4 ⊢ (𝑁 ∈ ℕ → ((Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) + (log‘((𝑁 + 1) + 1))) ≤ Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚))
102	12, 4	resubcld 11653	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ ℝ)
103		leaddsub 11701	. . . . 5 ⊢ (((Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ ℝ ∧ (log‘((𝑁 + 1) + 1)) ∈ ℝ ∧ Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) ∈ ℝ) → (((Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) + (log‘((𝑁 + 1) + 1))) ≤ Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) ↔ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ≤ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1)))))
104	102, 94, 72, 103	syl3anc 1398	. . . 4 ⊢ (𝑁 ∈ ℕ → (((Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) + (log‘((𝑁 + 1) + 1))) ≤ Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) ↔ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ≤ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1)))))
105	101, 104	mpbid 235	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ≤ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))))
106		fvoveq1 7439	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘(𝑛 + 1)) = (log‘(𝑁 + 1)))
107	85, 106	oveq12d 7434	. . . 4 ⊢ (𝑛 = 𝑁 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))))
108		emcl.2	. . . 4 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))))
109		ovex 7449	. . . 4 ⊢ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ V
110	107, 108, 109	fvmpt 6993	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐺‘𝑁) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))))
111		fvoveq1 7439	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → (log‘(𝑛 + 1)) = (log‘((𝑁 + 1) + 1)))
112	77, 111	oveq12d 7434	. . . . 5 ⊢ (𝑛 = (𝑁 + 1) → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))) = (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))))
113		ovex 7449	. . . . 5 ⊢ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))) ∈ V
114	112, 108, 113	fvmpt 6993	. . . 4 ⊢ ((𝑁 + 1) ∈ ℕ → (𝐺‘(𝑁 + 1)) = (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))))
115	1, 114	syl 18	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐺‘(𝑁 + 1)) = (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))))
116	105, 110, 115	3brtr4d 5145	. 2 ⊢ (𝑁 ∈ ℕ → (𝐺‘𝑁) ≤ (𝐺‘(𝑁 + 1)))
117	90, 116	jca 521	1 ⊢ (𝑁 ∈ ℕ → ((𝐹‘(𝑁 + 1)) ≤ (𝐹‘𝑁) ∧ (𝐺‘𝑁) ≤ (𝐺‘(𝑁 + 1))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 class class class wbr 5111 ↦ cmpt 5194 ‘cfv 6540 (class class class)co 7416 ℂcc 11109 ℝcr 11110 0cc0 11111 1c1 11112 + caddc 11114 < clt 11254 ≤ cle 11255 − cmin 11452 / cdiv 11882 ℕcn 12244 ℤ_≥cuz 12874 ℝ⁺crp 13028 ...cfz 13547 Σcsu 15757 expce 16133 logclog 26750

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-inf2 9613 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189 ax-addf 11190

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9325 df-fi 9374 df-sup 9405 df-inf 9406 df-oi 9475 df-card 9937 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-z 12603 df-dec 12724 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13149 df-xadd 13150 df-xmul 13151 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13548 df-fzo 13696 df-fl 13839 df-mod 13917 df-seq 14052 df-exp 14112 df-fac 14324 df-bc 14353 df-hash 14381 df-shft 15124 df-cj 15170 df-re 15171 df-im 15172 df-sqrt 15306 df-abs 15307 df-limsup 15542 df-clim 15559 df-rlim 15560 df-sum 15758 df-ef 16139 df-sin 16141 df-cos 16142 df-pi 16144 df-struct 17225 df-sets 17242 df-slot 17260 df-ndx 17272 df-base 17288 df-ress 17309 df-plusg 17341 df-mulr 17342 df-starv 17343 df-sca 17344 df-vsca 17345 df-ip 17346 df-tset 17347 df-ple 17348 df-ds 17350 df-unif 17351 df-hom 17352 df-cco 17353 df-rest 17493 df-topn 17494 df-0g 17512 df-gsum 17513 df-topgen 17514 df-pt 17515 df-prds 17518 df-xrs 17574 df-qtop 17579 df-imas 17580 df-xps 17582 df-mre 17656 df-mrc 17657 df-acs 17659 df-mgm 18716 df-sgrp 18799 df-mnd 18815 df-submnd 18866 df-mulg 19158 df-cntz 19411 df-cmn 19876 df-psmet 21544 df-xmet 21545 df-met 21546 df-bl 21547 df-mopn 21548 df-fbas 21549 df-fg 21550 df-cnfld 21553 df-top 23081 df-topon 23098 df-topsp 23120 df-bases 23133 df-cld 23206 df-ntr 23207 df-cls 23208 df-nei 23285 df-lp 23323 df-perf 23324 df-cn 23414 df-cnp 23415 df-haus 23502 df-tx 23750 df-hmeo 23943 df-fil 24034 df-fm 24126 df-flim 24127 df-flf 24128 df-xms 24508 df-ms 24509 df-tms 24510 df-cncf 25068 df-limc 26056 df-dv 26057 df-log 26752

This theorem is used by: emcllem6 27196 emcllem7 27197

W3C validator