tgoldbachgtde - Mathbox for Thierry Arnoux

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Theorem tgoldbachgtde 35155

Description: Lemma for tgoldbachgtd 35157. (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 35154	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
5	4	nnnn0d 12592	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6		3nn0 12549	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → 3 ∈ ℕ₀)
8		ssidd 3957	. . . . . . 7 ⊢ (𝜑 → ℕ ⊆ ℕ)
9	5, 7, 8	reprfi2 35118	. . . . . 6 ⊢ (𝜑 → (ℕ(repr‘3)𝑁) ∈ Fin)
10		diffi 9172	. . . . . 6 ⊢ ((ℕ(repr‘3)𝑁) ∈ Fin → ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) ∈ Fin)
11	9, 10	syl 18	. . . . 5 ⊢ (𝜑 → ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) ∈ Fin)
12		difssd 4087	. . . . . . 7 ⊢ (𝜑 → ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) ⊆ (ℕ(repr‘3)𝑁))
13	12	sselda 3934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
14		vmaf 27353	. . . . . . . . . 10 ⊢ Λ:ℕ⟶ℝ
15	14	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → Λ:ℕ⟶ℝ)
16		ssidd 3957	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ℕ ⊆ ℕ)
17	5	nn0zd 12643	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
18	17	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝑁 ∈ ℤ)
19	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 3 ∈ ℕ₀)
20		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
21	16, 18, 19, 20	reprf 35107	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝑛:(0..^3)⟶ℕ)
22		c0ex 11227	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
23	22	tpid1 4732	. . . . . . . . . . . 12 ⊢ 0 ∈ {0, 1, 2}
24		fzo0to3tp 13810	. . . . . . . . . . . 12 ⊢ (0..^3) = {0, 1, 2}
25	23, 24	eleqtrri 2861	. . . . . . . . . . 11 ⊢ 0 ∈ (0..^3)
26	25	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 0 ∈ (0..^3))
27	21, 26	ffvelcdmd 7081	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘0) ∈ ℕ)
28	15, 27	ffvelcdmd 7081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘0)) ∈ ℝ)
29		tgoldbachgtda.h	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻:ℕ⟶(0[,)+∞))
30		rge0ssre 13511	. . . . . . . . . . 11 ⊢ (0[,)+∞) ⊆ ℝ
31		fss 6723	. . . . . . . . . . 11 ⊢ ((𝐻:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐻:ℕ⟶ℝ)
32	29, 30, 31	sylancl 598	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻:ℕ⟶ℝ)
33	32	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝐻:ℕ⟶ℝ)
34	33, 27	ffvelcdmd 7081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐻‘(𝑛‘0)) ∈ ℝ)
35	28, 34	remulcld 11266	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) ∈ ℝ)
36		1eltp012 12338	. . . . . . . . . . . . 13 ⊢ 1 ∈ {0, 1, 2}
37	36, 24	eleqtrri 2861	. . . . . . . . . . . 12 ⊢ 1 ∈ (0..^3)
38	37	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 1 ∈ (0..^3))
39	21, 38	ffvelcdmd 7081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘1) ∈ ℕ)
40	15, 39	ffvelcdmd 7081	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘1)) ∈ ℝ)
41		tgoldbachgtda.k	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾:ℕ⟶(0[,)+∞))
42		fss 6723	. . . . . . . . . . . 12 ⊢ ((𝐾:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐾:ℕ⟶ℝ)
43	41, 30, 42	sylancl 598	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾:ℕ⟶ℝ)
44	43	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 𝐾:ℕ⟶ℝ)
45	44, 39	ffvelcdmd 7081	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐾‘(𝑛‘1)) ∈ ℝ)
