tgoldbachgtde - Mathbox for Thierry Arnoux

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Theorem tgoldbachgtde 34804

Description: Lemma for tgoldbachgtd 34806. (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 34803	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
5	4	nnnn0d 12498	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6		3nn0 12455	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → 3 ∈ ℕ₀)
8		ssidd 3945	. . . . . . 7 ⊢ (𝜑 → ℕ ⊆ ℕ)
9	5, 7, 8	reprfi2 34767	. . . . . 6 ⊢ (𝜑 → (ℕ(repr‘3)𝑁) ∈ Fin)
10		diffi 9109	. . . . . 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 4077	. . . . . . 7 ⊢ (𝜑 → ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) ⊆ (ℕ(repr‘3)𝑁))
13	12	sselda 3921	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
14		vmaf 27082	. . . . . . . . . 10 ⊢ Λ:ℕ⟶ℝ
15	14	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → Λ:ℕ⟶ℝ)
16		ssidd 3945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ℕ ⊆ ℕ)
17	5	nn0zd 12549	. . . . . . . . . . . 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 34756	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝑛:(0..^3)⟶ℕ)
22		c0ex 11138	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
23	22	tpid1 4712	. . . . . . . . . . . 12 ⊢ 0 ∈ {0, 1, 2}
24		fzo0to3tp 13707	. . . . . . . . . . . 12 ⊢ (0..^3) = {0, 1, 2}
25	23, 24	eleqtrri 2835	. . . . . . . . . . 11 ⊢ 0 ∈ (0..^3)
26	25	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 0 ∈ (0..^3))
27	21, 26	ffvelcdmd 7037	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘0) ∈ ℕ)
28	15, 27	ffvelcdmd 7037	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘0)) ∈ ℝ)
29		tgoldbachgtda.h	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻:ℕ⟶(0[,)+∞))
30		rge0ssre 13409	. . . . . . . . . . 11 ⊢ (0[,)+∞) ⊆ ℝ
31		fss 6684	. . . . . . . . . . 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 7037	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐻‘(𝑛‘0)) ∈ ℝ)
35	28, 34	remulcld 11175	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) ∈ ℝ)
36		1ex 11140	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
37	36	tpid2 4714	. . . . . . . . . . . . 13 ⊢ 1 ∈ {0, 1, 2}
38	37, 24	eleqtrri 2835	. . . . . . . . . . . 12 ⊢ 1 ∈ (0..^3)
39	38	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 1 ∈ (0..^3))
40	21, 39	ffvelcdmd 7037	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘1) ∈ ℕ)
41	15, 40	ffvelcdmd 7037	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘1)) ∈ ℝ)
42		tgoldbachgtda.k	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾:ℕ⟶(0[,)+∞))
43		fss 6684	. . . . . . . . . . . 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 7037	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐾‘(𝑛‘1)) ∈ ℝ)
47	41, 46	remulcld 11175	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) ∈ ℝ)
48		2ex 12258	. . . . . . . . . . . . . 14 ⊢ 2 ∈ V
49	48	tpid3 4717	. . . . . . . . . . . . 13 ⊢ 2 ∈ {0, 1, 2}
50	49, 24	eleqtrri 2835	. . . . . . . . . . . 12 ⊢ 2 ∈ (0..^3)
51	50	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 2 ∈ (0..^3))
52	21, 51	ffvelcdmd 7037	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘2) ∈ ℕ)
53	15, 52	ffvelcdmd 7037	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘2)) ∈ ℝ)
54	45, 52	ffvelcdmd 7037	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐾‘(𝑛‘2)) ∈ ℝ)
55	53, 54	remulcld 11175	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))) ∈ ℝ)
56	47, 55	remulcld 11175	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2)))) ∈ ℝ)
57	35, 56	remulcld 11175	. . . . . 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 15696	. . . 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 12452	. . . . . . 7 ⊢ 0 ∈ ℕ₀
61		qssre 12909	. . . . . . . 8 ⊢ ℚ ⊆ ℝ
62		4nn0 12456	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ₀
63		2nn0 12454	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
64		nn0ssq 12907	. . . . . . . . . . . . . . . 16 ⊢ ℕ₀ ⊆ ℚ
65		8nn0 12460	. . . . . . . . . . . . . . . 16 ⊢ 8 ∈ ℕ₀
66	64, 65	sselii 3918	. . . . . . . . . . . . . . 15 ⊢ 8 ∈ ℚ
67	62, 66	dp2clq 32940	. . . . . . . . . . . . . 14 ⊢ _48 ∈ ℚ
68	63, 67	dp2clq 32940	. . . . . . . . . . . . 13 ⊢ _2_48 ∈ ℚ
69	63, 68	dp2clq 32940	. . . . . . . . . . . 12 ⊢ _2_2_48 ∈ ℚ
70	62, 69	dp2clq 32940	. . . . . . . . . . 11 ⊢ _4_2_2_48 ∈ ℚ
71	60, 70	dp2clq 32940	. . . . . . . . . 10 ⊢ _0_4_2_2_48 ∈ ℚ
72	60, 71	dp2clq 32940	. . . . . . . . 9 ⊢ _0_0_4_2_2_48 ∈ ℚ
73	60, 72	dp2clq 32940	. . . . . . . 8 ⊢ _0_0_0_4_2_2_48 ∈ ℚ
74	61, 73	sselii 3918	. . . . . . 7 ⊢ _0_0_0_4_2_2_48 ∈ ℝ
75		dpcl 32950	. . . . . . 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 12189	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
79	78	resqcld 14087	. . . . 5 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ)
80	77, 79	remulcld 11175	. . . 4 ⊢ (𝜑 → ((0._0_0_0_4_2_2_48) · (𝑁↑2)) ∈ ℝ)
