tgoldbachgtde - Mathbox for Thierry Arnoux

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Theorem tgoldbachgtde 34653

Description: Lemma for tgoldbachgtd 34655. (Contributed by Thierry Arnoux, 15-Dec-2021.)

**Hypotheses**
Ref	Expression
tgoldbachgtda.o	⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}
tgoldbachgtda.n	⊢ (𝜑 → 𝑁 ∈ 𝑂)
tgoldbachgtda.0	⊢ (𝜑 → (;10↑;27) ≤ 𝑁)
tgoldbachgtda.h	⊢ (𝜑 → 𝐻:ℕ⟶(0[,)+∞))
tgoldbachgtda.k	⊢ (𝜑 → 𝐾:ℕ⟶(0[,)+∞))
tgoldbachgtda.1	⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐾‘𝑚) ≤ (1._0_7_9_9_55))
tgoldbachgtda.2	⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ≤ (1._4_14))
tgoldbachgtda.3	⊢ (𝜑 → ((0._0_0_0_4_2_2_48) · (𝑁↑2)) ≤ ∫(0(,)1)(((((Λ ∘_f · 𝐻)vts𝑁)‘𝑥) · ((((Λ ∘_f · 𝐾)vts𝑁)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥)

**Assertion**
Ref	Expression
tgoldbachgtde	⊢ (𝜑 → 0 < Σ𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))))

Distinct variable groups: 𝑚,𝐻,𝑛,𝑥 𝑚,𝐾,𝑛,𝑥 𝑚,𝑁,𝑛,𝑥,𝑧 𝑚,𝑂,𝑛,𝑧 𝜑,𝑚,𝑛,𝑥 Allowed substitution hints: 𝜑(𝑧) 𝐻(𝑧) 𝐾(𝑧) 𝑂(𝑥)

