emcllem7 - Metamath Proof Explorer

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Theorem emcllem7 26056

Description: Lemma for emcl 26057 and harmonicbnd 26058. 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 12550	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12281	. . . . 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 26055	. . . . . . 7 ⊢ (𝐹 ⇝ γ ∧ 𝐺 ⇝ γ)
8	7	simpri 485	. . . . . 6 ⊢ 𝐺 ⇝ γ
9	8	a1i 11	. . . . 5 ⊢ (⊤ → 𝐺 ⇝ γ)
10	3, 4	emcllem1 26050	. . . . . . . 8 ⊢ (𝐹:ℕ⟶ℝ ∧ 𝐺:ℕ⟶ℝ)
11	10	simpri 485	. . . . . . 7 ⊢ 𝐺:ℕ⟶ℝ
12	11	ffvelrni 6942	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ∈ ℝ)
13	12	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
14	1, 2, 9, 13	climrecl 15220	. . . 4 ⊢ (⊤ → γ ∈ ℝ)
15		1nn 11914	. . . . 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 26051	. . . . . . . . 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 15301	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ≤ γ)
23	22	ralrimiva 3107	. . . . 5 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐺‘𝑖) ≤ γ)
24		fveq2 6756	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝐺‘𝑖) = (𝐺‘1))
25		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
26	25	sumeq1d 15341	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...1)(1 / 𝑚))
27		1z 12280	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
28		ax-1cn 10860	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
29		oveq2 7263	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 1 → (1 / 𝑚) = (1 / 1))
30		1div1e1 11595	. . . . . . . . . . . . . . 15 ⊢ (1 / 1) = 1
31	29, 30	eqtrdi 2795	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (1 / 𝑚) = 1)
32	31	fsum1 15387	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℤ ∧ 1 ∈ ℂ) → Σ𝑚 ∈ (1...1)(1 / 𝑚) = 1)
33	27, 28, 32	mp2an 688	. . . . . . . . . . . 12 ⊢ Σ𝑚 ∈ (1...1)(1 / 𝑚) = 1
34	26, 33	eqtrdi 2795	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = 1)
35		oveq1 7262	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (𝑛 + 1) = (1 + 1))
36		df-2 11966	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
37	35, 36	eqtr4di 2797	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝑛 + 1) = 2)
38	37	fveq2d 6760	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (log‘(𝑛 + 1)) = (log‘2))
39	34, 38	oveq12d 7273	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))) = (1 − (log‘2)))
40		1re 10906	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
41		2rp 12664	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
42		relogcl 25636	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℝ⁺ → (log‘2) ∈ ℝ)
43	41, 42	ax-mp 5	. . . . . . . . . . . 12 ⊢ (log‘2) ∈ ℝ
44	40, 43	resubcli 11213	. . . . . . . . . . 11 ⊢ (1 − (log‘2)) ∈ ℝ
45	44	elexi 3441	. . . . . . . . . 10 ⊢ (1 − (log‘2)) ∈ V
46	39, 4, 45	fvmpt 6857	. . . . . . . . 9 ⊢ (1 ∈ ℕ → (𝐺‘1) = (1 − (log‘2)))
47	15, 46	ax-mp 5	. . . . . . . 8 ⊢ (𝐺‘1) = (1 − (log‘2))
48	24, 47	eqtrdi 2795	. . . . . . 7 ⊢ (𝑖 = 1 → (𝐺‘𝑖) = (1 − (log‘2)))
49	48	breq1d 5080	. . . . . 6 ⊢ (𝑖 = 1 → ((𝐺‘𝑖) ≤ γ ↔ (1 − (log‘2)) ≤ γ))
50	49	rspcva 3550	. . . . 5 ⊢ ((1 ∈ ℕ ∧ ∀𝑖 ∈ ℕ (𝐺‘𝑖) ≤ γ) → (1 − (log‘2)) ≤ γ)
51	15, 23, 50	sylancr 586	. . . 4 ⊢ (⊤ → (1 − (log‘2)) ≤ γ)
52		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑖 → (𝐹‘𝑥) = (𝐹‘𝑖))
53	52	negeqd 11145	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑖 → -(𝐹‘𝑥) = -(𝐹‘𝑖))
54		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) = (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))
55		negex 11149	. . . . . . . . . . 11 ⊢ -(𝐹‘𝑖) ∈ V
