tgoldbachgt - Mathbox for Thierry Arnoux

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Theorem tgoldbachgt 35292

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 12834	. . 3 ⊢ ;10 ∈ ℕ
2		2nn0 12623	. . . 4 ⊢ 2 ∈ ℕ₀
3		7nn0 12628	. . . 4 ⊢ 7 ∈ ℕ₀
4	2, 3	deccl 12829	. . 3 ⊢ ;27 ∈ ℕ₀
5		nnexpcl 14217	. . 3 ⊢ ((;10 ∈ ℕ ∧ ;27 ∈ ℕ₀) → (;10↑;27) ∈ ℕ)
6	1, 4, 5	mp2an 705	. 2 ⊢ (;10↑;27) ∈ ℕ
7	6	nnrei 12344	. . . 4 ⊢ (;10↑;27) ∈ ℝ
8	7	leidi 11850	. . 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 16851	. . . . . . . . . . . . . 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 2853	. . . . . . . . . . . . . 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 12624	. . . . . . . . . . . . 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 35241	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑐:(0..^3)⟶(𝑂 ∩ ℙ))
23		c0ex 11300	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
24	23	tpid1 4729	. . . . . . . . . . . . 13 ⊢ 0 ∈ {0, 1, 2}
25		fzo0to3tp 13887	. . . . . . . . . . . . 13 ⊢ (0..^3) = {0, 1, 2}
26	24, 25	eleqtrri 2860	. . . . . . . . . . . 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 7085	. . . . . . . . . 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 12413	. . . . . . . . . . . . 13 ⊢ 1 ∈ {0, 1, 2}
31	30, 25	eleqtrri 2860	. . . . . . . . . . . 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 7085	. . . . . . . . . 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 12420	. . . . . . . . . . . . . 14 ⊢ 2 ∈ V
36	35	tpid3 4734	. . . . . . . . . . . . 13 ⊢ 2 ∈ {0, 1, 2}
37	36, 25	eleqtrri 2860	. . . . . . . . . . . 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 7085	. . . . . . . . . 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 15867	. . . . . . . . . . 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 35242	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → Σ𝑖 ∈ (0..^3)(𝑐‘𝑖) = 𝑛)
48		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑐‘𝑖) = (𝑐‘0))
49		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝑐‘𝑖) = (𝑐‘1))
50		fveq2 6885	. . . . . . . . . . . 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 12351	. . . . . . . . . . . . 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 12351	. . . . . . . . . . . . 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 12351	. . . . . . . . . . . . 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 11303	. . . . . . . . . . . . . 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 12414	. . . . . . . . . . . . 13 ⊢ 0 ≠ 1
64	63	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ≠ 1)
65		0ne2 12552	. . . . . . . . . . . . 13 ⊢ 0 ≠ 2
66	65	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ≠ 2)
67		1ne2 12553	. . . . . . . . . . . . 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 15915	. . . . . . . . . . 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 2804	. . . . . . . . . 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 2849	. . . . . . . . . . . 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 7427	. . . . . . . . . . . . 13 ⊢ (𝑝 = (𝑐‘0) → (𝑝 + 𝑞) = ((𝑐‘0) + 𝑞))
75	74	oveq1d 7435	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝑐‘0) → ((𝑝 + 𝑞) + 𝑟) = (((𝑐‘0) + 𝑞) + 𝑟))
76	75	eqeq2d 2772	. . . . . . . . . . 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 2849	. . . . . . . . . . . 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 7428	. . . . . . . . . . . . 13 ⊢ (𝑞 = (𝑐‘1) → ((𝑐‘0) + 𝑞) = ((𝑐‘0) + (𝑐‘1)))
81	80	oveq1d 7435	. . . . . . . . . . . 12 ⊢ (𝑞 = (𝑐‘1) → (((𝑐‘0) + 𝑞) + 𝑟) = (((𝑐‘0) + (𝑐‘1)) + 𝑟))
82	81	eqeq2d 2772	. . . . . . . . . . 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 2849	. . . . . . . . . . . 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 7428	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑐‘2) → (((𝑐‘0) + (𝑐‘1)) + 𝑟) = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)))
87	86	eqeq2d 2772	. . . . . . . . . . 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 12350	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) ∈ ℝ)
94	17	zred 12803	. . . . . . . . . . . . . . . . 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 11458	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) ≤ 𝑛)
98	14, 9, 97	tgoldbachgtd 35291	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 0 < (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑛)))
99		ovex 7453	. . . . . . . . . . . . . . 15 ⊢ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ∈ V
100		hashneq0 14508	. . . . . . . . . . . . . . 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 2961	. . . . . . . . . . . 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 3088	. . . . . . . 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 3171	. . . . . 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 2853	. . . . . . . 8 ⊢ (𝑛 ∈ 𝐺 ↔ 𝑛 ∈ {𝑧 ∈ 𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))})
117		eqeq1 2765	. . . . . . . . . . . . 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 3187	. . . . . . . . . . 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 3187	. . . . . . . . . 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 3187	. . . . . . . . 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 3079	. . 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 3186	. . . 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 3577	. 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 2956 ∀wral 3077 ∃wrex 3087 {crab 3413 Vcvv 3451 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 {ctp 4588 class class class wbr 5103 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 < clt 11343 ≤ cle 11344 ℕcn 12335 2c2 12397 3c3 12398 7c7 12402 ℕ₀cn0 12606 ℤcz 12693 cdc 12814 ..^cfzo 13788 ↑cexp 14204 ♯chash 14474 Σcsu 15853 ∥ cdvds 16422 ℙcprime 16846 reprcrepr 35237

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 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-reg 9586 ax-inf2 9642 ax-cc 10513 ax-ac2 10541 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279 ax-hgt749 35273 ax-ros335 35274 ax-ros336 35275

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 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 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-oadd 8480 df-omul 8481 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-r1 9768 df-rank 9769 df-scott 9929 df-dju 9982 df-card 10020 df-acn 10023 df-ac 10195 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-xnn0 12680 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-word 14659 df-concat 14716 df-s1 14743 df-s2 14999 df-s3 15000 df-shft 15220 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-limsup 15638 df-clim 15655 df-rlim 15656 df-sum 15854 df-prod 16073 df-ef 16233 df-e 16234 df-sin 16235 df-cos 16236 df-tan 16237 df-pi 16238 df-dvds 16423 df-gcd 16665 df-prm 16847 df-pc 17015 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-pmtr 19656 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-cmp 23705 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-ovol 25785 df-vol 25786 df-mbf 25940 df-itg1 25941 df-itg2 25942 df-ibl 25943 df-itg 25944 df-0p 25991 df-limc 26186 df-dv 26187 df-ulm 26704 df-log 26884 df-cxp 26885 df-atan 27195 df-cht 27424 df-vma 27425 df-chp 27426 df-dp2 33438 df-dp 33455 df-repr 35238 df-vts 35265

This theorem is used by: (None)

W3C validator