tgoldbachgt - Mathbox for Thierry Arnoux

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Theorem tgoldbachgt 35172

Description: Odd integers greater than (10↑27) have at least a representation as a sum of three odd primes. Final statement in section 7.4 of [Helfgott] p. 70 , expressed using the set 𝐺 of odd numbers which can be written as a sum of three odd primes. (Contributed by Thierry Arnoux, 22-Dec-2021.)

**Hypotheses**
Ref	Expression
tgoldbachgt.o	⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}
tgoldbachgt.g	⊢ 𝐺 = {𝑧 ∈ 𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}

**Assertion**
Ref	Expression
tgoldbachgt	⊢ ∃𝑚 ∈ ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺))

Distinct variable groups: 𝑚,𝐺 𝑚,𝑂,𝑝,𝑞,𝑟,𝑧 𝑚,𝑛,𝑝,𝑞,𝑟,𝑧 Allowed substitution hints: 𝐺(𝑧, 𝑛, 𝑟, 𝑞, 𝑝) 𝑂(𝑛)

Proof of Theorem tgoldbachgt Dummy variables 𝑐 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		10nn 12757	. . 3 ⊢ ;10 ∈ ℕ
2		2nn0 12546	. . . 4 ⊢ 2 ∈ ℕ₀
3		7nn0 12551	. . . 4 ⊢ 7 ∈ ℕ₀
4	2, 3	deccl 12752	. . 3 ⊢ ;27 ∈ ℕ₀
5		nnexpcl 14139	. . 3 ⊢ ((;10 ∈ ℕ ∧ ;27 ∈ ℕ₀) → (;10↑;27) ∈ ℕ)
6	1, 4, 5	mp2an 705	. 2 ⊢ (;10↑;27) ∈ ℕ
7	6	nnrei 12267	. . . 4 ⊢ (;10↑;27) ∈ ℝ
8	7	leidi 11773	. . 3 ⊢ (;10↑;27) ≤ (;10↑;27)
9		simpl 488	. . . . . 6 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 𝑛 ∈ 𝑂)
10		inss2 4183	. . . . . . . . . . . . . 14 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ
11		prmssnn 16767	. . . . . . . . . . . . . 14 ⊢ ℙ ⊆ ℕ
12	10, 11	sstri 3940	. . . . . . . . . . . . 13 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ
13	12	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑂 ∩ ℙ) ⊆ ℕ)
14		tgoldbachgt.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}
15	14	eleq2i 2852	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑂 ↔ 𝑛 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧})
16		elrabi 3641	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} → 𝑛 ∈ ℤ)
17	15, 16	sylbi 220	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑂 → 𝑛 ∈ ℤ)
18	17	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑛 ∈ ℤ)
19		3nn0 12547	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℕ₀
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 3 ∈ ℕ₀)
21		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛))
22	13, 18, 20, 21	reprf 35121	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑐:(0..^3)⟶(𝑂 ∩ ℙ))
23		c0ex 11225	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
24	23	tpid1 4729	. . . . . . . . . . . . 13 ⊢ 0 ∈ {0, 1, 2}
25		fzo0to3tp 13809	. . . . . . . . . . . . 13 ⊢ (0..^3) = {0, 1, 2}
26	24, 25	eleqtrri 2859	. . . . . . . . . . . 12 ⊢ 0 ∈ (0..^3)
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ∈ (0..^3))
28	22, 27	ffvelcdmd 7079	. . . . . . . . . 10 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ (𝑂 ∩ ℙ))
29	28	elin2d 4151	. . . . . . . . 9 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ ℙ)
30		1eltp012 12336	. . . . . . . . . . . . 13 ⊢ 1 ∈ {0, 1, 2}
31	30, 25	eleqtrri 2859	. . . . . . . . . . . 12 ⊢ 1 ∈ (0..^3)
32	31	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 1 ∈ (0..^3))
33	22, 32	ffvelcdmd 7079	. . . . . . . . . 10 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ (𝑂 ∩ ℙ))
34	33	elin2d 4151	. . . . . . . . 9 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ ℙ)
35		2ex 12343	. . . . . . . . . . . . . 14 ⊢ 2 ∈ V
36	35	tpid3 4734	. . . . . . . . . . . . 13 ⊢ 2 ∈ {0, 1, 2}
37	36, 25	eleqtrri 2859	. . . . . . . . . . . 12 ⊢ 2 ∈ (0..^3)
38	37	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 2 ∈ (0..^3))
39	22, 38	ffvelcdmd 7079	. . . . . . . . . 10 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ (𝑂 ∩ ℙ))
40	39	elin2d 4151	. . . . . . . . 9 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ ℙ)
41	28	elin1d 4150	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ 𝑂)
42	33	elin1d 4150	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ 𝑂)
43	39	elin1d 4150	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ 𝑂)
44	41, 42, 43	3jca 1146	. . . . . . . . . 10 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → ((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ (𝑐‘2) ∈ 𝑂))
45	25	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (0..^3) = {0, 1, 2})
46	45	sumeq1d 15788	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → Σ𝑖 ∈ (0..^3)(𝑐‘𝑖) = Σ𝑖 ∈ {0, 1, 2} (𝑐‘𝑖))
47	13, 18, 20, 21	reprsum 35122	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → Σ𝑖 ∈ (0..^3)(𝑐‘𝑖) = 𝑛)
48		fveq2 6879	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑐‘𝑖) = (𝑐‘0))
49		fveq2 6879	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝑐‘𝑖) = (𝑐‘1))
50		fveq2 6879	. . . . . . . . . . . 12 ⊢ (𝑖 = 2 → (𝑐‘𝑖) = (𝑐‘2))
51	12, 28	sselid 3929	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ ℕ)
52	51	nncnd 12274	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ ℂ)
53	12, 33	sselid 3929	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ ℕ)
54	53	nncnd 12274	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ ℂ)
55	12, 39	sselid 3929	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ ℕ)
56	55	nncnd 12274	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ ℂ)