56	53, 54, 55	fvmpt 6857	. . . . . . . . . 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 10899	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ∈ ℂ)
61		nnex 11909	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
62	61	mptex 7081	. . . . . . . . . . . . 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	ffvelrni 6942	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ ℝ)
66	65	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
67	66	recnd 10934	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
68		fveq2 6756	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → (𝐹‘𝑥) = (𝐹‘𝑘))
69	68	negeqd 11145	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑘 → -(𝐹‘𝑥) = -(𝐹‘𝑘))
70		negex 11149	. . . . . . . . . . . . . . 15 ⊢ -(𝐹‘𝑘) ∈ V
71	69, 54, 70	fvmpt 6857	. . . . . . . . . . . . . 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 11138	. . . . . . . . . . . . 13 ⊢ -(𝐹‘𝑘) = (0 − (𝐹‘𝑘))
74	72, 73	eqtrdi 2795	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) = (0 − (𝐹‘𝑘)))
75	1, 2, 59, 60, 63, 67, 74	climsubc2 15276	. . . . . . . . . . 11 ⊢ (⊤ → (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) ⇝ (0 − γ))
76	75	adantr 480	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) ⇝ (0 − γ))
77	66	renegcld 11332	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → -(𝐹‘𝑘) ∈ ℝ)
78	72, 77	eqeltrd 2839	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) ∈ ℝ)
79	78	adantlr 711	. . . . . . . . . 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 11915	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
83	82	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
84	64	ffvelrni 6942	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ ℝ)
85	83, 84	syl 17	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℝ)
86	85, 66	lenegd 11484	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ -(𝐹‘𝑘) ≤ -(𝐹‘(𝑘 + 1))))
87	81, 86	mpbid 231	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → -(𝐹‘𝑘) ≤ -(𝐹‘(𝑘 + 1)))
88		fveq2 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑘 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑘 + 1)))
89	88	negeqd 11145	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑘 + 1) → -(𝐹‘𝑥) = -(𝐹‘(𝑘 + 1)))
90		negex 11149	. . . . . . . . . . . . . 14 ⊢ -(𝐹‘(𝑘 + 1)) ∈ V
91	89, 54, 90	fvmpt 6857	. . . . . . . . . . . . 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 5102	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) ≤ ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘(𝑘 + 1)))
94	93	adantlr 711	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) ≤ ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘(𝑘 + 1)))
95	1, 16, 76, 79, 94	climub 15301	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑖) ≤ (0 − γ))
96	57, 95	eqbrtrrd 5094	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → -(𝐹‘𝑖) ≤ (0 − γ))
97		df-neg 11138	. . . . . . . 8 ⊢ -γ = (0 − γ)
98	96, 97	breqtrrdi 5112	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → -(𝐹‘𝑖) ≤ -γ)
99	14	mptru 1546	. . . . . . . 8 ⊢ γ ∈ ℝ
100	64	ffvelrni 6942	. . . . . . . . 9 ⊢ (𝑖 ∈ ℕ → (𝐹‘𝑖) ∈ ℝ)
101	100	adantl 481	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ ℝ)
102		leneg 11408	. . . . . . . 8 ⊢ ((γ ∈ ℝ ∧ (𝐹‘𝑖) ∈ ℝ) → (γ ≤ (𝐹‘𝑖) ↔ -(𝐹‘𝑖) ≤ -γ))
103	99, 101, 102	sylancr 586	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (γ ≤ (𝐹‘𝑖) ↔ -(𝐹‘𝑖) ≤ -γ))
104	98, 103	mpbird 256	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → γ ≤ (𝐹‘𝑖))
105	104	ralrimiva 3107	. . . . 5 ⊢ (⊤ → ∀𝑖 ∈ ℕ γ ≤ (𝐹‘𝑖))
106		fveq2 6756	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = (𝐹‘1))
107		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (log‘𝑛) = (log‘1))
