tgoldbachgtde - Mathbox for Thierry Arnoux

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Theorem tgoldbachgtde 34796

Description: Lemma for tgoldbachgtd 34798. (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 34795	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
5	4	nnnn0d 12464	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6		3nn0 12421	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → 3 ∈ ℕ₀)
8		ssidd 3956	. . . . . . 7 ⊢ (𝜑 → ℕ ⊆ ℕ)
9	5, 7, 8	reprfi2 34759	. . . . . 6 ⊢ (𝜑 → (ℕ(repr‘3)𝑁) ∈ Fin)
10		diffi 9101	. . . . . 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 4088	. . . . . . 7 ⊢ (𝜑 → ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) ⊆ (ℕ(repr‘3)𝑁))
13	12	sselda 3932	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
14		vmaf 27087	. . . . . . . . . 10 ⊢ Λ:ℕ⟶ℝ
15	14	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → Λ:ℕ⟶ℝ)
16		ssidd 3956	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ℕ ⊆ ℕ)
17	5	nn0zd 12515	. . . . . . . . . . . 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 34748	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝑛:(0..^3)⟶ℕ)
22		c0ex 11128	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
23	22	tpid1 4724	. . . . . . . . . . . 12 ⊢ 0 ∈ {0, 1, 2}
24		fzo0to3tp 13670	. . . . . . . . . . . 12 ⊢ (0..^3) = {0, 1, 2}
25	23, 24	eleqtrri 2834	. . . . . . . . . . 11 ⊢ 0 ∈ (0..^3)
26	25	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 0 ∈ (0..^3))
27	21, 26	ffvelcdmd 7030	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘0) ∈ ℕ)
28	15, 27	ffvelcdmd 7030	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘0)) ∈ ℝ)
29		tgoldbachgtda.h	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻:ℕ⟶(0[,)+∞))
30		rge0ssre 13374	. . . . . . . . . . 11 ⊢ (0[,)+∞) ⊆ ℝ
31		fss 6677	. . . . . . . . . . 11 ⊢ ((𝐻:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐻:ℕ⟶ℝ)
32	29, 30, 31	sylancl 587	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻:ℕ⟶ℝ)
33	32	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝐻:ℕ⟶ℝ)
34	33, 27	ffvelcdmd 7030	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐻‘(𝑛‘0)) ∈ ℝ)
35	28, 34	remulcld 11164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) ∈ ℝ)
36		1ex 11130	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
37	36	tpid2 4726	. . . . . . . . . . . . 13 ⊢ 1 ∈ {0, 1, 2}
38	37, 24	eleqtrri 2834	. . . . . . . . . . . 12 ⊢ 1 ∈ (0..^3)
39	38	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 1 ∈ (0..^3))
40	21, 39	ffvelcdmd 7030	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘1) ∈ ℕ)
41	15, 40	ffvelcdmd 7030	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘1)) ∈ ℝ)
42		tgoldbachgtda.k	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾:ℕ⟶(0[,)+∞))
43		fss 6677	. . . . . . . . . . . 12 ⊢ ((𝐾:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐾:ℕ⟶ℝ)
44	42, 30, 43	sylancl 587	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾:ℕ⟶ℝ)
45	44	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝐾:ℕ⟶ℝ)
46	45, 40	ffvelcdmd 7030	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐾‘(𝑛‘1)) ∈ ℝ)
47	41, 46	remulcld 11164	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) ∈ ℝ)
48		2ex 12224	. . . . . . . . . . . . . 14 ⊢ 2 ∈ V
49	48	tpid3 4729	. . . . . . . . . . . . 13 ⊢ 2 ∈ {0, 1, 2}
50	49, 24	eleqtrri 2834	. . . . . . . . . . . 12 ⊢ 2 ∈ (0..^3)
51	50	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 2 ∈ (0..^3))
52	21, 51	ffvelcdmd 7030	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘2) ∈ ℕ)
53	15, 52	ffvelcdmd 7030	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘2)) ∈ ℝ)
54	45, 52	ffvelcdmd 7030	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐾‘(𝑛‘2)) ∈ ℝ)
55	53, 54	remulcld 11164	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))) ∈ ℝ)
56	47, 55	remulcld 11164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2)))) ∈ ℝ)
57	35, 56	remulcld 11164	. . . . . 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 592	. . . . 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 15659	. . . 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 12418	. . . . . . 7 ⊢ 0 ∈ ℕ₀
61		qssre 12874	. . . . . . . 8 ⊢ ℚ ⊆ ℝ
62		4nn0 12422	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ₀
63		2nn0 12420	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
64		nn0ssq 12872	. . . . . . . . . . . . . . . 16 ⊢ ℕ₀ ⊆ ℚ
65		8nn0 12426	. . . . . . . . . . . . . . . 16 ⊢ 8 ∈ ℕ₀
66	64, 65	sselii 3929	. . . . . . . . . . . . . . 15 ⊢ 8 ∈ ℚ
67	62, 66	dp2clq 32941	. . . . . . . . . . . . . 14 ⊢ _48 ∈ ℚ
