emcllem7 - Metamath Proof Explorer

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Theorem emcllem7 26982

Description: Lemma for emcl 26983 and harmonicbnd 26984. 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 12821	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12552	. . . . 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 26981	. . . . . . 7 ⊢ (𝐹 ⇝ γ ∧ 𝐺 ⇝ γ)
8	7	simpri 485	. . . . . 6 ⊢ 𝐺 ⇝ γ
9	8	a1i 11	. . . . 5 ⊢ (⊤ → 𝐺 ⇝ γ)
10	3, 4	emcllem1 26976	. . . . . . . 8 ⊢ (𝐹:ℕ⟶ℝ ∧ 𝐺:ℕ⟶ℝ)
11	10	simpri 485	. . . . . . 7 ⊢ 𝐺:ℕ⟶ℝ
12	11	ffvelcdmi 7030	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ∈ ℝ)
13	12	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
14	1, 2, 9, 13	climrecl 15539	. . . 4 ⊢ (⊤ → γ ∈ ℝ)
15		1nn 12179	. . . . 5 ⊢ 1 ∈ ℕ
16		simpr 484	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
17	8	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → 𝐺 ⇝ γ)
18	12	adantl 481	. . . . . . 7 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
19	3, 4	emcllem2 26977	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ∧ (𝐺‘𝑘) ≤ (𝐺‘(𝑘 + 1))))
20	19	simprd 495	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ≤ (𝐺‘(𝑘 + 1)))
21	20	adantl 481	. . . . . . 7 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ≤ (𝐺‘(𝑘 + 1)))
22	1, 16, 17, 18, 21	climub 15618	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ≤ γ)
23	22	ralrimiva 3130	. . . . 5 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐺‘𝑖) ≤ γ)
24		fveq2 6835	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝐺‘𝑖) = (𝐺‘1))
25		oveq2 7369	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
26	25	sumeq1d 15656	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...1)(1 / 𝑚))
27		1z 12551	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
28		ax-1cn 11090	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
29		oveq2 7369	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 1 → (1 / 𝑚) = (1 / 1))
30		1div1e1 11839	. . . . . . . . . . . . . . 15 ⊢ (1 / 1) = 1
31	29, 30	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (1 / 𝑚) = 1)
32	31	fsum1 15703	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℤ ∧ 1 ∈ ℂ) → Σ𝑚 ∈ (1...1)(1 / 𝑚) = 1)
33	27, 28, 32	mp2an 693	. . . . . . . . . . . 12 ⊢ Σ𝑚 ∈ (1...1)(1 / 𝑚) = 1
34	26, 33	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = 1)
35		oveq1 7368	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (𝑛 + 1) = (1 + 1))
36		df-2 12238	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
37	35, 36	eqtr4di 2790	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝑛 + 1) = 2)
38	37	fveq2d 6839	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (log‘(𝑛 + 1)) = (log‘2))
39	34, 38	oveq12d 7379	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))) = (1 − (log‘2)))
40		1re 11138	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
41		2rp 12941	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
42		relogcl 26555	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℝ⁺ → (log‘2) ∈ ℝ)
43	41, 42	ax-mp 5	. . . . . . . . . . . 12 ⊢ (log‘2) ∈ ℝ
44	40, 43	resubcli 11450	. . . . . . . . . . 11 ⊢ (1 − (log‘2)) ∈ ℝ
45	44	elexi 3453	. . . . . . . . . 10 ⊢ (1 − (log‘2)) ∈ V
46	39, 4, 45	fvmpt 6942	. . . . . . . . 9 ⊢ (1 ∈ ℕ → (𝐺‘1) = (1 − (log‘2)))
