tgoldbachgt - Mathbox for Thierry Arnoux

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Theorem tgoldbachgt 32044

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 12102	. . 3 ⊢ ;10 ∈ ℕ
2		2nn0 11902	. . . 4 ⊢ 2 ∈ ℕ₀
3		7nn0 11907	. . . 4 ⊢ 7 ∈ ℕ₀
4	2, 3	deccl 12101	. . 3 ⊢ ;27 ∈ ℕ₀
5		nnexpcl 13438	. . 3 ⊢ ((;10 ∈ ℕ ∧ ;27 ∈ ℕ₀) → (;10↑;27) ∈ ℕ)
6	1, 4, 5	mp2an 691	. 2 ⊢ (;10↑;27) ∈ ℕ
7	6	nnrei 11634	. . . 4 ⊢ (;10↑;27) ∈ ℝ
8	7	leidi 11163	. . 3 ⊢ (;10↑;27) ≤ (;10↑;27)
9		simpl 486	. . . . . 6 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 𝑛 ∈ 𝑂)
10		inss2 4156	. . . . . . . . . . . . . 14 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ
11		prmssnn 16010	. . . . . . . . . . . . . 14 ⊢ ℙ ⊆ ℕ
12	10, 11	sstri 3924	. . . . . . . . . . . . 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 2881	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑂 ↔ 𝑛 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧})
16		elrabi 3623	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} → 𝑛 ∈ ℤ)
17	15, 16	sylbi 220	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑂 → 𝑛 ∈ ℤ)
18	17	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑛 ∈ ℤ)
19		3nn0 11903	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℕ₀
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 3 ∈ ℕ₀)
21		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛))
22	13, 18, 20, 21	reprf 31993	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑐:(0..^3)⟶(𝑂 ∩ ℙ))
23		c0ex 10624	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
24	23	tpid1 4664	. . . . . . . . . . . . 13 ⊢ 0 ∈ {0, 1, 2}
25		fzo0to3tp 13118	. . . . . . . . . . . . 13 ⊢ (0..^3) = {0, 1, 2}
26	24, 25	eleqtrri 2889	. . . . . . . . . . . 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	ffvelrnd 6829	. . . . . . . . . 10 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ (𝑂 ∩ ℙ))
29	28	elin2d 4126	. . . . . . . . 9 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ ℙ)
30		1ex 10626	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
31	30	tpid2 4666	. . . . . . . . . . . . 13 ⊢ 1 ∈ {0, 1, 2}
32	31, 25	eleqtrri 2889	. . . . . . . . . . . 12 ⊢ 1 ∈ (0..^3)
33	32	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 1 ∈ (0..^3))
34	22, 33	ffvelrnd 6829	. . . . . . . . . 10 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ (𝑂 ∩ ℙ))
35	34	elin2d 4126	. . . . . . . . 9 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ ℙ)
36		2ex 11702	. . . . . . . . . . . . . 14 ⊢ 2 ∈ V
37	36	tpid3 4669	. . . . . . . . . . . . 13 ⊢ 2 ∈ {0, 1, 2}
38	37, 25	eleqtrri 2889	. . . . . . . . . . . 12 ⊢ 2 ∈ (0..^3)
39	38	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 2 ∈ (0..^3))
40	22, 39	ffvelrnd 6829	. . . . . . . . . 10 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ (𝑂 ∩ ℙ))
41	40	elin2d 4126	. . . . . . . . 9 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ ℙ)
42	28	elin1d 4125	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ 𝑂)
43	34	elin1d 4125	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ 𝑂)
44	40	elin1d 4125	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ 𝑂)
45	42, 43, 44	3jca 1125	. . . . . . . . . 10 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → ((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ (𝑐‘2) ∈ 𝑂))
46	25	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (0..^3) = {0, 1, 2})
47	46	sumeq1d 15050	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → Σ𝑖 ∈ (0..^3)(𝑐‘𝑖) = Σ𝑖 ∈ {0, 1, 2} (𝑐‘𝑖))
48	13, 18, 20, 21	reprsum 31994	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → Σ𝑖 ∈ (0..^3)(𝑐‘𝑖) = 𝑛)
49		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑐‘𝑖) = (𝑐‘0))
50		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝑐‘𝑖) = (𝑐‘1))
51		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑖 = 2 → (𝑐‘𝑖) = (𝑐‘2))
52	12, 28	sseldi 3913	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ ℕ)
53	52	nncnd 11641	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ ℂ)
54	12, 34	sseldi 3913	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ ℕ)
55	54	nncnd 11641	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ ℂ)
56	12, 40	sseldi 3913	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ ℕ)
57	56	nncnd 11641	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ ℂ)
58	53, 55, 57	3jca 1125	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → ((𝑐‘0) ∈ ℂ ∧ (𝑐‘1) ∈ ℂ ∧ (𝑐‘2) ∈ ℂ))
59	23	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ∈ V)
60	30	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 1 ∈ V)
61	36	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 2 ∈ V)
62	59, 60, 61	3jca 1125	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ V))
63		0ne1 11696	. . . . . . . . . . . . 13 ⊢ 0 ≠ 1
64	63	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ≠ 1)
65		0ne2 11832	. . . . . . . . . . . . 13 ⊢ 0 ≠ 2
66	65	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ≠ 2)
67		1ne2 11833	. . . . . . . . . . . . 13 ⊢ 1 ≠ 2
