tgoldbachgtde - Mathbox for Thierry Arnoux

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Theorem tgoldbachgtde 33970

Description: Lemma for tgoldbachgtd 33972. (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 33969	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
5	4	nnnn0d 12536	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6		3nn0 12494	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → 3 ∈ ℕ₀)
8		ssidd 4004	. . . . . . 7 ⊢ (𝜑 → ℕ ⊆ ℕ)
9	5, 7, 8	reprfi2 33933	. . . . . 6 ⊢ (𝜑 → (ℕ(repr‘3)𝑁) ∈ Fin)
10		diffi 9181	. . . . . 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 4131	. . . . . . 7 ⊢ (𝜑 → ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) ⊆ (ℕ(repr‘3)𝑁))
13	12	sselda 3981	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
14		vmaf 26859	. . . . . . . . . 10 ⊢ Λ:ℕ⟶ℝ
15	14	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → Λ:ℕ⟶ℝ)
16		ssidd 4004	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ℕ ⊆ ℕ)
17	5	nn0zd 12588	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
18	17	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝑁 ∈ ℤ)
19	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 3 ∈ ℕ₀)
20		simpr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
21	16, 18, 19, 20	reprf 33922	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝑛:(0..^3)⟶ℕ)
22		c0ex 11212	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
23	22	tpid1 4771	. . . . . . . . . . . 12 ⊢ 0 ∈ {0, 1, 2}
24		fzo0to3tp 13722	. . . . . . . . . . . 12 ⊢ (0..^3) = {0, 1, 2}
25	23, 24	eleqtrri 2830	. . . . . . . . . . 11 ⊢ 0 ∈ (0..^3)
26	25	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 0 ∈ (0..^3))
27	21, 26	ffvelcdmd 7086	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘0) ∈ ℕ)
28	15, 27	ffvelcdmd 7086	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘0)) ∈ ℝ)
29		tgoldbachgtda.h	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻:ℕ⟶(0[,)+∞))
30		rge0ssre 13437	. . . . . . . . . . 11 ⊢ (0[,)+∞) ⊆ ℝ
31		fss 6733	. . . . . . . . . . 11 ⊢ ((𝐻:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐻:ℕ⟶ℝ)
32	29, 30, 31	sylancl 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻:ℕ⟶ℝ)
33	32	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝐻:ℕ⟶ℝ)
34	33, 27	ffvelcdmd 7086	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐻‘(𝑛‘0)) ∈ ℝ)
35	28, 34	remulcld 11248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) ∈ ℝ)
36		1ex 11214	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
37	36	tpid2 4773	. . . . . . . . . . . . 13 ⊢ 1 ∈ {0, 1, 2}
38	37, 24	eleqtrri 2830	. . . . . . . . . . . 12 ⊢ 1 ∈ (0..^3)
39	38	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 1 ∈ (0..^3))
40	21, 39	ffvelcdmd 7086	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘1) ∈ ℕ)
41	15, 40	ffvelcdmd 7086	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘1)) ∈ ℝ)
42		tgoldbachgtda.k	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾:ℕ⟶(0[,)+∞))
43		fss 6733	. . . . . . . . . . . 12 ⊢ ((𝐾:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐾:ℕ⟶ℝ)
44	42, 30, 43	sylancl 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾:ℕ⟶ℝ)
45	44	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝐾:ℕ⟶ℝ)
46	45, 40	ffvelcdmd 7086	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐾‘(𝑛‘1)) ∈ ℝ)
47	41, 46	remulcld 11248	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) ∈ ℝ)
48		2ex 12293	. . . . . . . . . . . . . 14 ⊢ 2 ∈ V
49	48	tpid3 4776	. . . . . . . . . . . . 13 ⊢ 2 ∈ {0, 1, 2}
50	49, 24	eleqtrri 2830	. . . . . . . . . . . 12 ⊢ 2 ∈ (0..^3)