68	63, 67	dp2clq 32941	. . . . . . . . . . . . 13 ⊢ _2_48 ∈ ℚ
69	63, 68	dp2clq 32941	. . . . . . . . . . . 12 ⊢ _2_2_48 ∈ ℚ
70	62, 69	dp2clq 32941	. . . . . . . . . . 11 ⊢ _4_2_2_48 ∈ ℚ
71	60, 70	dp2clq 32941	. . . . . . . . . 10 ⊢ _0_4_2_2_48 ∈ ℚ
72	60, 71	dp2clq 32941	. . . . . . . . 9 ⊢ _0_0_4_2_2_48 ∈ ℚ
73	60, 72	dp2clq 32941	. . . . . . . 8 ⊢ _0_0_0_4_2_2_48 ∈ ℚ
74	61, 73	sselii 3929	. . . . . . 7 ⊢ _0_0_0_4_2_2_48 ∈ ℝ
75		dpcl 32951	. . . . . . 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 693	. . . . . 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 12162	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
79	78	resqcld 14050	. . . . 5 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ)
80	77, 79	remulcld 11164	. . . 4 ⊢ (𝜑 → ((0._0_0_0_4_2_2_48) · (𝑁↑2)) ∈ ℝ)
81	9, 57	fsumrecl 15659	. . . 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 12425	. . . . . . . . 9 ⊢ 7 ∈ ℕ₀
83	6, 67	dp2clq 32941	. . . . . . . . . 10 ⊢ _3_48 ∈ ℚ
84	61, 83	sselii 3929	. . . . . . . . 9 ⊢ _3_48 ∈ ℝ
85		dpcl 32951	. . . . . . . . 9 ⊢ ((7 ∈ ℕ₀ ∧ _3_48 ∈ ℝ) → (7._3_48) ∈ ℝ)
86	82, 84, 85	mp2an 693	. . . . . . . 8 ⊢ (7._3_48) ∈ ℝ
87	86	a1i 11	. . . . . . 7 ⊢ (𝜑 → (7._3_48) ∈ ℝ)
88	4	nnrpd 12949	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
89	88	relogcld 26590	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
90	5	nn0ge0d 12467	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝑁)
91	78, 90	resqrtcld 15343	. . . . . . . 8 ⊢ (𝜑 → (√‘𝑁) ∈ ℝ)
92	88	sqrtgt0d 15338	. . . . . . . . 9 ⊢ (𝜑 → 0 < (√‘𝑁))
93	92	gt0ne0d 11703	. . . . . . . 8 ⊢ (𝜑 → (√‘𝑁) ≠ 0)
94	89, 91, 93	redivcld 11971	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑁) / (√‘𝑁)) ∈ ℝ)
95	87, 94	remulcld 11164	. . . . . 6 ⊢ (𝜑 → ((7._3_48) · ((log‘𝑁) / (√‘𝑁))) ∈ ℝ)
96	95, 79	remulcld 11164	. . . . 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 34794	. . . . 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 12525	. . . . . . . 8 ⊢ 2 ∈ ℤ
101	100	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℤ)
102	88, 101	rpexpcld 14172	. . . . . 6 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ⁺)
103		hgt750lem 34787	. . . . . . 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 585	. . . . . 6 ⊢ (𝜑 → ((7._3_48) · ((log‘𝑁) / (√‘𝑁))) < (0._0_0_0_4_2_2_48))
105	95, 77, 102, 104	ltmul1dd 13006	. . . . 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 11293	. . . 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 34779	. . . . 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 5125	. . . 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 11295	. . 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 11726	. . 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 4189	. . . . . . . 8 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ
114		prmssnn 16605	. . . . . . . 8 ⊢ ℙ ⊆ ℕ
115	113, 114	sstri 3942	. . . . . . 7 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ
116	115	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ)
117	8, 17, 7, 116	reprss 34753	. . . . 5 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ⊆ (ℕ(repr‘3)𝑁))
118	9, 117	ssfid 9171	. . . 4 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ∈ Fin)
119	117	sselda 3932	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
120	57	recnd 11162	. . . . 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 592	. . . 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 15658	. . 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 11162	. . 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 4423	. . . . 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 4433	. . . . . 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 2741	. . . 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 15666	. . 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 11551	. 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 5123	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 1542 ∈ wcel 2114 {crab 3398 ∖ cdif 3897 ∪ cun 3898 ∩ cin 3899 ⊆ wss 3900 ∅c0 4284 {ctp 4583 class class class wbr 5097 ⟶wf 6487 ‘cfv 6491 (class class class)co 7358 ∘_f cof 7620 Fincfn 8885 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 ici 11030 · cmul 11033 +∞cpnf 11165 < clt 11168 ≤ cle 11169 − cmin 11366 -cneg 11367 / cdiv 11796 ℕcn 12147 2c2 12202 3c3 12203 4c4 12204 5c5 12205 7c7 12207 8c8 12208 9c9 12209 ℕ₀cn0 12403 ℤcz 12490 cdc 12609 ℚcq 12863 (,)cioo 13263 [,)cico 13265 ..^cfzo 13572 ↑cexp 13986 √csqrt 15158 Σcsu 15611 expce 15986 πcpi 15991 ∥ cdvds 16181 ℙcprime 16600 ∫citg 25577 logclog 26521 Λcvma 27060 cdp2 32931 .cdp 32948 reprcrepr 34744 vtscvts 34771

