emcllem2 - Metamath Proof Explorer

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Theorem emcllem2 26957

Description: Lemma for emcl 26963. 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 12250	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
2	1	nnrecred 12289	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ∈ ℝ)
3	1	nnrpd 13047	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ⁺)
4	3	relogcld 26582	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘(𝑁 + 1)) ∈ ℝ)
5		nnrp 13018	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
6	5	relogcld 26582	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℝ)
7	4, 6	resubcld 11663	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((log‘(𝑁 + 1)) − (log‘𝑁)) ∈ ℝ)
8		fzfid 13989	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
9		elfznn 13568	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...𝑁) → 𝑚 ∈ ℕ)
10	9	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ)
11	10	nnrecred 12289	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → (1 / 𝑚) ∈ ℝ)
12	8, 11	fsumrecl 15748	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) ∈ ℝ)
13	3	rpreccld 13059	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ∈ ℝ⁺)
14	13	rpge0d 13053	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 0 ≤ (1 / (𝑁 + 1)))
15		1div1e1 11930	. . . . . . . . . . . 12 ⊢ (1 / 1) = 1
16		1re 11233	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
17		ltaddrp 13044	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → 1 < (1 + 𝑁))
18	16, 5, 17	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 1 < (1 + 𝑁))
19		ax-1cn 11185	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
20		nncn 12246	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
21		addcom 11419	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (1 + 𝑁) = (𝑁 + 1))
22	19, 20, 21	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (1 + 𝑁) = (𝑁 + 1))
23	18, 22	breqtrd 5145	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 1 < (𝑁 + 1))
24	15, 23	eqbrtrid 5154	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (1 / 1) < (𝑁 + 1))
25	1	nnred 12253	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ)
26	1	nngt0d 12287	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1))
27		0lt1 11757	. . . . . . . . . . . . 13 ⊢ 0 < 1
28		ltrec1 12127	. . . . . . . . . . . . 13 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ ((𝑁 + 1) ∈ ℝ ∧ 0 < (𝑁 + 1))) → ((1 / 1) < (𝑁 + 1) ↔ (1 / (𝑁 + 1)) < 1))
29	16, 27, 28	mpanl12 702	. . . . . . . . . . . 12 ⊢ (((𝑁 + 1) ∈ ℝ ∧ 0 < (𝑁 + 1)) → ((1 / 1) < (𝑁 + 1) ↔ (1 / (𝑁 + 1)) < 1))
30	25, 26, 29	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((1 / 1) < (𝑁 + 1) ↔ (1 / (𝑁 + 1)) < 1))
31	24, 30	mpbid 232	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) < 1)
32	2, 14, 31	eflegeo 16137	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (exp‘(1 / (𝑁 + 1))) ≤ (1 / (1 − (1 / (𝑁 + 1)))))
33	25	recnd 11261	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℂ)
34		nnne0 12272	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
35	1	nnne0d 12288	. . . . . . . . . . 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 11228	. . . . . . . . . . . . 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 11486	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
40	20, 19, 39	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
41	40	oveq1d 7418	. . . . . . . . . . . 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 7418	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / (𝑁 + 1)) − (1 / (𝑁 + 1))) = (1 − (1 / (𝑁 + 1))))
44	38, 41, 43	3eqtr3rd 2779	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (1 − (1 / (𝑁 + 1))) = (𝑁 / (𝑁 + 1)))
45	44	oveq2d 7419	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / (1 − (1 / (𝑁 + 1)))) = (1 / (𝑁 / (𝑁 + 1))))
46	3, 5	rpdivcld 13066	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) / 𝑁) ∈ ℝ⁺)
47	46	reeflogd 26583	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (exp‘(log‘((𝑁 + 1) / 𝑁))) = ((𝑁 + 1) / 𝑁))
48	36, 45, 47	3eqtr4d 2780	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (1 / (1 − (1 / (𝑁 + 1)))) = (exp‘(log‘((𝑁 + 1) / 𝑁))))
49	32, 48	breqtrd 5145	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (exp‘(1 / (𝑁 + 1))) ≤ (exp‘(log‘((𝑁 + 1) / 𝑁))))
50	3, 5	relogdivd 26585	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) / 𝑁)) = ((log‘(𝑁 + 1)) − (log‘𝑁)))
51	50, 7	eqeltrd 2834	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) / 𝑁)) ∈ ℝ)
52		efle 16134	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((1 / (𝑁 + 1)) ≤ (log‘((𝑁 + 1) / 𝑁)) ↔ (exp‘(1 / (𝑁 + 1))) ≤ (exp‘(log‘((𝑁 + 1) / 𝑁)))))
54	49, 53	mpbird 257	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ≤ (log‘((𝑁 + 1) / 𝑁)))
55	54, 50	breqtrd 5145	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ≤ ((log‘(𝑁 + 1)) − (log‘𝑁)))
56	2, 7, 12, 55	leadd2dd 11850	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) + (1 / (𝑁 + 1))) ≤ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) + ((log‘(𝑁 + 1)) − (log‘𝑁))))
57		id 22	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ)
58		nnuz 12893	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
59	57, 58	eleqtrdi 2844	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘1))
60		elfznn 13568	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(𝑁 + 1)) → 𝑚 ∈ ℕ)
61	60	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → 𝑚 ∈ ℕ)
62	61	nnrecred 12289	. . . . . . 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 7411	. . . . . 6 ⊢ (𝑚 = (𝑁 + 1) → (1 / 𝑚) = (1 / (𝑁 + 1)))
65	59, 63, 64	fsump1 15770	. . . . 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 11615	. . . . 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 5151	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) ≤ ((log‘(𝑁 + 1)) + (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁))))
71		fzfid 13989	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 + 1)) ∈ Fin)
72	71, 62	fsumrecl 15748	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) ∈ ℝ)
73	12, 6	resubcld 11663	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ∈ ℝ)
74	72, 4, 73	lesubadd2d 11834	. . . 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 257	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘(𝑁 + 1))) ≤ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
76		oveq2 7411	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → (1...𝑛) = (1...(𝑁 + 1)))
77	76	sumeq1d 15714	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚))
78		fveq2 6875	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → (log‘𝑛) = (log‘(𝑁 + 1)))
79	77, 78	oveq12d 7421	. . . . 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 7436	. . . . 5 ⊢ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ V
82	79, 80, 81	fvmpt 6985	. . . 4 ⊢ ((𝑁 + 1) ∈ ℕ → (𝐹‘(𝑁 + 1)) = (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘(𝑁 + 1))))
83	1, 82	syl 17	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) = (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘(𝑁 + 1))))
84		oveq2 7411	. . . . . 6 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
85	84	sumeq1d 15714	. . . . 5 ⊢ (𝑛 = 𝑁 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...𝑁)(1 / 𝑚))
86		fveq2 6875	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘𝑛) = (log‘𝑁))
87	85, 86	oveq12d 7421	. . . 4 ⊢ (𝑛 = 𝑁 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
88		ovex 7436	. . . 4 ⊢ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ∈ V
89	87, 80, 88	fvmpt 6985	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
90	75, 83, 89	3brtr4d 5151	. 2 ⊢ (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) ≤ (𝐹‘𝑁))
91		peano2nn 12250	. . . . . . . . . 10 ⊢ ((𝑁 + 1) ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℕ)
92	1, 91	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℕ)
93	92	nnrpd 13047	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℝ⁺)
94	93	relogcld 26582	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) + 1)) ∈ ℝ)
95	94, 4	resubcld 11663	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((log‘((𝑁 + 1) + 1)) − (log‘(𝑁 + 1))) ∈ ℝ)
96		logdifbnd 26954	. . . . . . 7 ⊢ ((𝑁 + 1) ∈ ℝ⁺ → ((log‘((𝑁 + 1) + 1)) − (log‘(𝑁 + 1))) ≤ (1 / (𝑁 + 1)))
97	3, 96	syl 17	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((log‘((𝑁 + 1) + 1)) − (log‘(𝑁 + 1))) ≤ (1 / (𝑁 + 1)))
98	95, 2, 12, 97	leadd2dd 11850	. . . . 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 11614	. . . . 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 5151	. . . 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 11663	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ ℝ)
103		leaddsub 11711	. . . . 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 1373	. . . 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 232	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ≤ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))))
106		fvoveq1 7426	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘(𝑛 + 1)) = (log‘(𝑁 + 1)))
107	85, 106	oveq12d 7421	. . . 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 7436	. . . 4 ⊢ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ V
110	107, 108, 109	fvmpt 6985	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐺‘𝑁) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))))
111		fvoveq1 7426	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → (log‘(𝑛 + 1)) = (log‘((𝑁 + 1) + 1)))
112	77, 111	oveq12d 7421	. . . . 5 ⊢ (𝑛 = (𝑁 + 1) → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))) = (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))))
113		ovex 7436	. . . . 5 ⊢ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))) ∈ V
114	112, 108, 113	fvmpt 6985	. . . 4 ⊢ ((𝑁 + 1) ∈ ℕ → (𝐺‘(𝑁 + 1)) = (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))))
115	1, 114	syl 17	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐺‘(𝑁 + 1)) = (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))))
116	105, 110, 115	3brtr4d 5151	. 2 ⊢ (𝑁 ∈ ℕ → (𝐺‘𝑁) ≤ (𝐺‘(𝑁 + 1)))
117	90, 116	jca 511	1 ⊢ (𝑁 ∈ ℕ → ((𝐹‘(𝑁 + 1)) ≤ (𝐹‘𝑁) ∧ (𝐺‘𝑁) ≤ (𝐺‘(𝑁 + 1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5119 ↦ cmpt 5201 ‘cfv 6530 (class class class)co 7403 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 < clt 11267 ≤ cle 11268 − cmin 11464 / cdiv 11892 ℕcn 12238 ℤ_≥cuz 12850 ℝ⁺crp 13006 ...cfz 13522 Σcsu 15700 expce 16075 logclog 26513