46	40, 45	remulcld 11266	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) ∈ ℝ)
47		2ex 12345	. . . . . . . . . . . . . 14 ⊢ 2 ∈ V
48	47	tpid3 4737	. . . . . . . . . . . . 13 ⊢ 2 ∈ {0, 1, 2}
49	48, 24	eleqtrri 2861	. . . . . . . . . . . 12 ⊢ 2 ∈ (0..^3)
50	49	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → 2 ∈ (0..^3))
51	21, 50	ffvelcdmd 7081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝑛‘2) ∈ ℕ)
52	15, 51	ffvelcdmd 7081	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (Λ‘(𝑛‘2)) ∈ ℝ)
53	44, 51	ffvelcdmd 7081	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (𝐾‘(𝑛‘2)) ∈ ℝ)
54	52, 53	remulcld 11266	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))) ∈ ℝ)
55	46, 54	remulcld 11266	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ(repr‘3)𝑁)) → (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2)))) ∈ ℝ)
56	35, 55	remulcld 11266	. . . . . 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)
57	13, 56	syldan 603	. . . . 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)
58	11, 57	fsumrecl 15822	. . . 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)
59		0nn0 12546	. . . . . . 7 ⊢ 0 ∈ ℕ₀
60		qssre 13011	. . . . . . . 8 ⊢ ℚ ⊆ ℝ
61		4nn0 12550	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ₀
62		2nn0 12548	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
63		nn0ssq 13009	. . . . . . . . . . . . . . . 16 ⊢ ℕ₀ ⊆ ℚ
64		8nn0 12554	. . . . . . . . . . . . . . . 16 ⊢ 8 ∈ ℕ₀
65	63, 64	sselii 3931	. . . . . . . . . . . . . . 15 ⊢ 8 ∈ ℚ
66	61, 65	dp2clq 33313	. . . . . . . . . . . . . 14 ⊢ _48 ∈ ℚ
67	62, 66	dp2clq 33313	. . . . . . . . . . . . 13 ⊢ _2_48 ∈ ℚ
68	62, 67	dp2clq 33313	. . . . . . . . . . . 12 ⊢ _2_2_48 ∈ ℚ
69	61, 68	dp2clq 33313	. . . . . . . . . . 11 ⊢ _4_2_2_48 ∈ ℚ
70	59, 69	dp2clq 33313	. . . . . . . . . 10 ⊢ _0_4_2_2_48 ∈ ℚ
71	59, 70	dp2clq 33313	. . . . . . . . 9 ⊢ _0_0_4_2_2_48 ∈ ℚ
72	59, 71	dp2clq 33313	. . . . . . . 8 ⊢ _0_0_0_4_2_2_48 ∈ ℚ
73	60, 72	sselii 3931	. . . . . . 7 ⊢ _0_0_0_4_2_2_48 ∈ ℝ
74		dpcl 33323	. . . . . . 7 ⊢ ((0 ∈ ℕ₀ ∧ _0_0_0_4_2_2_48 ∈ ℝ) → (0._0_0_0_4_2_2_48) ∈ ℝ)
75	59, 73, 74	mp2an 705	. . . . . 6 ⊢ (0._0_0_0_4_2_2_48) ∈ ℝ
76	75	a1i 11	. . . . 5 ⊢ (𝜑 → (0._0_0_0_4_2_2_48) ∈ ℝ)
77	4	nnred 12275	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
78	77	resqcld 14191	. . . . 5 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ)
79	76, 78	remulcld 11266	. . . 4 ⊢ (𝜑 → ((0._0_0_0_4_2_2_48) · (𝑁↑2)) ∈ ℝ)
80	9, 56	fsumrecl 15822	. . . 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)
81		7nn0 12553	. . . . . . . . 9 ⊢ 7 ∈ ℕ₀
82	6, 66	dp2clq 33313	. . . . . . . . . 10 ⊢ _3_48 ∈ ℚ
83	60, 82	sselii 3931	. . . . . . . . 9 ⊢ _3_48 ∈ ℝ
84		dpcl 33323	. . . . . . . . 9 ⊢ ((7 ∈ ℕ₀ ∧ _3_48 ∈ ℝ) → (7._3_48) ∈ ℝ)
85	81, 83, 84	mp2an 705	. . . . . . . 8 ⊢ (7._3_48) ∈ ℝ
86	85	a1i 11	. . . . . . 7 ⊢ (𝜑 → (7._3_48) ∈ ℝ)
87	4	nnrpd 13086	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
88	87	relogcld 26858	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
89	5	nn0ge0d 12595	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝑁)
90	77, 89	resqrtcld 15507	. . . . . . . 8 ⊢ (𝜑 → (√‘𝑁) ∈ ℝ)
91	87	sqrtgt0d 15502	. . . . . . . . 9 ⊢ (𝜑 → 0 < (√‘𝑁))
92	91	gt0ne0d 11805	. . . . . . . 8 ⊢ (𝜑 → (√‘𝑁) ≠ 0)