81	9, 57	fsumrecl 15696	. . . 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 12459	. . . . . . . . 9 ⊢ 7 ∈ ℕ₀
83	6, 67	dp2clq 32940	. . . . . . . . . 10 ⊢ _3_48 ∈ ℚ
84	61, 83	sselii 3918	. . . . . . . . 9 ⊢ _3_48 ∈ ℝ
85		dpcl 32950	. . . . . . . . 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 12984	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
89	88	relogcld 26587	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
90	5	nn0ge0d 12501	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝑁)
91	78, 90	resqrtcld 15380	. . . . . . . 8 ⊢ (𝜑 → (√‘𝑁) ∈ ℝ)
92	88	sqrtgt0d 15375	. . . . . . . . 9 ⊢ (𝜑 → 0 < (√‘𝑁))
93	92	gt0ne0d 11714	. . . . . . . 8 ⊢ (𝜑 → (√‘𝑁) ≠ 0)
94	89, 91, 93	redivcld 11983	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑁) / (√‘𝑁)) ∈ ℝ)
95	87, 94	remulcld 11175	. . . . . 6 ⊢ (𝜑 → ((7._3_48) · ((log‘𝑁) / (√‘𝑁))) ∈ ℝ)
96	95, 79	remulcld 11175	. . . . 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 34802	. . . . 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 12559	. . . . . . . 8 ⊢ 2 ∈ ℤ
101	100	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℤ)
102	88, 101	rpexpcld 14209	. . . . . 6 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ⁺)
103		hgt750lem 34795	. . . . . . 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 13041	. . . . 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 11304	. . . 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 34787	. . . . 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 5113	. . . 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 11306	. . 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 11737	. . 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 4178	. . . . . . . 8 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ
114		prmssnn 16645	. . . . . . . 8 ⊢ ℙ ⊆ ℕ
115	113, 114	sstri 3931	. . . . . . 7 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ
116	115	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ)
117	8, 17, 7, 116	reprss 34761	. . . . 5 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ⊆ (ℕ(repr‘3)𝑁))
118	9, 117	ssfid 9179	. . . 4 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ∈ Fin)
119	117	sselda 3921	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
120	57	recnd 11173	. . . . 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 15695	. . 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 11173	. . 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 4412	. . . . 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 4422	. . . . . 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 2742	. . . 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 15703	. . 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 11562	. 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 5111	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 3389 ∖ cdif 3886 ∪ cun 3887 ∩ cin 3888 ⊆ wss 3889 ∅c0 4273 {ctp 4571 class class class wbr 5085 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ∘_f cof 7629 Fincfn 8893 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 ici 11040 · cmul 11043 +∞cpnf 11176 < clt 11179 ≤ cle 11180 − cmin 11377 -cneg 11378 / cdiv 11807 ℕcn 12174 2c2 12236 3c3 12237 4c4 12238 5c5 12239 7c7 12241 8c8 12242 9c9 12243 ℕ₀cn0 12437 ℤcz 12524 cdc 12644 ℚcq 12898 (,)cioo 13298 [,)cico 13300 ..^cfzo 13608 ↑cexp 14023 √csqrt 15195 Σcsu 15648 expce 16026 πcpi 16031 ∥ cdvds 16221 ℙcprime 16640 ∫citg 25585 logclog 26518 Λcvma 27055 cdp2 32930 .cdp 32947 reprcrepr 34752 vtscvts 34779

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 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-reg 9507 ax-inf2 9562 ax-cc 10357 ax-ac2 10385 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-ros335 34789 ax-ros336 34790

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-symdif 4193 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-ofr 7632 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-r1 9688 df-rank 9689 df-dju 9825 df-card 9863 df-acn 9866 df-ac 10038 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-word 14476 df-concat 14533 df-s1 14559 df-s2 14810 df-s3 14811 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-prod 15869 df-ef 16032 df-e 16033 df-sin 16034 df-cos 16035 df-tan 16036 df-pi 16037 df-dvds 16222 df-gcd 16464 df-prm 16641 df-pc 16808 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 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-pmtr 19417 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-ovol 25431 df-vol 25432 df-mbf 25586 df-itg1 25587 df-itg2 25588 df-ibl 25589 df-itg 25590 df-0p 25637 df-limc 25833 df-dv 25834 df-ulm 26342 df-log 26520 df-cxp 26521 df-atan 26831 df-cht 27060 df-vma 27061 df-chp 27062 df-dp2 32931 df-dp 32948 df-repr 34753 df-vts 34780

This theorem is referenced by: tgoldbachgtda 34805

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