emcllem7 - Metamath Proof Explorer

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Theorem emcllem7 27063

Description: Lemma for emcl 27064 and harmonicbnd 27065. Derive bounds on γ as 𝐹(1) and 𝐺(1). (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 9-Apr-2016.)

**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))))
emcl.3	⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛))))
emcl.4	⊢ 𝑇 = (𝑛 ∈ ℕ ↦ ((1 / 𝑛) − (log‘(1 + (1 / 𝑛)))))

**Assertion**
Ref	Expression
emcllem7	⊢ (γ ∈ ((1 − (log‘2))[,]1) ∧ 𝐹:ℕ⟶(γ[,]1) ∧ 𝐺:ℕ⟶((1 − (log‘2))[,]γ))

Distinct variable groups: 𝑚,𝐻 𝑚,𝑛,𝑇 Allowed substitution hints: 𝐹(𝑚,𝑛) 𝐺(𝑚,𝑛) 𝐻(𝑛)

Proof of Theorem emcllem7 Dummy variables 𝑖 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 12878	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12602	. . . . 5 ⊢ (⊤ → 1 ∈ ℤ)
3		emcl.1	. . . . . . . 8 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)))
4		emcl.2	. . . . . . . 8 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))))
5		emcl.3	. . . . . . . 8 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (log‘(1 + (1 / 𝑛))))
6		emcl.4	. . . . . . . 8 ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ ((1 / 𝑛) − (log‘(1 + (1 / 𝑛)))))
7	3, 4, 5, 6	emcllem6 27062	. . . . . . 7 ⊢ (𝐹 ⇝ γ ∧ 𝐺 ⇝ γ)
8	7	simpri 489	. . . . . 6 ⊢ 𝐺 ⇝ γ
9	8	a1i 11	. . . . 5 ⊢ (⊤ → 𝐺 ⇝ γ)
10	3, 4	emcllem1 27057	. . . . . . . 8 ⊢ (𝐹:ℕ⟶ℝ ∧ 𝐺:ℕ⟶ℝ)
11	10	simpri 489	. . . . . . 7 ⊢ 𝐺:ℕ⟶ℝ
12	11	ffvelcdmi 7064	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ∈ ℝ)
13	12	adantl 485	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
14	1, 2, 9, 13	climrecl 15610	. . . 4 ⊢ (⊤ → γ ∈ ℝ)
15		1nn 12221	. . . . 5 ⊢ 1 ∈ ℕ
16		simpr 488	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
17	8	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → 𝐺 ⇝ γ)
18	12	adantl 485	. . . . . . 7 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
19	3, 4	emcllem2 27058	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ∧ (𝐺‘𝑘) ≤ (𝐺‘(𝑘 + 1))))
20	19	simprd 499	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ≤ (𝐺‘(𝑘 + 1)))
21	20	adantl 485	. . . . . . 7 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ≤ (𝐺‘(𝑘 + 1)))
22	1, 16, 17, 18, 21	climub 15689	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ≤ γ)
23	22	ralrimiva 3154	. . . . 5 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐺‘𝑖) ≤ γ)
24		fveq2 6867	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝐺‘𝑖) = (𝐺‘1))
25		oveq2 7404	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
26	25	sumeq1d 15727	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...1)(1 / 𝑚))
27		1z 12601	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
28		ax-1cn 11131	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
29		oveq2 7404	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 1 → (1 / 𝑚) = (1 / 1))
30		1div1e1 11881	. . . . . . . . . . . . . . 15 ⊢ (1 / 1) = 1
31	29, 30	eqtrdi 2813	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (1 / 𝑚) = 1)
32	31	fsum1 15774	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℤ ∧ 1 ∈ ℂ) → Σ𝑚 ∈ (1...1)(1 / 𝑚) = 1)
33	27, 28, 32	mp2an 702	. . . . . . . . . . . 12 ⊢ Σ𝑚 ∈ (1...1)(1 / 𝑚) = 1
34	26, 33	eqtrdi 2813	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = 1)
35		oveq1 7403	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (𝑛 + 1) = (1 + 1))
36		df-2 12280	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
37	35, 36	eqtr4di 2815	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝑛 + 1) = 2)
38	37	fveq2d 6871	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (log‘(𝑛 + 1)) = (log‘2))
39	34, 38	oveq12d 7414	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))) = (1 − (log‘2)))
40		1re 11181	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
41		2rp 12998	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
42		relogcl 26637	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℝ⁺ → (log‘2) ∈ ℝ)
43	41, 42	ax-mp 5	. . . . . . . . . . . 12 ⊢ (log‘2) ∈ ℝ
44	40, 43	resubcli 11493	. . . . . . . . . . 11 ⊢ (1 − (log‘2)) ∈ ℝ
45	44	elexi 3476	. . . . . . . . . 10 ⊢ (1 − (log‘2)) ∈ V
46	39, 4, 45	fvmpt 6975	. . . . . . . . 9 ⊢ (1 ∈ ℕ → (𝐺‘1) = (1 − (log‘2)))
47	15, 46	ax-mp 5	. . . . . . . 8 ⊢ (𝐺‘1) = (1 − (log‘2))
48	24, 47	eqtrdi 2813	. . . . . . 7 ⊢ (𝑖 = 1 → (𝐺‘𝑖) = (1 − (log‘2)))
49	48	breq1d 5110	. . . . . 6 ⊢ (𝑖 = 1 → ((𝐺‘𝑖) ≤ γ ↔ (1 − (log‘2)) ≤ γ))
50	49	rspcva 3579	. . . . 5 ⊢ ((1 ∈ ℕ ∧ ∀𝑖 ∈ ℕ (𝐺‘𝑖) ≤ γ) → (1 − (log‘2)) ≤ γ)
51	15, 23, 50	sylancr 596	. . . 4 ⊢ (⊤ → (1 − (log‘2)) ≤ γ)