51	50	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 2 ∈ (0..^3))
52	21, 51	ffvelcdmd 7086	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘2) ∈ ℕ)
53	15, 52	ffvelcdmd 7086	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘2)) ∈ ℝ)
54	45, 52	ffvelcdmd 7086	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐾‘(𝑛‘2)) ∈ ℝ)
55	53, 54	remulcld 11248	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))) ∈ ℝ)
56	47, 55	remulcld 11248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2)))) ∈ ℝ)
57	35, 56	remulcld 11248	. . . . . 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 589	. . . . 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 15684	. . . 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 12491	. . . . . . 7 ⊢ 0 ∈ ℕ₀
61		qssre 12947	. . . . . . . 8 ⊢ ℚ ⊆ ℝ
62		4nn0 12495	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ₀
63		2nn0 12493	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
64		nn0ssq 12945	. . . . . . . . . . . . . . . 16 ⊢ ℕ₀ ⊆ ℚ
65		8nn0 12499	. . . . . . . . . . . . . . . 16 ⊢ 8 ∈ ℕ₀
66	64, 65	sselii 3978	. . . . . . . . . . . . . . 15 ⊢ 8 ∈ ℚ
67	62, 66	dp2clq 32314	. . . . . . . . . . . . . 14 ⊢ _48 ∈ ℚ
68	63, 67	dp2clq 32314	. . . . . . . . . . . . 13 ⊢ _2_48 ∈ ℚ
69	63, 68	dp2clq 32314	. . . . . . . . . . . 12 ⊢ _2_2_48 ∈ ℚ
70	62, 69	dp2clq 32314	. . . . . . . . . . 11 ⊢ _4_2_2_48 ∈ ℚ
71	60, 70	dp2clq 32314	. . . . . . . . . 10 ⊢ _0_4_2_2_48 ∈ ℚ
72	60, 71	dp2clq 32314	. . . . . . . . 9 ⊢ _0_0_4_2_2_48 ∈ ℚ
73	60, 72	dp2clq 32314	. . . . . . . 8 ⊢ _0_0_0_4_2_2_48 ∈ ℚ
74	61, 73	sselii 3978	. . . . . . 7 ⊢ _0_0_0_4_2_2_48 ∈ ℝ
75		dpcl 32324	. . . . . . 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 688	. . . . . 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 12231	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
79	78	resqcld 14094	. . . . 5 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ)
80	77, 79	remulcld 11248	. . . 4 ⊢ (𝜑 → ((0._0_0_0_4_2_2_48) · (𝑁↑2)) ∈ ℝ)
81	9, 57	fsumrecl 15684	. . . 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 12498	. . . . . . . . 9 ⊢ 7 ∈ ℕ₀
83	6, 67	dp2clq 32314	. . . . . . . . . 10 ⊢ _3_48 ∈ ℚ
84	61, 83	sselii 3978	. . . . . . . . 9 ⊢ _3_48 ∈ ℝ
85		dpcl 32324	. . . . . . . . 9 ⊢ ((7 ∈ ℕ₀ ∧ _3_48 ∈ ℝ) → (7._3_48) ∈ ℝ)
86	82, 84, 85	mp2an 688	. . . . . . . 8 ⊢ (7._3_48) ∈ ℝ
87	86	a1i 11	. . . . . . 7 ⊢ (𝜑 → (7._3_48) ∈ ℝ)
88	4	nnrpd 13018	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
89	88	relogcld 26367	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
90	5	nn0ge0d 12539	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝑁)
91	78, 90	resqrtcld 15368	. . . . . . . 8 ⊢ (𝜑 → (√‘𝑁) ∈ ℝ)
92	88	sqrtgt0d 15363	. . . . . . . . 9 ⊢ (𝜑 → 0 < (√‘𝑁))
93	92	gt0ne0d 11782	. . . . . . . 8 ⊢ (𝜑 → (√‘𝑁) ≠ 0)
94	89, 91, 93	redivcld 12046	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑁) / (√‘𝑁)) ∈ ℝ)
95	87, 94	remulcld 11248	. . . . . 6 ⊢ (𝜑 → ((7._3_48) · ((log‘𝑁) / (√‘𝑁))) ∈ ℝ)
96	95, 79	remulcld 11248	. . . . 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 33968	. . . . 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 12598	. . . . . . . 8 ⊢ 2 ∈ ℤ
101	100	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℤ)
102	88, 101	rpexpcld 14214	. . . . . 6 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ⁺)
103		hgt750lem 33961	. . . . . . 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 582	. . . . . 6 ⊢ (𝜑 → ((7._3_48) · ((log‘𝑁) / (√‘𝑁))) < (0._0_0_0_4_2_2_48))