47	15, 46	ax-mp 5	. . . . . . . 8 ⊢ (𝐺‘1) = (1 − (log‘2))
48	24, 47	eqtrdi 2788	. . . . . . 7 ⊢ (𝑖 = 1 → (𝐺‘𝑖) = (1 − (log‘2)))
49	48	breq1d 5096	. . . . . 6 ⊢ (𝑖 = 1 → ((𝐺‘𝑖) ≤ γ ↔ (1 − (log‘2)) ≤ γ))
50	49	rspcva 3563	. . . . 5 ⊢ ((1 ∈ ℕ ∧ ∀𝑖 ∈ ℕ (𝐺‘𝑖) ≤ γ) → (1 − (log‘2)) ≤ γ)
51	15, 23, 50	sylancr 588	. . . 4 ⊢ (⊤ → (1 − (log‘2)) ≤ γ)
52		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑖 → (𝐹‘𝑥) = (𝐹‘𝑖))
53	52	negeqd 11381	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑖 → -(𝐹‘𝑥) = -(𝐹‘𝑖))
54		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) = (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))
55		negex 11385	. . . . . . . . . . 11 ⊢ -(𝐹‘𝑖) ∈ V
56	53, 54, 55	fvmpt 6942	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℕ → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑖) = -(𝐹‘𝑖))
57	56	adantl 481	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑖) = -(𝐹‘𝑖))
58	7	simpli 483	. . . . . . . . . . . . 13 ⊢ 𝐹 ⇝ γ
59	58	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → 𝐹 ⇝ γ)
60		0cnd 11131	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ∈ ℂ)
61		nnex 12174	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
62	61	mptex 7172	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) ∈ V
63	62	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) ∈ V)
64	10	simpli 483	. . . . . . . . . . . . . . 15 ⊢ 𝐹:ℕ⟶ℝ
65	64	ffvelcdmi 7030	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ ℝ)
66	65	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
67	66	recnd 11167	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
68		fveq2 6835	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → (𝐹‘𝑥) = (𝐹‘𝑘))
69	68	negeqd 11381	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑘 → -(𝐹‘𝑥) = -(𝐹‘𝑘))
70		negex 11385	. . . . . . . . . . . . . . 15 ⊢ -(𝐹‘𝑘) ∈ V
71	69, 54, 70	fvmpt 6942	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) = -(𝐹‘𝑘))
72	71	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) = -(𝐹‘𝑘))
73		df-neg 11374	. . . . . . . . . . . . 13 ⊢ -(𝐹‘𝑘) = (0 − (𝐹‘𝑘))
74	72, 73	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) = (0 − (𝐹‘𝑘)))
75	1, 2, 59, 60, 63, 67, 74	climsubc2 15595	. . . . . . . . . . 11 ⊢ (⊤ → (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) ⇝ (0 − γ))
76	75	adantr 480	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) ⇝ (0 − γ))
77	66	renegcld 11571	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → -(𝐹‘𝑘) ∈ ℝ)
78	72, 77	eqeltrd 2837	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) ∈ ℝ)
79	78	adantlr 716	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) ∈ ℝ)
80	19	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
81	80	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
82		peano2nn 12180	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
83	82	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
84	64	ffvelcdmi 7030	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ ℝ)
85	83, 84	syl 17	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℝ)
86	85, 66	lenegd 11723	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ -(𝐹‘𝑘) ≤ -(𝐹‘(𝑘 + 1))))
87	81, 86	mpbid 232	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → -(𝐹‘𝑘) ≤ -(𝐹‘(𝑘 + 1)))
88		fveq2 6835	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑘 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑘 + 1)))
89	88	negeqd 11381	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑘 + 1) → -(𝐹‘𝑥) = -(𝐹‘(𝑘 + 1)))
90		negex 11385	. . . . . . . . . . . . . 14 ⊢ -(𝐹‘(𝑘 + 1)) ∈ V
91	89, 54, 90	fvmpt 6942	. . . . . . . . . . . . 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 5118	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) ≤ ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘(𝑘 + 1)))
94	93	adantlr 716	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) ≤ ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘(𝑘 + 1)))
95	1, 16, 76, 79, 94	climub 15618	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑖) ≤ (0 − γ))
96	57, 95	eqbrtrrd 5110	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → -(𝐹‘𝑖) ≤ (0 − γ))
97		df-neg 11374	. . . . . . . 8 ⊢ -γ = (0 − γ)
98	96, 97	breqtrrdi 5128	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → -(𝐹‘𝑖) ≤ -γ)
99	14	mptru 1549	. . . . . . . 8 ⊢ γ ∈ ℝ
100	64	ffvelcdmi 7030	. . . . . . . . 9 ⊢ (𝑖 ∈ ℕ → (𝐹‘𝑖) ∈ ℝ)
101	100	adantl 481	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ ℝ)
102		leneg 11647	. . . . . . . 8 ⊢ ((γ ∈ ℝ ∧ (𝐹‘𝑖) ∈ ℝ) → (γ ≤ (𝐹‘𝑖) ↔ -(𝐹‘𝑖) ≤ -γ))
103	99, 101, 102	sylancr 588	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (γ ≤ (𝐹‘𝑖) ↔ -(𝐹‘𝑖) ≤ -γ))
104	98, 103	mpbird 257	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → γ ≤ (𝐹‘𝑖))
105	104	ralrimiva 3130	. . . . 5 ⊢ (⊤ → ∀𝑖 ∈ ℕ γ ≤ (𝐹‘𝑖))
106		fveq2 6835	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = (𝐹‘1))
107		fveq2 6835	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (log‘𝑛) = (log‘1))
108		log1 26565	. . . . . . . . . . . . 13 ⊢ (log‘1) = 0
109	107, 108	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (log‘𝑛) = 0)
110	34, 109	oveq12d 7379	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = (1 − 0))
111		1m0e1 12291	. . . . . . . . . . 11 ⊢ (1 − 0) = 1
112	110, 111	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = 1)
113	40	elexi 3453	. . . . . . . . . 10 ⊢ 1 ∈ V
114	112, 3, 113	fvmpt 6942	. . . . . . . . 9 ⊢ (1 ∈ ℕ → (𝐹‘1) = 1)
115	15, 114	ax-mp 5	. . . . . . . 8 ⊢ (𝐹‘1) = 1
116	106, 115	eqtrdi 2788	. . . . . . 7 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = 1)
117	116	breq2d 5098	. . . . . 6 ⊢ (𝑖 = 1 → (γ ≤ (𝐹‘𝑖) ↔ γ ≤ 1))
118	117	rspcva 3563	. . . . 5 ⊢ ((1 ∈ ℕ ∧ ∀𝑖 ∈ ℕ γ ≤ (𝐹‘𝑖)) → γ ≤ 1)
119	15, 105, 118	sylancr 588	. . . 4 ⊢ (⊤ → γ ≤ 1)
120	44, 40	elicc2i 13359	. . . 4 ⊢ (γ ∈ ((1 − (log‘2))[,]1) ↔ (γ ∈ ℝ ∧ (1 − (log‘2)) ≤ γ ∧ γ ≤ 1))
121	14, 51, 119, 120	syl3anbrc 1345	. . 3 ⊢ (⊤ → γ ∈ ((1 − (log‘2))[,]1))
122		ffn 6663	. . . . 5 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ)
123	64, 122	mp1i 13	. . . 4 ⊢ (⊤ → 𝐹 Fn ℕ)
124	16, 1	eleqtrdi 2847	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ (ℤ_≥‘1))
125		elfznn 13501	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑖) → 𝑘 ∈ ℕ)
126	125	adantl 481	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ ℕ)
127	126, 65	syl 17	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑖)) → (𝐹‘𝑘) ∈ ℝ)
128		elfznn 13501	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑖 − 1)) → 𝑘 ∈ ℕ)
129	128	adantl 481	. . . . . . . . 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 13989	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ≤ (𝐹‘1))