57	52, 54, 56	3jca 1146	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → ((𝑐‘0) ∈ ℂ ∧ (𝑐‘1) ∈ ℂ ∧ (𝑐‘2) ∈ ℂ))
58	23	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ∈ V)
59		1ex 11228	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
60	59	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 1 ∈ V)
61	35	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 2 ∈ V)
62	58, 60, 61	3jca 1146	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ V))
63		0ne1 12337	. . . . . . . . . . . . 13 ⊢ 0 ≠ 1
64	63	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ≠ 1)
65		0ne2 12475	. . . . . . . . . . . . 13 ⊢ 0 ≠ 2
66	65	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ≠ 2)
67		1ne2 12476	. . . . . . . . . . . . 13 ⊢ 1 ≠ 2
68	67	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 1 ≠ 2)
69	48, 49, 50, 57, 62, 64, 66, 68	sumtp 15836	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → Σ𝑖 ∈ {0, 1, 2} (𝑐‘𝑖) = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)))
70	46, 47, 69	3eqtr3d 2803	. . . . . . . . . 10 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑛 = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)))
71	44, 70	jca 521	. . . . . . . . 9 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ (𝑐‘2) ∈ 𝑂) ∧ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2))))
72		eleq1 2848	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝑐‘0) → (𝑝 ∈ 𝑂 ↔ (𝑐‘0) ∈ 𝑂))
73	72	3anbi1d 1468	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑐‘0) → ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ↔ ((𝑐‘0) ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂)))
74		oveq1 7421	. . . . . . . . . . . . 13 ⊢ (𝑝 = (𝑐‘0) → (𝑝 + 𝑞) = ((𝑐‘0) + 𝑞))
75	74	oveq1d 7429	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝑐‘0) → ((𝑝 + 𝑞) + 𝑟) = (((𝑐‘0) + 𝑞) + 𝑟))
76	75	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑐‘0) → (𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑛 = (((𝑐‘0) + 𝑞) + 𝑟)))
77	73, 76	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑝 = (𝑐‘0) → (((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)) ↔ (((𝑐‘0) ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = (((𝑐‘0) + 𝑞) + 𝑟))))
78		eleq1 2848	. . . . . . . . . . . 12 ⊢ (𝑞 = (𝑐‘1) → (𝑞 ∈ 𝑂 ↔ (𝑐‘1) ∈ 𝑂))
79	78	3anbi2d 1469	. . . . . . . . . . 11 ⊢ (𝑞 = (𝑐‘1) → (((𝑐‘0) ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ↔ ((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ 𝑟 ∈ 𝑂)))
80		oveq2 7422	. . . . . . . . . . . . 13 ⊢ (𝑞 = (𝑐‘1) → ((𝑐‘0) + 𝑞) = ((𝑐‘0) + (𝑐‘1)))
81	80	oveq1d 7429	. . . . . . . . . . . 12 ⊢ (𝑞 = (𝑐‘1) → (((𝑐‘0) + 𝑞) + 𝑟) = (((𝑐‘0) + (𝑐‘1)) + 𝑟))
82	81	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (𝑞 = (𝑐‘1) → (𝑛 = (((𝑐‘0) + 𝑞) + 𝑟) ↔ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + 𝑟)))
83	79, 82	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑞 = (𝑐‘1) → ((((𝑐‘0) ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = (((𝑐‘0) + 𝑞) + 𝑟)) ↔ (((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + 𝑟))))
84		eleq1 2848	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑐‘2) → (𝑟 ∈ 𝑂 ↔ (𝑐‘2) ∈ 𝑂))
85	84	3anbi3d 1470	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑐‘2) → (((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ↔ ((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ (𝑐‘2) ∈ 𝑂)))
86		oveq2 7422	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑐‘2) → (((𝑐‘0) + (𝑐‘1)) + 𝑟) = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)))
87	86	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑐‘2) → (𝑛 = (((𝑐‘0) + (𝑐‘1)) + 𝑟) ↔ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2))))
88	85, 87	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑟 = (𝑐‘2) → ((((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + 𝑟)) ↔ (((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ (𝑐‘2) ∈ 𝑂) ∧ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)))))
89	77, 83, 88	rspc3ev 3593	. . . . . . . . 9 ⊢ ((((𝑐‘0) ∈ ℙ ∧ (𝑐‘1) ∈ ℙ ∧ (𝑐‘2) ∈ ℙ) ∧ (((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ (𝑐‘2) ∈ 𝑂) ∧ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)))) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
90	29, 34, 40, 71, 89	syl31anc 1400	. . . . . . . 8 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
91	90	adantr 486	. . . . . . 7 ⊢ ((((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) ∧ ⊤) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
92	6	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) ∈ ℕ)
93	92	nnred 12273	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) ∈ ℝ)
94	17	zred 12726	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑂 → 𝑛 ∈ ℝ)
95	94	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 𝑛 ∈ ℝ)
96		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) < 𝑛)
97	93, 95, 96	ltled 11383	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) ≤ 𝑛)
98	14, 9, 97	tgoldbachgtd 35171	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 0 < (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑛)))
99		ovex 7447	. . . . . . . . . . . . . . 15 ⊢ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ∈ V