**Proof of Theorem tgoldbachgtde**
Step	Hyp	Ref	Expression
1		tgoldbachgtda.o	. . . . . . . . 9 ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}
2		tgoldbachgtda.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ 𝑂)
3		tgoldbachgtda.0	. . . . . . . . 9 ⊢ (𝜑 → (;10↑;27) ≤ 𝑁)
4	1, 2, 3	tgoldbachgnn 34652	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
5	4	nnnn0d 12584	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6		3nn0 12541	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → 3 ∈ ℕ₀)
8		ssidd 4018	. . . . . . 7 ⊢ (𝜑 → ℕ ⊆ ℕ)
9	5, 7, 8	reprfi2 34616	. . . . . 6 ⊢ (𝜑 → (ℕ(repr‘3)𝑁) ∈ Fin)
10		diffi 9213	. . . . . 6 ⊢ ((ℕ(repr‘3)𝑁) ∈ Fin → ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) ∈ Fin)
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) ∈ Fin)
12		difssd 4146	. . . . . . 7 ⊢ (𝜑 → ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) ⊆ (ℕ(repr‘3)𝑁))
13	12	sselda 3994	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
14		vmaf 27176	. . . . . . . . . 10 ⊢ Λ:ℕ⟶ℝ
15	14	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → Λ:ℕ⟶ℝ)
16		ssidd 4018	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ℕ ⊆ ℕ)
17	5	nn0zd 12636	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
18	17	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝑁 ∈ ℤ)
19	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 3 ∈ ℕ₀)
20		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
21	16, 18, 19, 20	reprf 34605	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝑛:(0..^3)⟶ℕ)
22		c0ex 11252	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
23	22	tpid1 4772	. . . . . . . . . . . 12 ⊢ 0 ∈ {0, 1, 2}
24		fzo0to3tp 13787	. . . . . . . . . . . 12 ⊢ (0..^3) = {0, 1, 2}
25	23, 24	eleqtrri 2837	. . . . . . . . . . 11 ⊢ 0 ∈ (0..^3)
26	25	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 0 ∈ (0..^3))
27	21, 26	ffvelcdmd 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘0) ∈ ℕ)
28	15, 27	ffvelcdmd 7104	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘0)) ∈ ℝ)
29		tgoldbachgtda.h	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻:ℕ⟶(0[,)+∞))
30		rge0ssre 13492	. . . . . . . . . . 11 ⊢ (0[,)+∞) ⊆ ℝ
31		fss 6752	. . . . . . . . . . 11 ⊢ ((𝐻:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐻:ℕ⟶ℝ)
32	29, 30, 31	sylancl 586	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻:ℕ⟶ℝ)
33	32	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝐻:ℕ⟶ℝ)
34	33, 27	ffvelcdmd 7104	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐻‘(𝑛‘0)) ∈ ℝ)
35	28, 34	remulcld 11288	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) ∈ ℝ)
36		1ex 11254	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
37	36	tpid2 4774	. . . . . . . . . . . . 13 ⊢ 1 ∈ {0, 1, 2}
38	37, 24	eleqtrri 2837	. . . . . . . . . . . 12 ⊢ 1 ∈ (0..^3)
39	38	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 1 ∈ (0..^3))
40	21, 39	ffvelcdmd 7104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘1) ∈ ℕ)
41	15, 40	ffvelcdmd 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘1)) ∈ ℝ)
42		tgoldbachgtda.k	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾:ℕ⟶(0[,)+∞))
43		fss 6752	. . . . . . . . . . . 12 ⊢ ((𝐾:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐾:ℕ⟶ℝ)
44	42, 30, 43	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾:ℕ⟶ℝ)
45	44	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝐾:ℕ⟶ℝ)
46	45, 40	ffvelcdmd 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐾‘(𝑛‘1)) ∈ ℝ)
47	41, 46	remulcld 11288	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) ∈ ℝ)
48		2ex 12340	. . . . . . . . . . . . . 14 ⊢ 2 ∈ V
49	48	tpid3 4777	. . . . . . . . . . . . 13 ⊢ 2 ∈ {0, 1, 2}
50	49, 24	eleqtrri 2837	. . . . . . . . . . . 12 ⊢ 2 ∈ (0..^3)
51	50	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 2 ∈ (0..^3))
52	21, 51	ffvelcdmd 7104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘2) ∈ ℕ)
53	15, 52	ffvelcdmd 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘2)) ∈ ℝ)
54	45, 52	ffvelcdmd 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐾‘(𝑛‘2)) ∈ ℝ)
55	53, 54	remulcld 11288	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))) ∈ ℝ)
56	47, 55	remulcld 11288	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2)))) ∈ ℝ)
57	35, 56	remulcld 11288	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ∈ ℝ)
58	13, 57	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))) → (((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ∈ ℝ)
59	11, 58	fsumrecl 15766	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ∈ ℝ)