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 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-reg 9499 ax-inf2 9552 ax-cc 10347 ax-ac2 10375 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107 ax-ros335 34781 ax-ros336 34782

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-symdif 4204 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4902 df-iun 4947 df-iin 4948 df-disj 5065 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-map 8767 df-pm 8768 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-fi 9316 df-sup 9347 df-inf 9348 df-oi 9417 df-r1 9678 df-rank 9679 df-dju 9815 df-card 9853 df-acn 9856 df-ac 10028 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-xnn0 12477 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ioo 13267 df-ioc 13268 df-ico 13269 df-icc 13270 df-fz 13426 df-fzo 13573 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-word 14439 df-concat 14496 df-s1 14522 df-s2 14773 df-s3 14774 df-shft 14992 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-prod 15829 df-ef 15992 df-e 15993 df-sin 15994 df-cos 15995 df-tan 15996 df-pi 15997 df-dvds 16182 df-gcd 16424 df-prm 16601 df-pc 16767 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19248 df-pmtr 19373 df-cmn 19713 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-fbas 21308 df-fg 21309 df-cnfld 21312 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cld 22965 df-ntr 22966 df-cls 22967 df-nei 23044 df-lp 23082 df-perf 23083 df-cn 23173 df-cnp 23174 df-haus 23261 df-cmp 23333 df-tx 23508 df-hmeo 23701 df-fil 23792 df-fm 23884 df-flim 23885 df-flf 23886 df-xms 24266 df-ms 24267 df-tms 24268 df-cncf 24829 df-ovol 25423 df-vol 25424 df-mbf 25578 df-itg1 25579 df-itg2 25580 df-ibl 25581 df-itg 25582 df-0p 25629 df-limc 25825 df-dv 25826 df-ulm 26344 df-log 26523 df-cxp 26524 df-atan 26835 df-cht 27065 df-vma 27066 df-chp 27067 df-dp2 32932 df-dp 32949 df-repr 34745 df-vts 34772

This theorem is referenced by: tgoldbachgtda 34797

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