emcllem7 - Metamath Proof Explorer

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Theorem emcllem7 26939

Description: Lemma for emcl 26940 and harmonicbnd 26941. 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 12775	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12503	. . . . 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 26938	. . . . . . 7 ⊢ (𝐹 ⇝ γ ∧ 𝐺 ⇝ γ)
8	7	simpri 485	. . . . . 6 ⊢ 𝐺 ⇝ γ
9	8	a1i 11	. . . . 5 ⊢ (⊤ → 𝐺 ⇝ γ)
10	3, 4	emcllem1 26933	. . . . . . . 8 ⊢ (𝐹:ℕ⟶ℝ ∧ 𝐺:ℕ⟶ℝ)
11	10	simpri 485	. . . . . . 7 ⊢ 𝐺:ℕ⟶ℝ
12	11	ffvelcdmi 7016	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ∈ ℝ)
13	12	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
14	1, 2, 9, 13	climrecl 15490	. . . 4 ⊢ (⊤ → γ ∈ ℝ)
15		1nn 12136	. . . . 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 26934	. . . . . . . . 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 15569	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ≤ γ)
23	22	ralrimiva 3124	. . . . 5 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐺‘𝑖) ≤ γ)
24		fveq2 6822	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝐺‘𝑖) = (𝐺‘1))
25		oveq2 7354	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
26	25	sumeq1d 15607	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = Σ𝑚 ∈ (1...1)(1 / 𝑚))
27		1z 12502	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
28		ax-1cn 11064	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
29		oveq2 7354	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 1 → (1 / 𝑚) = (1 / 1))
30		1div1e1 11812	. . . . . . . . . . . . . . 15 ⊢ (1 / 1) = 1
31	29, 30	eqtrdi 2782	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (1 / 𝑚) = 1)
32	31	fsum1 15654	. . . . . . . . . . . . 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 2782	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) = 1)
35		oveq1 7353	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (𝑛 + 1) = (1 + 1))
36		df-2 12188	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
37	35, 36	eqtr4di 2784	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝑛 + 1) = 2)
38	37	fveq2d 6826	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (log‘(𝑛 + 1)) = (log‘2))
39	34, 38	oveq12d 7364	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘(𝑛 + 1))) = (1 − (log‘2)))
40		1re 11112	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
41		2rp 12895	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
42		relogcl 26511	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℝ⁺ → (log‘2) ∈ ℝ)
43	41, 42	ax-mp 5	. . . . . . . . . . . 12 ⊢ (log‘2) ∈ ℝ
44	40, 43	resubcli 11423	. . . . . . . . . . 11 ⊢ (1 − (log‘2)) ∈ ℝ
45	44	elexi 3459	. . . . . . . . . 10 ⊢ (1 − (log‘2)) ∈ V
46	39, 4, 45	fvmpt 6929	. . . . . . . . 9 ⊢ (1 ∈ ℕ → (𝐺‘1) = (1 − (log‘2)))
47	15, 46	ax-mp 5	. . . . . . . 8 ⊢ (𝐺‘1) = (1 − (log‘2))
48	24, 47	eqtrdi 2782	. . . . . . 7 ⊢ (𝑖 = 1 → (𝐺‘𝑖) = (1 − (log‘2)))
49	48	breq1d 5099	. . . . . 6 ⊢ (𝑖 = 1 → ((𝐺‘𝑖) ≤ γ ↔ (1 − (log‘2)) ≤ γ))
50	49	rspcva 3570	. . . . 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 6822	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑖 → (𝐹‘𝑥) = (𝐹‘𝑖))
53	52	negeqd 11354	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑖 → -(𝐹‘𝑥) = -(𝐹‘𝑖))
54		eqid 2731	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) = (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))
55		negex 11358	. . . . . . . . . . 11 ⊢ -(𝐹‘𝑖) ∈ V
56	53, 54, 55	fvmpt 6929	. . . . . . . . . 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 11105	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ∈ ℂ)
61		nnex 12131	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
62	61	mptex 7157	. . . . . . . . . . . . 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 7016	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ ℝ)
66	65	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
67	66	recnd 11140	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
68		fveq2 6822	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → (𝐹‘𝑥) = (𝐹‘𝑘))
69	68	negeqd 11354	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑘 → -(𝐹‘𝑥) = -(𝐹‘𝑘))
70		negex 11358	. . . . . . . . . . . . . . 15 ⊢ -(𝐹‘𝑘) ∈ V
71	69, 54, 70	fvmpt 6929	. . . . . . . . . . . . . 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 11347	. . . . . . . . . . . . 13 ⊢ -(𝐹‘𝑘) = (0 − (𝐹‘𝑘))
74	72, 73	eqtrdi 2782	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑘) = (0 − (𝐹‘𝑘)))
75	1, 2, 59, 60, 63, 67, 74	climsubc2 15546	. . . . . . . . . . 11 ⊢ (⊤ → (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) ⇝ (0 − γ))
76	75	adantr 480	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝑥 ∈ ℕ ↦ -(𝐹‘𝑥)) ⇝ (0 − γ))
