emcllem2 - Metamath Proof Explorer

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Theorem emcllem2 26884

Description: Lemma for emcl 26890. 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 12228	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
2	1	nnrecred 12267	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ∈ ℝ)
3	1	nnrpd 13020	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ⁺)
4	3	relogcld 26512	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘(𝑁 + 1)) ∈ ℝ)
5		nnrp 12991	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
6	5	relogcld 26512	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℝ)
7	4, 6	resubcld 11646	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((log‘(𝑁 + 1)) − (log‘𝑁)) ∈ ℝ)
8		fzfid 13944	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
9		elfznn 13536	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...𝑁) → 𝑚 ∈ ℕ)
10	9	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ)
11	10	nnrecred 12267	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...𝑁)) → (1 / 𝑚) ∈ ℝ)
12	8, 11	fsumrecl 15686	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) ∈ ℝ)
13	3	rpreccld 13032	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ∈ ℝ⁺)
14	13	rpge0d 13026	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 0 ≤ (1 / (𝑁 + 1)))
15		1div1e1 11908	. . . . . . . . . . . 12 ⊢ (1 / 1) = 1
16		1re 11218	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
17		ltaddrp 13017	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → 1 < (1 + 𝑁))
18	16, 5, 17	sylancr 586	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 1 < (1 + 𝑁))
19		ax-1cn 11170	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
20		nncn 12224	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
21		addcom 11404	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (1 + 𝑁) = (𝑁 + 1))
22	19, 20, 21	sylancr 586	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (1 + 𝑁) = (𝑁 + 1))
23	18, 22	breqtrd 5167	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 1 < (𝑁 + 1))
24	15, 23	eqbrtrid 5176	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (1 / 1) < (𝑁 + 1))
25	1	nnred 12231	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ)
26	1	nngt0d 12265	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1))
27		0lt1 11740	. . . . . . . . . . . . 13 ⊢ 0 < 1
28		ltrec1 12105	. . . . . . . . . . . . 13 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ ((𝑁 + 1) ∈ ℝ ∧ 0 < (𝑁 + 1))) → ((1 / 1) < (𝑁 + 1) ↔ (1 / (𝑁 + 1)) < 1))
29	16, 27, 28	mpanl12 699	. . . . . . . . . . . 12 ⊢ (((𝑁 + 1) ∈ ℝ ∧ 0 < (𝑁 + 1)) → ((1 / 1) < (𝑁 + 1) ↔ (1 / (𝑁 + 1)) < 1))
30	25, 26, 29	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((1 / 1) < (𝑁 + 1) ↔ (1 / (𝑁 + 1)) < 1))
31	24, 30	mpbid 231	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) < 1)
32	2, 14, 31	eflegeo 16071	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (exp‘(1 / (𝑁 + 1))) ≤ (1 / (1 − (1 / (𝑁 + 1)))))
33	25	recnd 11246	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℂ)
34		nnne0 12250	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
35	1	nnne0d 12266	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ≠ 0)
36	20, 33, 34, 35	recdivd 12011	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 / (𝑁 + 1))) = ((𝑁 + 1) / 𝑁))
37		1cnd 11213	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
38	33, 37, 33, 35	divsubdird 12033	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) − 1) / (𝑁 + 1)) = (((𝑁 + 1) / (𝑁 + 1)) − (1 / (𝑁 + 1))))
39		pncan 11470	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
40	20, 19, 39	sylancl 585	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
41	40	oveq1d 7420	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) − 1) / (𝑁 + 1)) = (𝑁 / (𝑁 + 1)))
42	33, 35	dividd 11992	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) / (𝑁 + 1)) = 1)
43	42	oveq1d 7420	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / (𝑁 + 1)) − (1 / (𝑁 + 1))) = (1 − (1 / (𝑁 + 1))))
44	38, 41, 43	3eqtr3rd 2775	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (1 − (1 / (𝑁 + 1))) = (𝑁 / (𝑁 + 1)))
45	44	oveq2d 7421	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / (1 − (1 / (𝑁 + 1)))) = (1 / (𝑁 / (𝑁 + 1))))
46	3, 5	rpdivcld 13039	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) / 𝑁) ∈ ℝ⁺)
47	46	reeflogd 26513	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (exp‘(log‘((𝑁 + 1) / 𝑁))) = ((𝑁 + 1) / 𝑁))
48	36, 45, 47	3eqtr4d 2776	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (1 / (1 − (1 / (𝑁 + 1)))) = (exp‘(log‘((𝑁 + 1) / 𝑁))))
49	32, 48	breqtrd 5167	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (exp‘(1 / (𝑁 + 1))) ≤ (exp‘(log‘((𝑁 + 1) / 𝑁))))
50	3, 5	relogdivd 26515	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) / 𝑁)) = ((log‘(𝑁 + 1)) − (log‘𝑁)))
51	50, 7	eqeltrd 2827	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) / 𝑁)) ∈ ℝ)
52		efle 16068	. . . . . . . . 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 583	. . . . . . . 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 5167	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 / (𝑁 + 1)) ≤ ((log‘(𝑁 + 1)) − (log‘𝑁)))
