emcllem7 - Metamath Proof Explorer

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Theorem emcllem7 26912

Description: Lemma for emcl 26913 and harmonicbnd 26914. 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 12836	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12564	. . . . 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 26911	. . . . . . 7 ⊢ (𝐹 ⇝ γ ∧ 𝐺 ⇝ γ)
8	7	simpri 485	. . . . . 6 ⊢ 𝐺 ⇝ γ
9	8	a1i 11	. . . . 5 ⊢ (⊤ → 𝐺 ⇝ γ)
10	3, 4	emcllem1 26906	. . . . . . . 8 ⊢ (𝐹:ℕ⟶ℝ ∧ 𝐺:ℕ⟶ℝ)
11	10	simpri 485	. . . . . . 7 ⊢ 𝐺:ℕ⟶ℝ
12	11	ffvelcdmi 7055	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ∈ ℝ)
13	12	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
14	1, 2, 9, 13	climrecl 15549	. . . 4 ⊢ (⊤ → γ ∈ ℝ)
15		1nn 12197	. . . . 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 26907	. . . . . . . . 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 15628	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ≤ γ)
23	22	ralrimiva 3125	. . . . 5 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐺‘𝑖) ≤ γ)
24		fveq2 6858	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝐺‘𝑖) = (𝐺‘1))
25		oveq2 7395	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
26	25	sumeq1d 15666	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...1)(1 / 𝑚))
27		1z 12563	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
28		ax-1cn 11126	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
29		oveq2 7395	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 1 → (1 / 𝑚) = (1 / 1))
30		1div1e1 11873	. . . . . . . . . . . . . . 15 ⊢ (1 / 1) = 1
31	29, 30	eqtrdi 2780	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (1 / 𝑚) = 1)
32	31	fsum1 15713	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℤ ∧ 1 ∈ ℂ) → Σ𝑚 ∈ (1...1)(1 / 𝑚) = 1)
33	27, 28, 32	mp2an 692	. . . . . . . . . . . 12 ⊢ Σ𝑚 ∈ (1...1)(1 / 𝑚) = 1
34	26, 33	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = 1)
35		oveq1 7394	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (𝑛 + 1) = (1 + 1))
36		df-2 12249	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
37	35, 36	eqtr4di 2782	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝑛 + 1) = 2)
38	37	fveq2d 6862	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (log‘(𝑛 + 1)) = (log‘2))
39	34, 38	oveq12d 7405	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))) = (1 − (log‘2)))
40		1re 11174	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
41		2rp 12956	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
42		relogcl 26484	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℝ⁺ → (log‘2) ∈ ℝ)
43	41, 42	ax-mp 5	. . . . . . . . . . . 12 ⊢ (log‘2) ∈ ℝ
44	40, 43	resubcli 11484	. . . . . . . . . . 11 ⊢ (1 − (log‘2)) ∈ ℝ
45	44	elexi 3470	. . . . . . . . . 10 ⊢ (1 − (log‘2)) ∈ V
46	39, 4, 45	fvmpt 6968	. . . . . . . . 9 ⊢ (1 ∈ ℕ → (𝐺‘1) = (1 − (log‘2)))
47	15, 46	ax-mp 5	. . . . . . . 8 ⊢ (𝐺‘1) = (1 − (log‘2))
48	24, 47	eqtrdi 2780	. . . . . . 7 ⊢ (𝑖 = 1 → (𝐺‘𝑖) = (1 − (log‘2)))
49	48	breq1d 5117	. . . . . 6 ⊢ (𝑖 = 1 → ((𝐺‘𝑖) ≤ γ ↔ (1 − (log‘2)) ≤ γ))
50	49	rspcva 3586	. . . . 5 ⊢ ((1 ∈ ℕ ∧ ∀𝑖 ∈ ℕ (𝐺‘𝑖) ≤ γ) → (1 − (log‘2)) ≤ γ)
51	15, 23, 50	sylancr 587	. . . 4 ⊢ (⊤ → (1 − (log‘2)) ≤ γ)
52		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑖 → (𝐹‘𝑥) = (𝐹‘𝑖))
53	52	negeqd 11415	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑖 → -(𝐹‘𝑥) = -(𝐹‘𝑖))
54		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) = (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))
55		negex 11419	. . . . . . . . . . 11 ⊢ -(𝐹‘𝑖) ∈ V
