emcllem2 - Metamath Proof Explorer

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Theorem emcllem2 27233

Description: Lemma for emcl 27239. 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 12269	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
2	1	nnrecred 12311	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ∈ ℝ)
3	1	nnrpd 13084	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ⁺)
4	3	relogcld 26860	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘(𝑁 + 1)) ∈ ℝ)
5		nnrp 13054	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
6	5	relogcld 26860	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℝ)
7	4, 6	resubcld 11666	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((log‘(𝑁 + 1)) − (log‘𝑁)) ∈ ℝ)
8		fzfid 14037	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
9		elfznn 13608	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...𝑁) → 𝑚 ∈ ℕ)
10	9	adantl 487	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ)
11	10	nnrecred 12311	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → (1 / 𝑚) ∈ ℝ)
12	8, 11	fsumrecl 15820	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) ∈ ℝ)
13	3	rpreccld 13096	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ∈ ℝ⁺)
14	13	rpge0d 13090	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 0 ≤ (1 / (𝑁 + 1)))
15		1div1e1 11929	. . . . . . . . . . . 12 ⊢ (1 / 1) = 1
16		1re 11232	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
17		ltaddrp 13081	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → 1 < (1 + 𝑁))
18	16, 5, 17	sylancr 599	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 1 < (1 + 𝑁))
19		ax-1cn 11182	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
20		nncn 12265	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
21		addcom 11420	. . . . . . . . . . . . . 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 5131	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 1 < (𝑁 + 1))
24	15, 23	eqbrtrid 5140	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (1 / 1) < (𝑁 + 1))
25	1	nnred 12272	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ)
26	1	nngt0d 12309	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1))
27		0lt1 11760	. . . . . . . . . . . . 13 ⊢ 0 < 1
28		ltrec1 12126	. . . . . . . . . . . . 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 16209	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (exp‘(1 / (𝑁 + 1))) ≤ (1 / (1 − (1 / (𝑁 + 1)))))
33	25	recnd 11261	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℂ)
34		nnne0 12294	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
35	1	nnne0d 12310	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ≠ 0)
36	20, 33, 34, 35	recdivd 12032	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 / (𝑁 + 1))) = ((𝑁 + 1) / 𝑁))
37		1cnd 11226	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
38	33, 37, 33, 35	divsubdird 12054	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) − 1) / (𝑁 + 1)) = (((𝑁 + 1) / (𝑁 + 1)) − (1 / (𝑁 + 1))))
39		pncan 11487	. . . . . . . . . . . . . 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 7428	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) − 1) / (𝑁 + 1)) = (𝑁 / (𝑁 + 1)))
42	33, 35	dividd 12013	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) / (𝑁 + 1)) = 1)
43	42	oveq1d 7428	. . . . . . . . . . . 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 7429	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / (1 − (1 / (𝑁 + 1)))) = (1 / (𝑁 / (𝑁 + 1))))
46	3, 5	rpdivcld 13103	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) / 𝑁) ∈ ℝ⁺)
47	46	reeflogd 26861	. . . . . . . . . 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 5131	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (exp‘(1 / (𝑁 + 1))) ≤ (exp‘(log‘((𝑁 + 1) / 𝑁))))
50	3, 5	relogdivd 26863	. . . . . . . . . 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 16206	. . . . . . . . 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 5131	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ≤ ((log‘(𝑁 + 1)) − (log‘𝑁)))
56	2, 7, 12, 55	leadd2dd 11853	. . . . 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 12926	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
59	57, 58	eleqtrdi 2870	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘1))
60		elfznn 13608	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(𝑁 + 1)) → 𝑚 ∈ ℕ)
61	60	adantl 487	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → 𝑚 ∈ ℕ)
62	61	nnrecred 12311	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → (1 / 𝑚) ∈ ℝ)
63	62	recnd 11261	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → (1 / 𝑚) ∈ ℂ)
64		oveq2 7421	. . . . . 6 ⊢ (𝑚 = (𝑁 + 1) → (1 / 𝑚) = (1 / (𝑁 + 1)))
65	59, 63, 64	fsump1 15842	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) + (1 / (𝑁 + 1))))
66	4	recnd 11261	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘(𝑁 + 1)) ∈ ℂ)
67	12	recnd 11261	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) ∈ ℂ)