93	88, 90, 92	redivcld 12070	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑁) / (√‘𝑁)) ∈ ℝ)
94	86, 93	remulcld 11266	. . . . . 6 ⊢ (𝜑 → ((7._3_48) · ((log‘𝑁) / (√‘𝑁))) ∈ ℝ)
95	94, 78	remulcld 11266	. . . . 5 ⊢ (𝜑 → (((7._3_48) · ((log‘𝑁) / (√‘𝑁))) · (𝑁↑2)) ∈ ℝ)
96		tgoldbachgtda.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐾‘𝑚) ≤ (1._0_7_9_9_55))
97		tgoldbachgtda.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ≤ (1._4_14))
98	1, 4, 3, 29, 41, 96, 97	hgt750leme 35153	. . . . 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)))
99		2z 12653	. . . . . . . 8 ⊢ 2 ∈ ℤ
100	99	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℤ)
101	87, 100	rpexpcld 14313	. . . . . 6 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ⁺)
102		hgt750lem 35146	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (;10↑;27) ≤ 𝑁) → ((7._3_48) · ((log‘𝑁) / (√‘𝑁))) < (0._0_0_0_4_2_2_48))
103	5, 3, 102	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ((7._3_48) · ((log‘𝑁) / (√‘𝑁))) < (0._0_0_0_4_2_2_48))
104	94, 76, 101, 103	ltmul1dd 13143	. . . . 5 ⊢ (𝜑 → (((7._3_48) · ((log‘𝑁) / (√‘𝑁))) · (𝑁↑2)) < ((0._0_0_0_4_2_2_48) · (𝑁↑2)))
105	58, 95, 79, 98, 104	lelttrd 11395	. . . 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)))
106		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)
107	32, 43, 5	circlemethhgt 35138	. . . . 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)
108	106, 107	breqtrrd 5137	. . . 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))))))
109	58, 79, 80, 105, 108	ltletrd 11397	. . 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))))))
110	58, 80	posdifd 11828	. . 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))))))))
111	109, 110	mpbid 235	. 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)))))))
112		inss2 4186	. . . . . . . 8 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ
113		prmssnn 16770	. . . . . . . 8 ⊢ ℙ ⊆ ℕ
114	112, 113	sstri 3943	. . . . . . 7 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ
115	114	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ)
116	8, 17, 7, 115	reprss 35112	. . . . 5 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ⊆ (ℕ(repr‘3)𝑁))
117	9, 116	ssfid 9242	. . . 4 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ∈ Fin)
118	116	sselda 3934	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
119	56	recnd 11264	. . . . 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)
120	118, 119	syldan 603	. . . 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)
121	117, 120	fsumcl 15821	. . 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)
122	58	recnd 11264	. . 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)
123		disjdif 4429	. . . . 5 ⊢ (((𝑂 ∩ ℙ)(repr‘3)𝑁) ∩ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))) = ∅
124	123	a1i 11	. . . 4 ⊢ (𝜑 → (((𝑂 ∩ ℙ)(repr‘3)𝑁) ∩ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))) = ∅)
125		undif 4441	. . . . . 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))
126	116, 125	sylib 221	. . . . 5 ⊢ (𝜑 → (((𝑂 ∩ ℙ)(repr‘3)𝑁) ∪ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))) = (ℕ(repr‘3)𝑁))
127	126	eqcomd 2768	. . . 4 ⊢ (𝜑 → (ℕ(repr‘3)𝑁) = (((𝑂 ∩ ℙ)(repr‘3)𝑁) ∪ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))))
128	124, 127, 9, 119	fsumsplit 15829	. . 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)))))))