56	53, 54, 55	fvmpt 6968	. . . . . . . . . 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 11167	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ∈ ℂ)
61		nnex 12192	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
62	61	mptex 7197	. . . . . . . . . . . . 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 7055	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ ℝ)
66	65	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
67	66	recnd 11202	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
68		fveq2 6858	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → (𝐹‘𝑥) = (𝐹‘𝑘))
69	68	negeqd 11415	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑘 → -(𝐹‘𝑥) = -(𝐹‘𝑘))
70		negex 11419	. . . . . . . . . . . . . . 15 ⊢ -(𝐹‘𝑘) ∈ V
71	69, 54, 70	fvmpt 6968	. . . . . . . . . . . . . 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 11408	. . . . . . . . . . . . 13 ⊢ -(𝐹‘𝑘) = (0 − (𝐹‘𝑘))
74	72, 73	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) = (0 − (𝐹‘𝑘)))
75	1, 2, 59, 60, 63, 67, 74	climsubc2 15605	. . . . . . . . . . 11 ⊢ (⊤ → (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) ⇝ (0 − γ))
76	75	adantr 480	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) ⇝ (0 − γ))
77	66	renegcld 11605	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → -(𝐹‘𝑘) ∈ ℝ)
78	72, 77	eqeltrd 2828	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) ∈ ℝ)
79	78	adantlr 715	. . . . . . . . . 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 12198	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
83	82	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
84	64	ffvelcdmi 7055	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ ℝ)
85	83, 84	syl 17	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℝ)
86	85, 66	lenegd 11757	. . . . . . . . . . . . 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 6858	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑘 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑘 + 1)))
89	88	negeqd 11415	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑘 + 1) → -(𝐹‘𝑥) = -(𝐹‘(𝑘 + 1)))
90		negex 11419	. . . . . . . . . . . . . 14 ⊢ -(𝐹‘(𝑘 + 1)) ∈ V
91	89, 54, 90	fvmpt 6968	. . . . . . . . . . . . 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 5139	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) ≤ ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘(𝑘 + 1)))
94	93	adantlr 715	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑖 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) ≤ ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘(𝑘 + 1)))
95	1, 16, 76, 79, 94	climub 15628	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑖) ≤ (0 − γ))
96	57, 95	eqbrtrrd 5131	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → -(𝐹‘𝑖) ≤ (0 − γ))
97		df-neg 11408	. . . . . . . 8 ⊢ -γ = (0 − γ)
98	96, 97	breqtrrdi 5149	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → -(𝐹‘𝑖) ≤ -γ)
99	14	mptru 1547	. . . . . . . 8 ⊢ γ ∈ ℝ
100	64	ffvelcdmi 7055	. . . . . . . . 9 ⊢ (𝑖 ∈ ℕ → (𝐹‘𝑖) ∈ ℝ)
101	100	adantl 481	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ ℝ)
102		leneg 11681	. . . . . . . 8 ⊢ ((γ ∈ ℝ ∧ (𝐹‘𝑖) ∈ ℝ) → (γ ≤ (𝐹‘𝑖) ↔ -(𝐹‘𝑖) ≤ -γ))
103	99, 101, 102	sylancr 587	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (γ ≤ (𝐹‘𝑖) ↔ -(𝐹‘𝑖) ≤ -γ))
104	98, 103	mpbird 257	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → γ ≤ (𝐹‘𝑖))
105	104	ralrimiva 3125	. . . . 5 ⊢ (⊤ → ∀𝑖 ∈ ℕ γ ≤ (𝐹‘𝑖))
106		fveq2 6858	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = (𝐹‘1))
107		fveq2 6858	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (log‘𝑛) = (log‘1))