60		0nn0 12538	. . . . . . 7 ⊢ 0 ∈ ℕ₀
61		qssre 12998	. . . . . . . 8 ⊢ ℚ ⊆ ℝ
62		4nn0 12542	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ₀
63		2nn0 12540	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
64		nn0ssq 12996	. . . . . . . . . . . . . . . 16 ⊢ ℕ₀ ⊆ ℚ
65		8nn0 12546	. . . . . . . . . . . . . . . 16 ⊢ 8 ∈ ℕ₀
66	64, 65	sselii 3991	. . . . . . . . . . . . . . 15 ⊢ 8 ∈ ℚ
67	62, 66	dp2clq 32847	. . . . . . . . . . . . . 14 ⊢ _48 ∈ ℚ
68	63, 67	dp2clq 32847	. . . . . . . . . . . . 13 ⊢ _2_48 ∈ ℚ
69	63, 68	dp2clq 32847	. . . . . . . . . . . 12 ⊢ _2_2_48 ∈ ℚ
70	62, 69	dp2clq 32847	. . . . . . . . . . 11 ⊢ _4_2_2_48 ∈ ℚ
71	60, 70	dp2clq 32847	. . . . . . . . . 10 ⊢ _0_4_2_2_48 ∈ ℚ
72	60, 71	dp2clq 32847	. . . . . . . . 9 ⊢ _0_0_4_2_2_48 ∈ ℚ
73	60, 72	dp2clq 32847	. . . . . . . 8 ⊢ _0_0_0_4_2_2_48 ∈ ℚ
74	61, 73	sselii 3991	. . . . . . 7 ⊢ _0_0_0_4_2_2_48 ∈ ℝ
75		dpcl 32857	. . . . . . 7 ⊢ ((0 ∈ ℕ₀ ∧ _0_0_0_4_2_2_48 ∈ ℝ) → (0._0_0_0_4_2_2_48) ∈ ℝ)
76	60, 74, 75	mp2an 692	. . . . . 6 ⊢ (0._0_0_0_4_2_2_48) ∈ ℝ
77	76	a1i 11	. . . . 5 ⊢ (𝜑 → (0._0_0_0_4_2_2_48) ∈ ℝ)
78	4	nnred 12278	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
79	78	resqcld 14161	. . . . 5 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ)
80	77, 79	remulcld 11288	. . . 4 ⊢ (𝜑 → ((0._0_0_0_4_2_2_48) · (𝑁↑2)) ∈ ℝ)
81	9, 57	fsumrecl 15766	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (ℕ(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ∈ ℝ)
82		7nn0 12545	. . . . . . . . 9 ⊢ 7 ∈ ℕ₀
83	6, 67	dp2clq 32847	. . . . . . . . . 10 ⊢ _3_48 ∈ ℚ
84	61, 83	sselii 3991	. . . . . . . . 9 ⊢ _3_48 ∈ ℝ
85		dpcl 32857	. . . . . . . . 9 ⊢ ((7 ∈ ℕ₀ ∧ _3_48 ∈ ℝ) → (7._3_48) ∈ ℝ)
86	82, 84, 85	mp2an 692	. . . . . . . 8 ⊢ (7._3_48) ∈ ℝ
87	86	a1i 11	. . . . . . 7 ⊢ (𝜑 → (7._3_48) ∈ ℝ)
88	4	nnrpd 13072	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
89	88	relogcld 26679	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
90	5	nn0ge0d 12587	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝑁)
91	78, 90	resqrtcld 15452	. . . . . . . 8 ⊢ (𝜑 → (√‘𝑁) ∈ ℝ)
92	88	sqrtgt0d 15447	. . . . . . . . 9 ⊢ (𝜑 → 0 < (√‘𝑁))
93	92	gt0ne0d 11824	. . . . . . . 8 ⊢ (𝜑 → (√‘𝑁) ≠ 0)
94	89, 91, 93	redivcld 12092	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑁) / (√‘𝑁)) ∈ ℝ)
95	87, 94	remulcld 11288	. . . . . 6 ⊢ (𝜑 → ((7._3_48) · ((log‘𝑁) / (√‘𝑁))) ∈ ℝ)
96	95, 79	remulcld 11288	. . . . 5 ⊢ (𝜑 → (((7._3_48) · ((log‘𝑁) / (√‘𝑁))) · (𝑁↑2)) ∈ ℝ)
97		tgoldbachgtda.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐾‘𝑚) ≤ (1._0_7_9_9_55))
98		tgoldbachgtda.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ≤ (1._4_14))
99	1, 4, 3, 29, 42, 97, 98	hgt750leme 34651	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ≤ (((7._3_48) · ((log‘𝑁) / (√‘𝑁))) · (𝑁↑2)))
100		2z 12646	. . . . . . . 8 ⊢ 2 ∈ ℤ
101	100	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℤ)
102	88, 101	rpexpcld 14282	. . . . . 6 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ⁺)
103		hgt750lem 34644	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (;10↑;27) ≤ 𝑁) → ((7._3_48) · ((log‘𝑁) / (√‘𝑁))) < (0._0_0_0_4_2_2_48))
104	5, 3, 103	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((7._3_48) · ((log‘𝑁) / (√‘𝑁))) < (0._0_0_0_4_2_2_48))
105	95, 77, 102, 104	ltmul1dd 13129	. . . . 5 ⊢ (𝜑 → (((7._3_48) · ((log‘𝑁) / (√‘𝑁))) · (𝑁↑2)) < ((0._0_0_0_4_2_2_48) · (𝑁↑2)))
106	59, 96, 80, 99, 105	lelttrd 11416	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) < ((0._0_0_0_4_2_2_48) · (𝑁↑2)))
107		tgoldbachgtda.3	. . . . 5 ⊢ (𝜑 → ((0._0_0_0_4_2_2_48) · (𝑁↑2)) ≤ ∫(0(,)1)(((((Λ ∘_f · 𝐻)vts𝑁)‘𝑥) · ((((Λ ∘_f · 𝐾)vts𝑁)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥)
108	32, 44, 5	circlemethhgt 34636	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (ℕ(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) = ∫(0(,)1)(((((Λ ∘_f · 𝐻)vts𝑁)‘𝑥) · ((((Λ ∘_f · 𝐾)vts𝑁)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥)
109	107, 108	breqtrrd 5175	. . . 4 ⊢ (𝜑 → ((0._0_0_0_4_2_2_48) · (𝑁↑2)) ≤ Σ𝑛 ∈ (ℕ(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))))
110	59, 80, 81, 106, 109	ltletrd 11418	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) < Σ𝑛 ∈ (ℕ(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))))
111	59, 81	posdifd 11847	. . 3 ⊢ (𝜑 → (Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) < Σ𝑛 ∈ (ℕ(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ↔ 0 < (Σ𝑛 ∈ (ℕ(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) − Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))))))