108		log1 25646	. . . . . . . . . . . . 13 ⊢ (log‘1) = 0
109	107, 108	eqtrdi 2795	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (log‘𝑛) = 0)
110	34, 109	oveq12d 7273	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = (1 − 0))
111		1m0e1 12024	. . . . . . . . . . 11 ⊢ (1 − 0) = 1
112	110, 111	eqtrdi 2795	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = 1)
113	40	elexi 3441	. . . . . . . . . 10 ⊢ 1 ∈ V
114	112, 3, 113	fvmpt 6857	. . . . . . . . 9 ⊢ (1 ∈ ℕ → (𝐹‘1) = 1)
115	15, 114	ax-mp 5	. . . . . . . 8 ⊢ (𝐹‘1) = 1
116	106, 115	eqtrdi 2795	. . . . . . 7 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = 1)
117	116	breq2d 5082	. . . . . 6 ⊢ (𝑖 = 1 → (γ ≤ (𝐹‘𝑖) ↔ γ ≤ 1))
118	117	rspcva 3550	. . . . 5 ⊢ ((1 ∈ ℕ ∧ ∀𝑖 ∈ ℕ γ ≤ (𝐹‘𝑖)) → γ ≤ 1)
119	15, 105, 118	sylancr 586	. . . 4 ⊢ (⊤ → γ ≤ 1)
120	44, 40	elicc2i 13074	. . . 4 ⊢ (γ ∈ ((1 − (log‘2))[,]1) ↔ (γ ∈ ℝ ∧ (1 − (log‘2)) ≤ γ ∧ γ ≤ 1))
121	14, 51, 119, 120	syl3anbrc 1341	. . 3 ⊢ (⊤ → γ ∈ ((1 − (log‘2))[,]1))
122		ffn 6584	. . . . 5 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ)
123	64, 122	mp1i 13	. . . 4 ⊢ (⊤ → 𝐹 Fn ℕ)
124	16, 1	eleqtrdi 2849	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ (ℤ_≥‘1))
125		elfznn 13214	. . . . . . . . . 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 13214	. . . . . . . . . 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 13682	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ≤ (𝐹‘1))
132	131, 115	breqtrdi 5111	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ≤ 1)
133	99, 40	elicc2i 13074	. . . . . 6 ⊢ ((𝐹‘𝑖) ∈ (γ[,]1) ↔ ((𝐹‘𝑖) ∈ ℝ ∧ γ ≤ (𝐹‘𝑖) ∧ (𝐹‘𝑖) ≤ 1))
134	101, 104, 132, 133	syl3anbrc 1341	. . . . 5 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ (γ[,]1))
135	134	ralrimiva 3107	. . . 4 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐹‘𝑖) ∈ (γ[,]1))
136		ffnfv 6974	. . . 4 ⊢ (𝐹:ℕ⟶(γ[,]1) ↔ (𝐹 Fn ℕ ∧ ∀𝑖 ∈ ℕ (𝐹‘𝑖) ∈ (γ[,]1)))
137	123, 135, 136	sylanbrc 582	. . 3 ⊢ (⊤ → 𝐹:ℕ⟶(γ[,]1))
138		ffn 6584	. . . . 5 ⊢ (𝐺:ℕ⟶ℝ → 𝐺 Fn ℕ)
139	11, 138	mp1i 13	. . . 4 ⊢ (⊤ → 𝐺 Fn ℕ)
140	11	ffvelrni 6942	. . . . . . 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 13681	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘1) ≤ (𝐺‘𝑖))
145	47, 144	eqbrtrrid 5106	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (1 − (log‘2)) ≤ (𝐺‘𝑖))
146	44, 99	elicc2i 13074	. . . . . 6 ⊢ ((𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ) ↔ ((𝐺‘𝑖) ∈ ℝ ∧ (1 − (log‘2)) ≤ (𝐺‘𝑖) ∧ (𝐺‘𝑖) ≤ γ))
147	141, 145, 22, 146	syl3anbrc 1341	. . . . 5 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ))
148	147	ralrimiva 3107	. . . 4 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ))
149		ffnfv 6974	. . . 4 ⊢ (𝐺:ℕ⟶((1 − (log‘2))[,]γ) ↔ (𝐺 Fn ℕ ∧ ∀𝑖 ∈ ℕ (𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ)))
150	139, 148, 149	sylanbrc 582	. . 3 ⊢ (⊤ → 𝐺:ℕ⟶((1 − (log‘2))[,]γ))
151	121, 137, 150	3jca 1126	. 2 ⊢ (⊤ → (γ ∈ ((1 − (log‘2))[,]1) ∧ 𝐹:ℕ⟶(γ[,]1) ∧ 𝐺:ℕ⟶((1 − (log‘2))[,]γ)))
152	151	mptru 1546	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 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ⊤wtru 1540 ∈ wcel 2108 ∀wral 3063 Vcvv 3422 class class class wbr 5070 ↦ cmpt 5153 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 ≤ cle 10941 − cmin 11135 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659 [,]cicc 13011 ...cfz 13168 ⇝ cli 15121 Σcsu 15325 logclog 25615 γcem 26046

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617 df-em 26047

This theorem is referenced by: emcl 26057 harmonicbnd 26058 harmonicbnd2 26059

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