129	121, 122, 128	mvrraddd 11653	. 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))))))
130	111, 129	breqtrd 5135	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {crab 3414 ∖ cdif 3899 ∪ cun 3900 ∩ cin 3901 ⊆ wss 3902 ∅c0 4282 {ctp 4591 class class class wbr 5107 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 ∘_f cof 7679 Fincfn 8955 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 ici 11129 · cmul 11132 +∞cpnf 11267 < clt 11270 ≤ cle 11271 − cmin 11468 -cneg 11469 / cdiv 11898 ℕcn 12260 2c2 12322 3c3 12323 4c4 12324 5c5 12325 7c7 12327 8c8 12328 9c9 12329 ℕ₀cn0 12531 ℤcz 12618 cdc 12739 ℚcq 13000 (,)cioo 13400 [,)cico 13402 ..^cfzo 13711 ↑cexp 14127 √csqrt 15322 Σcsu 15775 expce 16151 πcpi 16156 ∥ cdvds 16346 ℙcprime 16765 ∫citg 25847 logclog 26789 Λcvma 27326 cdp2 33303 .cdp 33320 reprcrepr 35103 vtscvts 35130

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-reg 9567 ax-inf2 9623 ax-cc 10440 ax-ac2 10468 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206 ax-ros335 35140 ax-ros336 35141

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-symdif 4202 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-disj 5075 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-ofr 7682 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-oadd 8462 df-omul 8463 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-r1 9749 df-rank 9750 df-scott 9871 df-dju 9909 df-card 9947 df-acn 9950 df-ac 10122 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-xnn0 12605 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13045 df-xneg 13165 df-xadd 13166 df-xmul 13167 df-ioo 13404 df-ioc 13405 df-ico 13406 df-icc 13407 df-fz 13564 df-fzo 13712 df-fl 13855 df-mod 13933 df-seq 14068 df-exp 14128 df-fac 14340 df-bc 14369 df-hash 14397 df-word 14581 df-concat 14638 df-s1 14665 df-s2 14921 df-s3 14922 df-shft 15142 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-limsup 15560 df-clim 15577 df-rlim 15578 df-sum 15776 df-prod 15995 df-ef 16157 df-e 16158 df-sin 16159 df-cos 16160 df-tan 16161 df-pi 16162 df-dvds 16347 df-gcd 16589 df-prm 16766 df-pc 16933 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-starv 17361 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ds 17368 df-unif 17369 df-hom 17370 df-cco 17371 df-rest 17511 df-topn 17512 df-0g 17530 df-gsum 17531 df-topgen 17532 df-pt 17533 df-prds 17536 df-xrs 17592 df-qtop 17597 df-imas 17598 df-xps 17600 df-mre 17674 df-mrc 17675 df-acs 17677 df-mgm 18734 df-sgrp 18823 df-mnd 18839 df-submnd 18893 df-mulg 19192 df-cntz 19445 df-pmtr 19570 df-cmn 19910 df-psmet 21578 df-xmet 21579 df-met 21580 df-bl 21581 df-mopn 21582 df-fbas 21583 df-fg 21584 df-cnfld 21587 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-cld 23245 df-ntr 23246 df-cls 23247 df-nei 23324 df-lp 23362 df-perf 23363 df-cn 23453 df-cnp 23454 df-haus 23541 df-cmp 23613 df-tx 23789 df-hmeo 23982 df-fil 24073 df-fm 24165 df-flim 24166 df-flf 24167 df-xms 24547 df-ms 24548 df-tms 24549 df-cncf 25107 df-ovol 25693 df-vol 25694 df-mbf 25848 df-itg1 25849 df-itg2 25850 df-ibl 25851 df-itg 25852 df-0p 25899 df-limc 26095 df-dv 26096 df-ulm 26610 df-log 26791 df-cxp 26792 df-atan 27102 df-cht 27331 df-vma 27332 df-chp 27333 df-dp2 33304 df-dp 33321 df-repr 35104 df-vts 35131

This theorem is used by: tgoldbachgtda 35156

W3C validator