100		hashneq0 14429	. . . . . . . . . . . . . . 15 ⊢ (((𝑂 ∩ ℙ)(repr‘3)𝑛) ∈ V → (0 < (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑛)) ↔ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ≠ ∅))
101	99, 100	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (0 < (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑛)) ↔ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ≠ ∅)
102	98, 101	sylib 221	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ((𝑂 ∩ ℙ)(repr‘3)𝑛) ≠ ∅)
103	102	neneqd 2960	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ¬ ((𝑂 ∩ ℙ)(repr‘3)𝑛) = ∅)
104		neq0 4299	. . . . . . . . . . . 12 ⊢ (¬ ((𝑂 ∩ ℙ)(repr‘3)𝑛) = ∅ ↔ ∃𝑐 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛))
105	103, 104	sylib 221	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ∃𝑐 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛))
106		tru 1574	. . . . . . . . . . 11 ⊢ ⊤
107	105, 106	jctil 529	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (⊤ ∧ ∃𝑐 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)))
108		19.42v 1986	. . . . . . . . . 10 ⊢ (∃𝑐(⊤ ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) ↔ (⊤ ∧ ∃𝑐 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)))
109	107, 108	sylibr 237	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ∃𝑐(⊤ ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)))
110		exancom 1894	. . . . . . . . 9 ⊢ (∃𝑐(⊤ ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) ↔ ∃𝑐(𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ∧ ⊤))
111	109, 110	sylib 221	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ∃𝑐(𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ∧ ⊤))
112		df-rex 3087	. . . . . . . 8 ⊢ (∃𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)⊤ ↔ ∃𝑐(𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ∧ ⊤))
113	111, 112	sylibr 237	. . . . . . 7 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ∃𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)⊤)
114	91, 113	r19.29a 3170	. . . . . 6 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
115		tgoldbachgt.g	. . . . . . . . 9 ⊢ 𝐺 = {𝑧 ∈ 𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
116	115	eleq2i 2852	. . . . . . . 8 ⊢ (𝑛 ∈ 𝐺 ↔ 𝑛 ∈ {𝑧 ∈ 𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))})
117		eqeq1 2764	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑛 → (𝑧 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
118	117	anbi2d 642	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑛 → (((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟)) ↔ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
119	118	rexbidv 3186	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑛 → (∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟)) ↔ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
120	119	rexbidv 3186	. . . . . . . . . 10 ⊢ (𝑧 = 𝑛 → (∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟)) ↔ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
121	120	rexbidv 3186	. . . . . . . . 9 ⊢ (𝑧 = 𝑛 → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟)) ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
122	121	elrab3 3646	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑂 → (𝑛 ∈ {𝑧 ∈ 𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))} ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
123	116, 122	bitrid 286	. . . . . . 7 ⊢ (𝑛 ∈ 𝑂 → (𝑛 ∈ 𝐺 ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
124	123	biimpar 483	. . . . . 6 ⊢ ((𝑛 ∈ 𝑂 ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟))) → 𝑛 ∈ 𝐺)
125	9, 114, 124	syl2anc 596	. . . . 5 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 𝑛 ∈ 𝐺)
126	125	ex 418	. . . 4 ⊢ (𝑛 ∈ 𝑂 → ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺))
127	126	rgen 3078	. . 3 ⊢ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺)
128	8, 127	pm3.2i 476	. 2 ⊢ ((;10↑;27) ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺))
129		breq1 5106	. . . 4 ⊢ (𝑚 = (;10↑;27) → (𝑚 ≤ (;10↑;27) ↔ (;10↑;27) ≤ (;10↑;27)))
130		breq1 5106	. . . . . 6 ⊢ (𝑚 = (;10↑;27) → (𝑚 < 𝑛 ↔ (;10↑;27) < 𝑛))
131	130	imbi1d 344	. . . . 5 ⊢ (𝑚 = (;10↑;27) → ((𝑚 < 𝑛 → 𝑛 ∈ 𝐺) ↔ ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺)))
132	131	ralbidv 3185	. . . 4 ⊢ (𝑚 = (;10↑;27) → (∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺) ↔ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺)))
133	129, 132	anbi12d 644	. . 3 ⊢ (𝑚 = (;10↑;27) → ((𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺)) ↔ ((;10↑;27) ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺))))
134	133	rspcev 3576	. 2 ⊢ (((;10↑;27) ∈ ℕ ∧ ((;10↑;27) ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺))) → ∃𝑚 ∈ ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺)))
135	6, 128, 134	mp2an 705	1 ⊢ ∃𝑚 ∈ ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ⊤wtru 1571 ∃wex 1812 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 {crab 3412 Vcvv 3450 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 {ctp 4588 class class class wbr 5103 ‘cfv 6533 (class class class)co 7414 ℂcc 11123 ℝcr 11124 0cc0 11125 1c1 11126 + caddc 11128 < clt 11268 ≤ cle 11269 ℕcn 12258 2c2 12320 3c3 12321 7c7 12325 ℕ₀cn0 12529 ℤcz 12616 cdc 12737 ..^cfzo 13710 ↑cexp 14126 ♯chash 14395 Σcsu 15774 ∥ cdvds 16343 ℙcprime 16762 reprcrepr 35117