68	6	recnd 11261	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℂ)
69	66, 67, 68	addsub12d 11616	. . . . 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 5137	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) ≤ ((log‘(𝑁 + 1)) + (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁))))
71		fzfid 14037	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 + 1)) ∈ Fin)
72	71, 62	fsumrecl 15820	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) ∈ ℝ)
73	12, 6	resubcld 11666	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ∈ ℝ)
74	72, 4, 73	lesubadd2d 11837	. . . 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 7421	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → (1...𝑛) = (1...(𝑁 + 1)))
77	76	sumeq1d 15787	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚))
78		fveq2 6878	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → (log‘𝑛) = (log‘(𝑁 + 1)))
79	77, 78	oveq12d 7431	. . . . 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 7446	. . . . 5 ⊢ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ V
82	79, 80, 81	fvmpt 6986	. . . 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 7421	. . . . . 6 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
85	84	sumeq1d 15787	. . . . 5 ⊢ (𝑛 = 𝑁 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...𝑁)(1 / 𝑚))
86		fveq2 6878	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘𝑛) = (log‘𝑁))
87	85, 86	oveq12d 7431	. . . 4 ⊢ (𝑛 = 𝑁 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
88		ovex 7446	. . . 4 ⊢ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ∈ V
89	87, 80, 88	fvmpt 6986	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
90	75, 83, 89	3brtr4d 5137	. 2 ⊢ (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) ≤ (𝐹‘𝑁))
91		peano2nn 12269	. . . . . . . . . 10 ⊢ ((𝑁 + 1) ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℕ)
92	1, 91	syl 18	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℕ)
93	92	nnrpd 13084	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℝ⁺)
94	93	relogcld 26860	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) + 1)) ∈ ℝ)
95	94, 4	resubcld 11666	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((log‘((𝑁 + 1) + 1)) − (log‘(𝑁 + 1))) ∈ ℝ)
96		logdifbnd 27230	. . . . . . 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 11853	. . . . 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 11261	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) + 1)) ∈ ℂ)
100	67, 66, 99	subadd23d 11615	. . . . 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 5137	. . . 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 11666	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ ℝ)
103		leaddsub 11714	. . . . 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 7436	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘(𝑛 + 1)) = (log‘(𝑁 + 1)))
107	85, 106	oveq12d 7431	. . . 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 7446	. . . 4 ⊢ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ V
110	107, 108, 109	fvmpt 6986	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐺‘𝑁) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))))
111		fvoveq1 7436	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → (log‘(𝑛 + 1)) = (log‘((𝑁 + 1) + 1)))
112	77, 111	oveq12d 7431	. . . . 5 ⊢ (𝑛 = (𝑁 + 1) → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))) = (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))))
113		ovex 7446	. . . . 5 ⊢ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))) ∈ V
114	112, 108, 113	fvmpt 6986	. . . 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 5137	. 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 2145 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 < clt 11267 ≤ cle 11268 − cmin 11465 / cdiv 11895 ℕcn 12257 ℤ_≥cuz 12887 ℝ⁺crp 13042 ...cfz 13561 Σcsu 15773 expce 16147 logclog 26791

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

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 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 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 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 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ioc 13403 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-fac 14338 df-bc 14367 df-hash 14395 df-shft 15140 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-limsup 15558 df-clim 15575 df-rlim 15576 df-sum 15774 df-ef 16153 df-sin 16155 df-cos 16156 df-pi 16158 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-fbas 21582 df-fg 21583 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-ntr 23245 df-cls 23246 df-nei 23323 df-lp 23361 df-perf 23362 df-cn 23452 df-cnp 23453 df-haus 23540 df-tx 23788 df-hmeo 23981 df-fil 24072 df-fm 24164 df-flim 24165 df-flf 24166 df-xms 24546 df-ms 24547 df-tms 24548 df-cncf 25106 df-limc 26093 df-dv 26094 df-log 26793

This theorem is used by: emcllem6 27237 emcllem7 27238

W3C validator