emcllem2 - Metamath Proof Explorer

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Theorem emcllem2 27330

Description: Lemma for emcl 27336. 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 12351	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
2	1	nnrecred 12393	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ∈ ℝ)
3	1	nnrpd 13166	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ⁺)
4	3	relogcld 26957	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘(𝑁 + 1)) ∈ ℝ)
5		nnrp 13136	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
6	5	relogcld 26957	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℝ)
7	4, 6	resubcld 11748	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((log‘(𝑁 + 1)) − (log‘𝑁)) ∈ ℝ)
8		fzfid 14120	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
9		elfznn 13691	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...𝑁) → 𝑚 ∈ ℕ)
10	9	adantl 487	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ)
11	10	nnrecred 12393	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → (1 / 𝑚) ∈ ℝ)
12	8, 11	fsumrecl 15906	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) ∈ ℝ)
13	3	rpreccld 13178	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ∈ ℝ⁺)
14	13	rpge0d 13172	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 0 ≤ (1 / (𝑁 + 1)))
15		1div1e1 12011	. . . . . . . . . . . 12 ⊢ (1 / 1) = 1
16		1re 11312	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
17		ltaddrp 13163	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → 1 < (1 + 𝑁))
18	16, 5, 17	sylancr 599	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 1 < (1 + 𝑁))
19		ax-1cn 11262	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
20		nncn 12347	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
21		addcom 11500	. . . . . . . . . . . . . 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 5130	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 1 < (𝑁 + 1))
24	15, 23	eqbrtrid 5139	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (1 / 1) < (𝑁 + 1))
25	1	nnred 12354	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ)
26	1	nngt0d 12391	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1))
27		0lt1 11842	. . . . . . . . . . . . 13 ⊢ 0 < 1
28		ltrec1 12208	. . . . . . . . . . . . 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 16295	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (exp‘(1 / (𝑁 + 1))) ≤ (1 / (1 − (1 / (𝑁 + 1)))))
33	25	recnd 11341	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℂ)
34		nnne0 12376	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
35	1	nnne0d 12392	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ≠ 0)
36	20, 33, 34, 35	recdivd 12114	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 / (𝑁 + 1))) = ((𝑁 + 1) / 𝑁))
37		1cnd 11306	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
38	33, 37, 33, 35	divsubdird 12136	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) − 1) / (𝑁 + 1)) = (((𝑁 + 1) / (𝑁 + 1)) − (1 / (𝑁 + 1))))
39		pncan 11567	. . . . . . . . . . . . . 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 7436	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) − 1) / (𝑁 + 1)) = (𝑁 / (𝑁 + 1)))
42	33, 35	dividd 12095	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) / (𝑁 + 1)) = 1)
43	42	oveq1d 7436	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / (𝑁 + 1)) − (1 / (𝑁 + 1))) = (1 − (1 / (𝑁 + 1))))
44	38, 41, 43	3eqtr3rd 2804	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (1 − (1 / (𝑁 + 1))) = (𝑁 / (𝑁 + 1)))
45	44	oveq2d 7437	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / (1 − (1 / (𝑁 + 1)))) = (1 / (𝑁 / (𝑁 + 1))))
46	3, 5	rpdivcld 13185	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) / 𝑁) ∈ ℝ⁺)
47	46	reeflogd 26958	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (exp‘(log‘((𝑁 + 1) / 𝑁))) = ((𝑁 + 1) / 𝑁))
48	36, 45, 47	3eqtr4d 2805	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (1 / (1 − (1 / (𝑁 + 1)))) = (exp‘(log‘((𝑁 + 1) / 𝑁))))
49	32, 48	breqtrd 5130	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (exp‘(1 / (𝑁 + 1))) ≤ (exp‘(log‘((𝑁 + 1) / 𝑁))))
50	3, 5	relogdivd 26960	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) / 𝑁)) = ((log‘(𝑁 + 1)) − (log‘𝑁)))
51	50, 7	eqeltrd 2860	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) / 𝑁)) ∈ ℝ)
52		efle 16292	. . . . . . . . 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 5130	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ≤ ((log‘(𝑁 + 1)) − (log‘𝑁)))
56	2, 7, 12, 55	leadd2dd 11935	. . . . 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 13008	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
59	57, 58	eleqtrdi 2870	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘1))
60		elfznn 13691	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(𝑁 + 1)) → 𝑚 ∈ ℕ)
61	60	adantl 487	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → 𝑚 ∈ ℕ)
62	61	nnrecred 12393	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → (1 / 𝑚) ∈ ℝ)
63	62	recnd 11341	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → (1 / 𝑚) ∈ ℂ)
64		oveq2 7429	. . . . . 6 ⊢ (𝑚 = (𝑁 + 1) → (1 / 𝑚) = (1 / (𝑁 + 1)))
65	59, 63, 64	fsump1 15928	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) + (1 / (𝑁 + 1))))
66	4	recnd 11341	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘(𝑁 + 1)) ∈ ℂ)
67	12	recnd 11341	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) ∈ ℂ)
68	6	recnd 11341	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℂ)
69	66, 67, 68	addsub12d 11696	. . . . 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 5136	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) ≤ ((log‘(𝑁 + 1)) + (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁))))
71		fzfid 14120	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 + 1)) ∈ Fin)
72	71, 62	fsumrecl 15906	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) ∈ ℝ)
73	12, 6	resubcld 11748	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ∈ ℝ)
74	72, 4, 73	lesubadd2d 11919	. . . 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 7429	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → (1...𝑛) = (1...(𝑁 + 1)))
77	76	sumeq1d 15873	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚))
78		fveq2 6886	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → (log‘𝑛) = (log‘(𝑁 + 1)))
79	77, 78	oveq12d 7439	. . . . 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 7454	. . . . 5 ⊢ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ V
82	79, 80, 81	fvmpt 6994	. . . 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 7429	. . . . . 6 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
85	84	sumeq1d 15873	. . . . 5 ⊢ (𝑛 = 𝑁 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...𝑁)(1 / 𝑚))
86		fveq2 6886	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘𝑛) = (log‘𝑁))
87	85, 86	oveq12d 7439	. . . 4 ⊢ (𝑛 = 𝑁 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
88		ovex 7454	. . . 4 ⊢ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ∈ V
89	87, 80, 88	fvmpt 6994	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
90	75, 83, 89	3brtr4d 5136	. 2 ⊢ (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) ≤ (𝐹‘𝑁))
91		peano2nn 12351	. . . . . . . . . 10 ⊢ ((𝑁 + 1) ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℕ)
92	1, 91	syl 18	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℕ)
93	92	nnrpd 13166	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℝ⁺)
94	93	relogcld 26957	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) + 1)) ∈ ℝ)
95	94, 4	resubcld 11748	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((log‘((𝑁 + 1) + 1)) − (log‘(𝑁 + 1))) ∈ ℝ)
96		logdifbnd 27327	. . . . . . 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 11935	. . . . 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 11341	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) + 1)) ∈ ℂ)
100	67, 66, 99	subadd23d 11695	. . . . 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 5136	. . . 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 11748	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ ℝ)
103		leaddsub 11796	. . . . 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 7444	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘(𝑛 + 1)) = (log‘(𝑁 + 1)))
107	85, 106	oveq12d 7439	. . . 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 7454	. . . 4 ⊢ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ V
110	107, 108, 109	fvmpt 6994	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐺‘𝑁) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))))
111		fvoveq1 7444	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → (log‘(𝑛 + 1)) = (log‘((𝑁 + 1) + 1)))
112	77, 111	oveq12d 7439	. . . . 5 ⊢ (𝑛 = (𝑁 + 1) → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))) = (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))))
113		ovex 7454	. . . . 5 ⊢ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))) ∈ V
114	112, 108, 113	fvmpt 6994	. . . 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 5136	. 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 1569 ∈ wcel 2144 class class class wbr 5102 ↦ cmpt 5185 ‘cfv 6539 (class class class)co 7421 ℂcc 11202 ℝcr 11203 0cc0 11204 1c1 11205 + caddc 11207 < clt 11347 ≤ cle 11348 − cmin 11545 / cdiv 11977 ℕcn 12339 ℤ_≥cuz 12969 ℝ⁺crp 13124 ...cfz 13643 Σcsu 15859 expce 16233 logclog 26888