105	95, 77, 102, 104	ltmul1dd 13075	. . . . 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 11376	. . . 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 33953	. . . . 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 11378	. . 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 11805	. . 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 231	. 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 4228	. . . . . . . 8 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ
114		prmssnn 16617	. . . . . . . 8 ⊢ ℙ ⊆ ℕ
115	113, 114	sstri 3990	. . . . . . 7 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ
116	115	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ)
117	8, 17, 7, 116	reprss 33927	. . . . 5 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ⊆ (ℕ(repr‘3)𝑁))
118	9, 117	ssfid 9269	. . . 4 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ∈ Fin)
119	117	sselda 3981	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
120	57	recnd 11246	. . . . 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 589	. . . 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 15683	. . 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 11246	. . 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 4470	. . . . 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 4480	. . . . . 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 217	. . . . 5 ⊢ (𝜑 → (((𝑂 ∩ ℙ)(repr‘3)𝑁) ∪ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))) = (ℕ(repr‘3)𝑁))
128	127	eqcomd 2736	. . . 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 15691	. . 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 11630	. 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 394 = wceq 1539 ∈ wcel 2104 {crab 3430 ∖ cdif 3944 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 {ctp 4631 class class class wbr 5147 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ∘_f cof 7670 Fincfn 8941 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 ici 11114 · cmul 11117 +∞cpnf 11249 < clt 11252 ≤ cle 11253 − cmin 11448 -cneg 11449 / cdiv 11875 ℕcn 12216 2c2 12271 3c3 12272 4c4 12273 5c5 12274 7c7 12276 8c8 12277 9c9 12278 ℕ₀cn0 12476 ℤcz 12562 cdc 12681 ℚcq 12936 (,)cioo 13328 [,)cico 13330 ..^cfzo 13631 ↑cexp 14031 √csqrt 15184 Σcsu 15636 expce 16009 πcpi 16014 ∥ cdvds 16201 ℙcprime 16612 ∫citg 25367 logclog 26299 Λcvma 26832 cdp2 32304 .cdp 32321 reprcrepr 33918 vtscvts 33945

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-reg 9589 ax-inf2 9638 ax-cc 10432 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 ax-ros335 33955 ax-ros336 33956

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-symdif 4241 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-r1 9761 df-rank 9762 df-dju 9898 df-card 9936 df-acn 9939 df-ac 10113 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-word 14469 df-concat 14525 df-s1 14550 df-s2 14803 df-s3 14804 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-prod 15854 df-ef 16015 df-e 16016 df-sin 16017 df-cos 16018 df-tan 16019 df-pi 16020 df-dvds 16202 df-gcd 16440 df-prm 16613 df-pc 16774 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-pmtr 19351 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-cmp 23111 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-ovol 25213 df-vol 25214 df-mbf 25368 df-itg1 25369 df-itg2 25370 df-ibl 25371 df-itg 25372 df-0p 25419 df-limc 25615 df-dv 25616 df-ulm 26125 df-log 26301 df-cxp 26302 df-atan 26608 df-cht 26837 df-vma 26838 df-chp 26839 df-dp2 32305 df-dp 32322 df-repr 33919 df-vts 33946

This theorem is referenced by: tgoldbachgtda 33971

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