77	66	renegcld 11544	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → -(𝐹‘𝑘) ∈ ℝ)
78	72, 77	eqeltrd 2831	. . . . . . . . . . 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 12137	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
83	82	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
84	64	ffvelcdmi 7016	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ ℝ)
85	83, 84	syl 17	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℝ)
86	85, 66	lenegd 11696	. . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑘 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑘 + 1)))
89	88	negeqd 11354	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑘 + 1) → -(𝐹‘𝑥) = -(𝐹‘(𝑘 + 1)))
90		negex 11358	. . . . . . . . . . . . . 14 ⊢ -(𝐹‘(𝑘 + 1)) ∈ V
91	89, 54, 90	fvmpt 6929	. . . . . . . . . . . . 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 5121	. . . . . . . . . . 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 15569	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ -(𝐹‘𝑥))‘𝑖) ≤ (0 − γ))
96	57, 95	eqbrtrrd 5113	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → -(𝐹‘𝑖) ≤ (0 − γ))
97		df-neg 11347	. . . . . . . 8 ⊢ -γ = (0 − γ)
98	96, 97	breqtrrdi 5131	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → -(𝐹‘𝑖) ≤ -γ)
99	14	mptru 1548	. . . . . . . 8 ⊢ γ ∈ ℝ
100	64	ffvelcdmi 7016	. . . . . . . . 9 ⊢ (𝑖 ∈ ℕ → (𝐹‘𝑖) ∈ ℝ)
101	100	adantl 481	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ ℝ)
102		leneg 11620	. . . . . . . 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 3124	. . . . 5 ⊢ (⊤ → ∀𝑖 ∈ ℕ γ ≤ (𝐹‘𝑖))
106		fveq2 6822	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = (𝐹‘1))
107		fveq2 6822	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (log‘𝑛) = (log‘1))
108		log1 26521	. . . . . . . . . . . . 13 ⊢ (log‘1) = 0
109	107, 108	eqtrdi 2782	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (log‘𝑛) = 0)
110	34, 109	oveq12d 7364	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = (1 − 0))
111		1m0e1 12241	. . . . . . . . . . 11 ⊢ (1 − 0) = 1
112	110, 111	eqtrdi 2782	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (Σ𝑚 ∈ (1...𝑛)(1 / 𝑚) − (log‘𝑛)) = 1)
113	40	elexi 3459	. . . . . . . . . 10 ⊢ 1 ∈ V
114	112, 3, 113	fvmpt 6929	. . . . . . . . 9 ⊢ (1 ∈ ℕ → (𝐹‘1) = 1)
115	15, 114	ax-mp 5	. . . . . . . 8 ⊢ (𝐹‘1) = 1
116	106, 115	eqtrdi 2782	. . . . . . 7 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = 1)
117	116	breq2d 5101	. . . . . 6 ⊢ (𝑖 = 1 → (γ ≤ (𝐹‘𝑖) ↔ γ ≤ 1))
118	117	rspcva 3570	. . . . 5 ⊢ ((1 ∈ ℕ ∧ ∀𝑖 ∈ ℕ γ ≤ (𝐹‘𝑖)) → γ ≤ 1)
119	15, 105, 118	sylancr 587	. . . 4 ⊢ (⊤ → γ ≤ 1)
120	44, 40	elicc2i 13312	. . . 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 6651	. . . . 5 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ)
123	64, 122	mp1i 13	. . . 4 ⊢ (⊤ → 𝐹 Fn ℕ)
124	16, 1	eleqtrdi 2841	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ (ℤ_≥‘1))
125		elfznn 13453	. . . . . . . . . 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 13453	. . . . . . . . . 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 13940	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ≤ (𝐹‘1))
132	131, 115	breqtrdi 5130	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ≤ 1)
133	99, 40	elicc2i 13312	. . . . . 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 3124	. . . 4 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐹‘𝑖) ∈ (γ[,]1))
136		ffnfv 7052	. . . 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 6651	. . . . 5 ⊢ (𝐺:ℕ⟶ℝ → 𝐺 Fn ℕ)
139	11, 138	mp1i 13	. . . 4 ⊢ (⊤ → 𝐺 Fn ℕ)
140	11	ffvelcdmi 7016	. . . . . . 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 13939	. . . . . . 7 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (𝐺‘1) ≤ (𝐺‘𝑖))
145	47, 144	eqbrtrrid 5125	. . . . . 6 ⊢ ((⊤ ∧ 𝑖 ∈ ℕ) → (1 − (log‘2)) ≤ (𝐺‘𝑖))
146	44, 99	elicc2i 13312	. . . . . 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 3124	. . . 4 ⊢ (⊤ → ∀𝑖 ∈ ℕ (𝐺‘𝑖) ∈ ((1 − (log‘2))[,]γ))
149		ffnfv 7052	. . . 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 1548	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 1541 ⊤wtru 1542 ∈ wcel 2111 ∀wral 3047 Vcvv 3436 class class class wbr 5089 ↦ cmpt 5170 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 ≤ cle 11147 − cmin 11344 -cneg 11345 / cdiv 11774 ℕcn 12125 2c2 12180 ℤcz 12468 ℤ_≥cuz 12732 ℝ⁺crp 12890 [,]cicc 13248 ...cfz 13407 ⇝ cli 15391 Σcsu 15593 logclog 26490 γcem 26929

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-haus 23230 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-limc 25794 df-dv 25795 df-log 26492 df-em 26930

This theorem is referenced by: emcl 26940 harmonicbnd 26941 harmonicbnd2 26942

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