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-inf2 9646 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282 ax-addf 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-map 8853 df-pm 8854 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-fi 9407 df-sup 9438 df-inf 9439 df-oi 9508 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-q 13080 df-rp 13125 df-xneg 13245 df-xadd 13246 df-xmul 13247 df-ioo 13484 df-ioc 13485 df-ico 13486 df-icc 13487 df-fz 13644 df-fzo 13793 df-fl 13936 df-mod 14014 df-seq 14149 df-exp 14209 df-fac 14422 df-bc 14451 df-hash 14479 df-shft 15226 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-limsup 15644 df-clim 15661 df-rlim 15662 df-sum 15860 df-ef 16239 df-sin 16241 df-cos 16242 df-pi 16244 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-starv 17449 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-unif 17457 df-hom 17458 df-cco 17459 df-rest 17599 df-topn 17600 df-0g 17618 df-gsum 17619 df-topgen 17620 df-pt 17621 df-prds 17624 df-xrs 17680 df-qtop 17685 df-imas 17686 df-xps 17688 df-mre 17762 df-mrc 17763 df-acs 17765 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-mulg 19284 df-cntz 19537 df-cmn 20002 df-psmet 21676 df-xmet 21677 df-met 21678 df-bl 21679 df-mopn 21680 df-fbas 21681 df-fg 21682 df-cnfld 21685 df-top 23218 df-topon 23235 df-topsp 23257 df-bases 23270 df-cld 23343 df-ntr 23344 df-cls 23345 df-nei 23422 df-lp 23460 df-perf 23461 df-cn 23551 df-cnp 23552 df-haus 23639 df-tx 23887 df-hmeo 24080 df-fil 24171 df-fm 24263 df-flim 24264 df-flf 24265 df-xms 24645 df-ms 24646 df-tms 24647 df-cncf 25205 df-limc 26192 df-dv 26193 df-log 26890

This theorem is used by: emcllem6 27334 emcllem7 27335

W3C validator