52		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑖 → (𝐹‘𝑥) = (𝐹‘𝑖))
53	52	negeqd 11424	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑖 → -(𝐹‘𝑥) = -(𝐹‘𝑖))
54		eqid 2762	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) = (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))
55		negex 11428	. . . . . . . . . . 11 ⊢ -(𝐹‘𝑖) ∈ V
56	53, 54, 55	fvmpt 6975	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℕ → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑖) = -(𝐹‘𝑖))
57	56	adantl 485	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑖) = -(𝐹‘𝑖))
58	7	simpli 487	. . . . . . . . . . . . 13 ⊢ 𝐹 ⇝ γ
59	58	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → 𝐹 ⇝ γ)
60		0cnd 11172	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ∈ ℂ)
61		nnex 12216	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
62	61	mptex 7207	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) ∈ V
63	62	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) ∈ V)
64	10	simpli 487	. . . . . . . . . . . . . . 15 ⊢ 𝐹:ℕ⟶ℝ
65	64	ffvelcdmi 7064	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ ℝ)
66	65	adantl 485	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
67	66	recnd 11210	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
68		fveq2 6867	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → (𝐹‘𝑥) = (𝐹‘𝑘))
69	68	negeqd 11424	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑘 → -(𝐹‘𝑥) = -(𝐹‘𝑘))
70		negex 11428	. . . . . . . . . . . . . . 15 ⊢ -(𝐹‘𝑘) ∈ V
71	69, 54, 70	fvmpt 6975	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) = -(𝐹‘𝑘))
72	71	adantl 485	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) = -(𝐹‘𝑘))
73		df-neg 11417	. . . . . . . . . . . . 13 ⊢ -(𝐹‘𝑘) = (0 − (𝐹‘𝑘))
74	72, 73	eqtrdi 2813	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) = (0 − (𝐹‘𝑘)))
75	1, 2, 59, 60, 63, 67, 74	climsubc2 15666	. . . . . . . . . . 11 ⊢ (⊤ → (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) ⇝ (0 − γ))
76	75	adantr 484	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) ⇝ (0 − γ))
77	66	renegcld 11614	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → -(𝐹‘𝑘) ∈ ℝ)
78	72, 77	eqeltrd 2862	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) ∈ ℝ)
79	78	adantlr 725	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) ∈ ℝ)
80	19	simpld 498	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
81	80	adantl 485	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
82		peano2nn 12222	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
83	82	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
84	64	ffvelcdmi 7064	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ ℝ)
85	83, 84	syl 17	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℝ)
86	85, 66	lenegd 11766	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ -(𝐹‘𝑘) ≤ -(𝐹‘(𝑘 + 1))))
87	81, 86	mpbid 234	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → -(𝐹‘𝑘) ≤ -(𝐹‘(𝑘 + 1)))
88		fveq2 6867	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑘 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑘 + 1)))
89	88	negeqd 11424	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑘 + 1) → -(𝐹‘𝑥) = -(𝐹‘(𝑘 + 1)))
90		negex 11428	. . . . . . . . . . . . . 14 ⊢ -(𝐹‘(𝑘 + 1)) ∈ V
91	89, 54, 90	fvmpt 6975	. . . . . . . . . . . . 13 ⊢ ((𝑘 + 1) ∈ ℕ → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘(𝑘 + 1)) = -(𝐹‘(𝑘 + 1)))
92	83, 91	syl 17	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘(𝑘 + 1)) = -(𝐹‘(𝑘 + 1)))
93	87, 72, 92	3brtr4d 5132	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) ≤ ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘(𝑘 + 1)))
94	93	adantlr 725	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) ≤ ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘(𝑘 + 1)))
95	1, 16, 76, 79, 94	climub 15689	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑖) ≤ (0 − γ))
96	57, 95	eqbrtrrd 5124	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → -(𝐹‘𝑖) ≤ (0 − γ))
97		df-neg 11417	. . . . . . . 8 ⊢ -γ = (0 − γ)
98	96, 97	breqtrrdi 5142	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → -(𝐹‘𝑖) ≤ -γ)
99	14	mptru 1567	. . . . . . . 8 ⊢ γ ∈ ℝ
100	64	ffvelcdmi 7064	. . . . . . . . 9 ⊢ (𝑖 ∈ ℕ → (𝐹‘𝑖) ∈ ℝ)
101	100	adantl 485	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ ℝ)
102		leneg 11690	. . . . . . . 8 ⊢ ((γ ∈ ℝ ∧ (𝐹‘𝑖) ∈ ℝ) → (γ ≤ (𝐹‘𝑖) ↔ -(𝐹‘𝑖) ≤ -γ))
103	99, 101, 102	sylancr 596	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (γ ≤ (𝐹‘𝑖) ↔ -(𝐹‘𝑖) ≤ -γ))