108		log1 26494	. . . . . . . . . . . . 13 ⊢ (log‘1) = 0
109	107, 108	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (log‘𝑛) = 0)
110	34, 109	oveq12d 7405	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = (1 − 0))
111		1m0e1 12302	. . . . . . . . . . 11 ⊢ (1 − 0) = 1
112	110, 111	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = 1)
113	40	elexi 3470	. . . . . . . . . 10 ⊢ 1 ∈ V
114	112, 3, 113	fvmpt 6968	. . . . . . . . 9 ⊢ (1 ∈ ℕ → (𝐹‘1) = 1)
115	15, 114	ax-mp 5	. . . . . . . 8 ⊢ (𝐹‘1) = 1
116	106, 115	eqtrdi 2780	. . . . . . 7 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = 1)
117	116	breq2d 5119	. . . . . 6 ⊢ (𝑖 = 1 → (γ ≤ (𝐹‘𝑖) ↔ γ ≤ 1))
118	117	rspcva 3586	. . . . 5 ⊢ ((1 ∈ ℕ ∧ ∀𝑖 ∈ ℕ γ ≤ (𝐹‘𝑖)) → γ ≤ 1)
119	15, 105, 118	sylancr 587	. . . 4 ⊢ (⊤ → γ ≤ 1)
120	44, 40	elicc2i 13373	. . . 4 ⊢ (γ ∈ ((1 − (log‘2))[,]1) ↔ (γ ∈ ℝ ∧ (1 − (log‘2)) ≤ γ ∧ γ ≤ 1))
121	14, 51, 119, 120	syl3anbrc 1344	. . 3 ⊢ (⊤ → γ ∈ ((1 − (log‘2))[,]1))
122		ffn 6688	. . . . 5 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ)
123	64, 122	mp1i 13	. . . 4 ⊢ (⊤ → 𝐹 Fn ℕ)
124	16, 1	eleqtrdi 2838	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ (ℤ_≥‘1))
125		elfznn 13514	. . . . . . . . . 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 13514	. . . . . . . . . 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 13998	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ≤ (𝐹‘1))
132	131, 115	breqtrdi 5148	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ≤ 1)
133	99, 40	elicc2i 13373	. . . . . 6 ⊢ ((𝐹‘𝑖) ∈ (γ[,]1) ↔ ((𝐹‘𝑖) ∈ ℝ ∧ γ ≤ (𝐹‘𝑖) ∧ (𝐹‘𝑖) ≤ 1))
134	101, 104, 132, 133	syl3anbrc 1344	. . . . 5 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ (γ[,]1))
135	134	ralrimiva 3125	. . . 4 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐹‘𝑖) ∈ (γ[,]1))
136		ffnfv 7091	. . . 4 ⊢ (𝐹:ℕ⟶(γ[,]1) ↔ (𝐹 Fn ℕ ∧ ∀𝑖 ∈ ℕ (𝐹‘𝑖) ∈ (γ[,]1)))
137	123, 135, 136	sylanbrc 583	. . 3 ⊢ (⊤ → 𝐹:ℕ⟶(γ[,]1))
138		ffn 6688	. . . . 5 ⊢ (𝐺:ℕ⟶ℝ → 𝐺 Fn ℕ)
139	11, 138	mp1i 13	. . . 4 ⊢ (⊤ → 𝐺 Fn ℕ)
140	11	ffvelcdmi 7055	. . . . . . 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 13997	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘1) ≤ (𝐺‘𝑖))
145	47, 144	eqbrtrrid 5143	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (1 − (log‘2)) ≤ (𝐺‘𝑖))
146	44, 99	elicc2i 13373	. . . . . 6 ⊢ ((𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ) ↔ ((𝐺‘𝑖) ∈ ℝ ∧ (1 − (log‘2)) ≤ (𝐺‘𝑖) ∧ (𝐺‘𝑖) ≤ γ))
147	141, 145, 22, 146	syl3anbrc 1344	. . . . 5 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ))
148	147	ralrimiva 3125	. . . 4 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ))
149		ffnfv 7091	. . . 4 ⊢ (𝐺:ℕ⟶((1 − (log‘2))[,]γ) ↔ (𝐺 Fn ℕ ∧ ∀𝑖 ∈ ℕ (𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ)))
150	139, 148, 149	sylanbrc 583	. . 3 ⊢ (⊤ → 𝐺:ℕ⟶((1 − (log‘2))[,]γ))
151	121, 137, 150	3jca 1128	. 2 ⊢ (⊤ → (γ ∈ ((1 − (log‘2))[,]1) ∧ 𝐹:ℕ⟶(γ[,]1) ∧ 𝐺:ℕ⟶((1 − (log‘2))[,]γ)))
152	151	mptru 1547	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 1086 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ∀wral 3044 Vcvv 3447 class class class wbr 5107 ↦ cmpt 5188 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 ≤ cle 11209 − cmin 11405 -cneg 11406 / cdiv 11835 ℕcn 12186 2c2 12241 ℤcz 12529 ℤ_≥cuz 12793 ℝ⁺crp 12951 [,]cicc 13309 ...cfz 13468 ⇝ cli 15450 Σcsu 15652 logclog 26463 γcem 26902

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-log 26465 df-em 26903

This theorem is referenced by: emcl 26913 harmonicbnd 26914 harmonicbnd2 26915

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