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 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-reg 9565 ax-inf2 9621 ax-cc 10438 ax-ac2 10466 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-addf 11204 ax-hgt749 35153 ax-ros335 35154 ax-ros336 35155

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4199 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-oadd 8460 df-omul 8461 df-er 8697 df-map 8829 df-pm 8830 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9333 df-fi 9382 df-sup 9413 df-inf 9414 df-oi 9483 df-r1 9747 df-rank 9748 df-scott 9869 df-dju 9907 df-card 9945 df-acn 9948 df-ac 10120 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-xnn0 12603 df-z 12617 df-dec 12738 df-uz 12889 df-q 12999 df-rp 13044 df-xneg 13164 df-xadd 13165 df-xmul 13166 df-ioo 13403 df-ioc 13404 df-ico 13405 df-icc 13406 df-fz 13563 df-fzo 13711 df-fl 13854 df-mod 13932 df-seq 14067 df-exp 14127 df-fac 14339 df-bc 14368 df-hash 14396 df-word 14580 df-concat 14637 df-s1 14664 df-s2 14920 df-s3 14921 df-shft 15141 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324 df-limsup 15559 df-clim 15576 df-rlim 15577 df-sum 15775 df-prod 15994 df-ef 16154 df-e 16155 df-sin 16156 df-cos 16157 df-tan 16158 df-pi 16159 df-dvds 16344 df-gcd 16586 df-prm 16763 df-pc 16930 df-struct 17240 df-sets 17257 df-slot 17275 df-ndx 17287 df-base 17303 df-ress 17324 df-plusg 17356 df-mulr 17357 df-starv 17358 df-sca 17359 df-vsca 17360 df-ip 17361 df-tset 17362 df-ple 17363 df-ds 17365 df-unif 17366 df-hom 17367 df-cco 17368 df-rest 17508 df-topn 17509 df-0g 17527 df-gsum 17528 df-topgen 17529 df-pt 17530 df-prds 17533 df-xrs 17589 df-qtop 17594 df-imas 17595 df-xps 17597 df-mre 17671 df-mrc 17672 df-acs 17674 df-mgm 18731 df-sgrp 18822 df-mnd 18838 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 26094 df-dv 26095 df-ulm 26614 df-log 26794 df-cxp 26795 df-atan 27105 df-cht 27334 df-vma 27335 df-chp 27336 df-dp2 33318 df-dp 33335 df-repr 35118 df-vts 35145

This theorem is used by: (None)

W3C validator