56	2, 7, 12, 55	leadd2dd 11833	. . . . 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 12869	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
59	57, 58	eleqtrdi 2837	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘1))
60		elfznn 13536	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(𝑁 + 1)) → 𝑚 ∈ ℕ)
61	60	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → 𝑚 ∈ ℕ)
62	61	nnrecred 12267	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → (1 / 𝑚) ∈ ℝ)
63	62	recnd 11246	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → (1 / 𝑚) ∈ ℂ)
64		oveq2 7413	. . . . . 6 ⊢ (𝑚 = (𝑁 + 1) → (1 / 𝑚) = (1 / (𝑁 + 1)))
65	59, 63, 64	fsump1 15708	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) + (1 / (𝑁 + 1))))
66	4	recnd 11246	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘(𝑁 + 1)) ∈ ℂ)
67	12	recnd 11246	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) ∈ ℂ)
68	6	recnd 11246	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℂ)
69	66, 67, 68	addsub12d 11598	. . . . 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 5173	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) ≤ ((log‘(𝑁 + 1)) + (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁))))
71		fzfid 13944	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 + 1)) ∈ Fin)
72	71, 62	fsumrecl 15686	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) ∈ ℝ)
73	12, 6	resubcld 11646	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ∈ ℝ)
74	72, 4, 73	lesubadd2d 11817	. . . 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 7413	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → (1...𝑛) = (1...(𝑁 + 1)))
77	76	sumeq1d 15653	. . . . . 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 7423	. . . . 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 7438	. . . . 5 ⊢ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ V
82	79, 80, 81	fvmpt 6992	. . . 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 7413	. . . . . 6 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
85	84	sumeq1d 15653	. . . . 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 7423	. . . 4 ⊢ (𝑛 = 𝑁 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
88		ovex 7438	. . . 4 ⊢ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)) ∈ V
89	87, 80, 88	fvmpt 6992	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘𝑁)))
90	75, 83, 89	3brtr4d 5173	. 2 ⊢ (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) ≤ (𝐹‘𝑁))
91		peano2nn 12228	. . . . . . . . . 10 ⊢ ((𝑁 + 1) ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℕ)
92	1, 91	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℕ)
93	92	nnrpd 13020	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) + 1) ∈ ℝ⁺)
94	93	relogcld 26512	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) + 1)) ∈ ℝ)
95	94, 4	resubcld 11646	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((log‘((𝑁 + 1) + 1)) − (log‘(𝑁 + 1))) ∈ ℝ)
96		logdifbnd 26881	. . . . . . 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 11833	. . . . 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 11246	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘((𝑁 + 1) + 1)) ∈ ℂ)
100	67, 66, 99	subadd23d 11597	. . . . 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 5173	. . . 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 11646	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ ℝ)
103		leaddsub 11694	. . . . 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 1368	. . . 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 231	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ≤ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))))
106		fvoveq1 7428	. . . . 5 ⊢ (𝑛 = 𝑁 → (log‘(𝑛 + 1)) = (log‘(𝑁 + 1)))
107	85, 106	oveq12d 7423	. . . 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 7438	. . . 4 ⊢ (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))) ∈ V
110	107, 108, 109	fvmpt 6992	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐺‘𝑁) = (Σ𝑚 ∈ (1...𝑁)(1 / 𝑚) − (log‘(𝑁 + 1))))
111		fvoveq1 7428	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → (log‘(𝑛 + 1)) = (log‘((𝑁 + 1) + 1)))
112	77, 111	oveq12d 7423	. . . . 5 ⊢ (𝑛 = (𝑁 + 1) → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))) = (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))))
113		ovex 7438	. . . . 5 ⊢ (Σ𝑚 ∈ (1...(𝑁 + 1))(1 / 𝑚) − (log‘((𝑁 + 1) + 1))) ∈ V
114	112, 108, 113	fvmpt 6992	. . . 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 5173	. 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 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 ↦ cmpt 5224 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 < clt 11252 ≤ cle 11253 − cmin 11448 / cdiv 11875 ℕcn 12216 ℤ_≥cuz 12826 ℝ⁺crp 12980 ...cfz 13490 Σcsu 15638 expce 16011 logclog 26443

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-ef 16017 df-sin 16019 df-cos 16020 df-pi 16022 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-lp 22995 df-perf 22996 df-cn 23086 df-cnp 23087 df-haus 23174 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-xms 24181 df-ms 24182 df-tms 24183 df-cncf 24753 df-limc 25750 df-dv 25751 df-log 26445

This theorem is referenced by: emcllem6 26888 emcllem7 26889

W3C validator