132	131, 115	breqtrdi 5127	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ≤ 1)
133	99, 40	elicc2i 13359	. . . . . 6 ⊢ ((𝐹‘𝑖) ∈ (γ[,]1) ↔ ((𝐹‘𝑖) ∈ ℝ ∧ γ ≤ (𝐹‘𝑖) ∧ (𝐹‘𝑖) ≤ 1))
134	101, 104, 132, 133	syl3anbrc 1345	. . . . 5 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ (γ[,]1))
135	134	ralrimiva 3130	. . . 4 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐹‘𝑖) ∈ (γ[,]1))
136		ffnfv 7066	. . . 4 ⊢ (𝐹:ℕ⟶(γ[,]1) ↔ (𝐹 Fn ℕ ∧ ∀𝑖 ∈ ℕ (𝐹‘𝑖) ∈ (γ[,]1)))
137	123, 135, 136	sylanbrc 584	. . 3 ⊢ (⊤ → 𝐹:ℕ⟶(γ[,]1))
138		ffn 6663	. . . . 5 ⊢ (𝐺:ℕ⟶ℝ → 𝐺 Fn ℕ)
139	11, 138	mp1i 13	. . . 4 ⊢ (⊤ → 𝐺 Fn ℕ)
140	11	ffvelcdmi 7030	. . . . . . 7 ⊢ (𝑖 ∈ ℕ → (𝐺‘𝑖) ∈ ℝ)
141	140	adantl 481	. . . . . 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 13988	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘1) ≤ (𝐺‘𝑖))
145	47, 144	eqbrtrrid 5122	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (1 − (log‘2)) ≤ (𝐺‘𝑖))
146	44, 99	elicc2i 13359	. . . . . 6 ⊢ ((𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ) ↔ ((𝐺‘𝑖) ∈ ℝ ∧ (1 − (log‘2)) ≤ (𝐺‘𝑖) ∧ (𝐺‘𝑖) ≤ γ))
147	141, 145, 22, 146	syl3anbrc 1345	. . . . 5 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ))
148	147	ralrimiva 3130	. . . 4 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ))
149		ffnfv 7066	. . . 4 ⊢ (𝐺:ℕ⟶((1 − (log‘2))[,]γ) ↔ (𝐺 Fn ℕ ∧ ∀𝑖 ∈ ℕ (𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ)))
150	139, 148, 149	sylanbrc 584	. . 3 ⊢ (⊤ → 𝐺:ℕ⟶((1 − (log‘2))[,]γ))
151	121, 137, 150	3jca 1129	. 2 ⊢ (⊤ → (γ ∈ ((1 − (log‘2))[,]1) ∧ 𝐹:ℕ⟶(γ[,]1) ∧ 𝐺:ℕ⟶((1 − (log‘2))[,]γ)))
152	151	mptru 1549	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ∀wral 3052 Vcvv 3430 class class class wbr 5086 ↦ cmpt 5167 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 ℂcc 11030 ℝcr 11031 0cc0 11032 1c1 11033 + caddc 11035 ≤ cle 11174 − cmin 11371 -cneg 11372 / cdiv 11801 ℕcn 12168 2c2 12230 ℤcz 12518 ℤ_≥cuz 12782 ℝ⁺crp 12936 [,]cicc 13295 ...cfz 13455 ⇝ cli 15440 Σcsu 15642 logclog 26534 γcem 26972

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13296 df-ioc 13297 df-ico 13298 df-icc 13299 df-fz 13456 df-fzo 13603 df-fl 13745 df-mod 13823 df-seq 13958 df-exp 14018 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15023 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-limsup 15427 df-clim 15444 df-rlim 15445 df-sum 15643 df-ef 16026 df-sin 16028 df-cos 16029 df-pi 16031 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-hom 17238 df-cco 17239 df-rest 17379 df-topn 17380 df-0g 17398 df-gsum 17399 df-topgen 17400 df-pt 17401 df-prds 17404 df-xrs 17460 df-qtop 17465 df-imas 17466 df-xps 17468 df-mre 17542 df-mrc 17543 df-acs 17545 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-mulg 19038 df-cntz 19286 df-cmn 19751 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22872 df-topon 22889 df-topsp 22911 df-bases 22924 df-cld 22997 df-ntr 22998 df-cls 22999 df-nei 23076 df-lp 23114 df-perf 23115 df-cn 23205 df-cnp 23206 df-haus 23293 df-tx 23540 df-hmeo 23733 df-fil 23824 df-fm 23916 df-flim 23917 df-flf 23918 df-xms 24298 df-ms 24299 df-tms 24300 df-cncf 24858 df-limc 25846 df-dv 25847 df-log 26536 df-em 26973

This theorem is referenced by: emcl 26983 harmonicbnd 26984 harmonicbnd2 26985

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