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-inf2 9653 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8717 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-fi 9421 df-sup 9452 df-inf 9453 df-oi 9522 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-ioo 13364 df-ioc 13365 df-ico 13366 df-icc 13367 df-fz 13523 df-fzo 13670 df-fl 13807 df-mod 13885 df-seq 14018 df-exp 14078 df-fac 14290 df-bc 14319 df-hash 14347 df-shft 15084 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15485 df-clim 15502 df-rlim 15503 df-sum 15701 df-ef 16081 df-sin 16083 df-cos 16084 df-pi 16086 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17514 df-qtop 17519 df-imas 17520 df-xps 17522 df-mre 17596 df-mrc 17597 df-acs 17599 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-submnd 18760 df-mulg 19049 df-cntz 19298 df-cmn 19761 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-cld 22955 df-ntr 22956 df-cls 22957 df-nei 23034 df-lp 23072 df-perf 23073 df-cn 23163 df-cnp 23164 df-haus 23251 df-tx 23498 df-hmeo 23691 df-fil 23782 df-fm 23874 df-flim 23875 df-flf 23876 df-xms 24257 df-ms 24258 df-tms 24259 df-cncf 24820 df-limc 25817 df-dv 25818 df-log 26515

This theorem is referenced by: emcllem6 26961 emcllem7 26962

W3C validator