112	110, 111	mpbid 232	. 2 ⊢ (𝜑 → 0 < (Σ𝑛 ∈ (ℕ(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) − Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2)))))))
113		inss2 4245	. . . . . . . 8 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ
114		prmssnn 16709	. . . . . . . 8 ⊢ ℙ ⊆ ℕ
115	113, 114	sstri 4004	. . . . . . 7 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ
116	115	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ)
117	8, 17, 7, 116	reprss 34610	. . . . 5 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ⊆ (ℕ(repr‘3)𝑁))
118	9, 117	ssfid 9298	. . . 4 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ∈ Fin)
119	117	sselda 3994	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
120	57	recnd 11286	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ∈ ℂ)
121	119, 120	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) → (((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ∈ ℂ)
122	118, 121	fsumcl 15765	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ∈ ℂ)
123	59	recnd 11286	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ∈ ℂ)
124		disjdif 4477	. . . . 5 ⊢ (((𝑂 ∩ ℙ)(repr‘3)𝑁) ∩ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))) = ∅
125	124	a1i 11	. . . 4 ⊢ (𝜑 → (((𝑂 ∩ ℙ)(repr‘3)𝑁) ∩ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))) = ∅)
126		undif 4487	. . . . . 6 ⊢ (((𝑂 ∩ ℙ)(repr‘3)𝑁) ⊆ (ℕ(repr‘3)𝑁) ↔ (((𝑂 ∩ ℙ)(repr‘3)𝑁) ∪ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))) = (ℕ(repr‘3)𝑁))
127	117, 126	sylib 218	. . . . 5 ⊢ (𝜑 → (((𝑂 ∩ ℙ)(repr‘3)𝑁) ∪ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))) = (ℕ(repr‘3)𝑁))
128	127	eqcomd 2740	. . . 4 ⊢ (𝜑 → (ℕ(repr‘3)𝑁) = (((𝑂 ∩ ℙ)(repr‘3)𝑁) ∪ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))))
129	125, 128, 9, 120	fsumsplit 15773	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (ℕ(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) = (Σ𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) + Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2)))))))
130	122, 123, 129	mvrraddd 11672	. 2 ⊢ (𝜑 → (Σ𝑛 ∈ (ℕ(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) − Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2)))))) = Σ𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))))
131	112, 130	breqtrd 5173	1 ⊢ (𝜑 → 0 < Σ𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {crab 3432 ∖ cdif 3959 ∪ cun 3960 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 {ctp 4634 class class class wbr 5147 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ∘_f cof 7694 Fincfn 8983 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 ici 11154 · cmul 11157 +∞cpnf 11289 < clt 11292 ≤ cle 11293 − cmin 11489 -cneg 11490 / cdiv 11917 ℕcn 12263 2c2 12318 3c3 12319 4c4 12320 5c5 12321 7c7 12323 8c8 12324 9c9 12325 ℕ₀cn0 12523 ℤcz 12610 cdc 12730 ℚcq 12987 (,)cioo 13383 [,)cico 13385 ..^cfzo 13690 ↑cexp 14098 √csqrt 15268 Σcsu 15718 expce 16093 πcpi 16098 ∥ cdvds 16286 ℙcprime 16704 ∫citg 25666 logclog 26610 Λcvma 27149 cdp2 32837 .cdp 32854 reprcrepr 34601 vtscvts 34628

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-reg 9629 ax-inf2 9678 ax-cc 10472 ax-ac2 10500 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-ros335 34638 ax-ros336 34639

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-symdif 4258 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-disj 5115 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-ofr 7697 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-omul 8509 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-r1 9801 df-rank 9802 df-dju 9938 df-card 9976 df-acn 9979 df-ac 10153 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-word 14549 df-concat 14605 df-s1 14630 df-s2 14883 df-s3 14884 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-prod 15936 df-ef 16099 df-e 16100 df-sin 16101 df-cos 16102 df-tan 16103 df-pi 16104 df-dvds 16287 df-gcd 16528 df-prm 16705 df-pc 16870 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-pmtr 19474 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-cmp 23410 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-ovol 25512 df-vol 25513 df-mbf 25667 df-itg1 25668 df-itg2 25669 df-ibl 25670 df-itg 25671 df-0p 25718 df-limc 25915 df-dv 25916 df-ulm 26434 df-log 26612 df-cxp 26613 df-atan 26924 df-cht 27154 df-vma 27155 df-chp 27156 df-dp2 32838 df-dp 32855 df-repr 34602 df-vts 34629

This theorem is referenced by: tgoldbachgtda 34654

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