104	98, 103	mpbird 259	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → γ ≤ (𝐹‘𝑖))
105	104	ralrimiva 3154	. . . . 5 ⊢ (⊤ → ∀𝑖 ∈ ℕ γ ≤ (𝐹‘𝑖))
106		fveq2 6867	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = (𝐹‘1))
107		fveq2 6867	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (log‘𝑛) = (log‘1))
108		log1 26647	. . . . . . . . . . . . 13 ⊢ (log‘1) = 0
109	107, 108	eqtrdi 2813	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (log‘𝑛) = 0)
110	34, 109	oveq12d 7414	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = (1 − 0))
111		1m0e1 12337	. . . . . . . . . . 11 ⊢ (1 − 0) = 1
112	110, 111	eqtrdi 2813	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = 1)
113	40	elexi 3476	. . . . . . . . . 10 ⊢ 1 ∈ V
114	112, 3, 113	fvmpt 6975	. . . . . . . . 9 ⊢ (1 ∈ ℕ → (𝐹‘1) = 1)
115	15, 114	ax-mp 5	. . . . . . . 8 ⊢ (𝐹‘1) = 1
116	106, 115	eqtrdi 2813	. . . . . . 7 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = 1)
117	116	breq2d 5112	. . . . . 6 ⊢ (𝑖 = 1 → (γ ≤ (𝐹‘𝑖) ↔ γ ≤ 1))
118	117	rspcva 3579	. . . . 5 ⊢ ((1 ∈ ℕ ∧ ∀𝑖 ∈ ℕ γ ≤ (𝐹‘𝑖)) → γ ≤ 1)
119	15, 105, 118	sylancr 596	. . . 4 ⊢ (⊤ → γ ≤ 1)
120	44, 40	elicc2i 13416	. . . 4 ⊢ (γ ∈ ((1 − (log‘2))[,]1) ↔ (γ ∈ ℝ ∧ (1 − (log‘2)) ≤ γ ∧ γ ≤ 1))
121	14, 51, 119, 120	syl3anbrc 1357	. . 3 ⊢ (⊤ → γ ∈ ((1 − (log‘2))[,]1))
122		ffn 6691	. . . . 5 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ)
123	64, 122	mp1i 13	. . . 4 ⊢ (⊤ → 𝐹 Fn ℕ)
124	16, 1	eleqtrdi 2872	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ (ℤ_≥‘1))
125		elfznn 13558	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑖) → 𝑘 ∈ ℕ)
126	125	adantl 485	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ ℕ)
127	126, 65	syl 17	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑖)) → (𝐹‘𝑘) ∈ ℝ)
128		elfznn 13558	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑖 − 1)) → 𝑘 ∈ ℕ)
129	128	adantl 485	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ (1...(𝑖 − 1))) → 𝑘 ∈ ℕ)
130	129, 80	syl 17	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ (1...(𝑖 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
131	124, 127, 130	monoord2 14046	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ≤ (𝐹‘1))
132	131, 115	breqtrdi 5141	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ≤ 1)
133	99, 40	elicc2i 13416	. . . . . 6 ⊢ ((𝐹‘𝑖) ∈ (γ[,]1) ↔ ((𝐹‘𝑖) ∈ ℝ ∧ γ ≤ (𝐹‘𝑖) ∧ (𝐹‘𝑖) ≤ 1))
134	101, 104, 132, 133	syl3anbrc 1357	. . . . 5 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ (γ[,]1))
135	134	ralrimiva 3154	. . . 4 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐹‘𝑖) ∈ (γ[,]1))
136		ffnfv 7100	. . . 4 ⊢ (𝐹:ℕ⟶(γ[,]1) ↔ (𝐹 Fn ℕ ∧ ∀𝑖 ∈ ℕ (𝐹‘𝑖) ∈ (γ[,]1)))
137	123, 135, 136	sylanbrc 592	. . 3 ⊢ (⊤ → 𝐹:ℕ⟶(γ[,]1))
138		ffn 6691	. . . . 5 ⊢ (𝐺:ℕ⟶ℝ → 𝐺 Fn ℕ)
139	11, 138	mp1i 13	. . . 4 ⊢ (⊤ → 𝐺 Fn ℕ)
140	11	ffvelcdmi 7064	. . . . . . 7 ⊢ (𝑖 ∈ ℕ → (𝐺‘𝑖) ∈ ℝ)
141	140	adantl 485	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ ℝ)
142	126, 12	syl 17	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑖)) → (𝐺‘𝑘) ∈ ℝ)
143	129, 20	syl 17	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ (1...(𝑖 − 1))) → (𝐺‘𝑘) ≤ (𝐺‘(𝑘 + 1)))
144	124, 142, 143	monoord 14045	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘1) ≤ (𝐺‘𝑖))
145	47, 144	eqbrtrrid 5136	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (1 − (log‘2)) ≤ (𝐺‘𝑖))
146	44, 99	elicc2i 13416	. . . . . 6 ⊢ ((𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ) ↔ ((𝐺‘𝑖) ∈ ℝ ∧ (1 − (log‘2)) ≤ (𝐺‘𝑖) ∧ (𝐺‘𝑖) ≤ γ))
147	141, 145, 22, 146	syl3anbrc 1357	. . . . 5 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ))
148	147	ralrimiva 3154	. . . 4 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ))
149		ffnfv 7100	. . . 4 ⊢ (𝐺:ℕ⟶((1 − (log‘2))[,]γ) ↔ (𝐺 Fn ℕ ∧ ∀𝑖 ∈ ℕ (𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ)))
150	139, 148, 149	sylanbrc 592	. . 3 ⊢ (⊤ → 𝐺:ℕ⟶((1 − (log‘2))[,]γ))
151	121, 137, 150	3jca 1141	. 2 ⊢ (⊤ → (γ ∈ ((1 − (log‘2))[,]1) ∧ 𝐹:ℕ⟶(γ[,]1) ∧ 𝐺:ℕ⟶((1 − (log‘2))[,]γ)))
152	151	mptru 1567	1 ⊢ (γ ∈ ((1 − (log‘2))[,]1) ∧ 𝐹:ℕ⟶(γ[,]1) ∧ 𝐺:ℕ⟶((1 − (log‘2))[,]γ))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 399 ∧ w3a 1098 = wceq 1560 ⊤wtru 1561 ∈ wcel 2142 ∀wral 3076 Vcvv 3454 class class class wbr 5100 ↦ cmpt 5181 Fn wfn 6516 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ℂcc 11071 ℝcr 11072 0cc0 11073 1c1 11074 + caddc 11076 ≤ cle 11217 − cmin 11414 -cneg 11415 / cdiv 11844 ℕcn 12210 2c2 12272 ℤcz 12568 ℤ_≥cuz 12839 ℝ⁺crp 12993 [,]cicc 13352 ...cfz 13512 ⇝ cli 15511 Σcsu 15713 logclog 26616 γcem 27053

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-ioc 13354 df-ico 13355 df-icc 13356 df-fz 13513 df-fzo 13660 df-fl 13802 df-mod 13880 df-seq 14015 df-exp 14075 df-fac 14287 df-bc 14316 df-hash 14344 df-shft 15080 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-limsup 15498 df-clim 15515 df-rlim 15516 df-sum 15714 df-ef 16097 df-sin 16099 df-cos 16100 df-pi 16102 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-pt 17473 df-prds 17476 df-xrs 17532 df-qtop 17537 df-imas 17538 df-xps 17540 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-mulg 19110 df-cntz 19357 df-cmn 19822 df-psmet 21413 df-xmet 21414 df-met 21415 df-bl 21416 df-mopn 21417 df-fbas 21418 df-fg 21419 df-cnfld 21422 df-top 22951 df-topon 22968 df-topsp 22990 df-bases 23003 df-cld 23076 df-ntr 23077 df-cls 23078 df-nei 23155 df-lp 23193 df-perf 23194 df-cn 23284 df-cnp 23285 df-haus 23372 df-tx 23619 df-hmeo 23812 df-fil 23903 df-fm 23995 df-flim 23996 df-flf 23997 df-xms 24377 df-ms 24378 df-tms 24379 df-cncf 24937 df-limc 25925 df-dv 25926 df-log 26618 df-em 27054

This theorem is referenced by: emcl 27064 harmonicbnd 27065 harmonicbnd2 27066

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