68	67	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 1 ≠ 2)
69	49, 50, 51, 58, 62, 64, 66, 68	sumtp 15096	. . . . . . . . . . 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	47, 48, 69	3eqtr3d 2841	. . . . . . . . . 10 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑛 = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)))
71	45, 70	jca 515	. . . . . . . . 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 2877	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝑐‘0) → (𝑝 ∈ 𝑂 ↔ (𝑐‘0) ∈ 𝑂))
73	72	3anbi1d 1437	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑐‘0) → ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ↔ ((𝑐‘0) ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂)))
74		oveq1 7142	. . . . . . . . . . . . 13 ⊢ (𝑝 = (𝑐‘0) → (𝑝 + 𝑞) = ((𝑐‘0) + 𝑞))
75	74	oveq1d 7150	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝑐‘0) → ((𝑝 + 𝑞) + 𝑟) = (((𝑐‘0) + 𝑞) + 𝑟))
76	75	eqeq2d 2809	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑐‘0) → (𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑛 = (((𝑐‘0) + 𝑞) + 𝑟)))
77	73, 76	anbi12d 633	. . . . . . . . . 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 2877	. . . . . . . . . . . 12 ⊢ (𝑞 = (𝑐‘1) → (𝑞 ∈ 𝑂 ↔ (𝑐‘1) ∈ 𝑂))
79	78	3anbi2d 1438	. . . . . . . . . . 11 ⊢ (𝑞 = (𝑐‘1) → (((𝑐‘0) ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ↔ ((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ 𝑟 ∈ 𝑂)))
80		oveq2 7143	. . . . . . . . . . . . 13 ⊢ (𝑞 = (𝑐‘1) → ((𝑐‘0) + 𝑞) = ((𝑐‘0) + (𝑐‘1)))
81	80	oveq1d 7150	. . . . . . . . . . . 12 ⊢ (𝑞 = (𝑐‘1) → (((𝑐‘0) + 𝑞) + 𝑟) = (((𝑐‘0) + (𝑐‘1)) + 𝑟))
82	81	eqeq2d 2809	. . . . . . . . . . 11 ⊢ (𝑞 = (𝑐‘1) → (𝑛 = (((𝑐‘0) + 𝑞) + 𝑟) ↔ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + 𝑟)))
83	79, 82	anbi12d 633	. . . . . . . . . 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 2877	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑐‘2) → (𝑟 ∈ 𝑂 ↔ (𝑐‘2) ∈ 𝑂))
85	84	3anbi3d 1439	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑐‘2) → (((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ↔ ((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ (𝑐‘2) ∈ 𝑂)))
86		oveq2 7143	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑐‘2) → (((𝑐‘0) + (𝑐‘1)) + 𝑟) = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)))
87	86	eqeq2d 2809	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑐‘2) → (𝑛 = (((𝑐‘0) + (𝑐‘1)) + 𝑟) ↔ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2))))
88	85, 87	anbi12d 633	. . . . . . . . . 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 3585	. . . . . . . . 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, 35, 41, 71, 89	syl31anc 1370	. . . . . . . 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 484	. . . . . . 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 11640	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) ∈ ℝ)
94	17	zred 12075	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑂 → 𝑛 ∈ ℝ)
95	94	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 𝑛 ∈ ℝ)
96		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) < 𝑛)
97	93, 95, 96	ltled 10777	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) ≤ 𝑛)
98	14, 9, 97	tgoldbachgtd 32043	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 0 < (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑛)))
99		ovex 7168	. . . . . . . . . . . . . . 15 ⊢ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ∈ V
100		hashneq0 13721	. . . . . . . . . . . . . . 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 2992	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ¬ ((𝑂 ∩ ℙ)(repr‘3)𝑛) = ∅)
104		neq0 4259	. . . . . . . . . . . 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 1542	. . . . . . . . . . 11 ⊢ ⊤
107	105, 106	jctil 523	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (⊤ ∧ ∃𝑐 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)))
108		19.42v 1954	. . . . . . . . . 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 1862	. . . . . . . . 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 3112	. . . . . . . 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 3248	. . . . . 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 2881	. . . . . . . 8 ⊢ (𝑛 ∈ 𝐺 ↔ 𝑛 ∈ {𝑧 ∈ 𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))})
117		eqeq1 2802	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑛 → (𝑧 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
118	117	anbi2d 631	. . . . . . . . . . . 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 3256	. . . . . . . . . . 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 3256	. . . . . . . . . 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 3256	. . . . . . . . 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 3629	. . . . . . . 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	syl5bb 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 481	. . . . . 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 587	. . . . 5 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 𝑛 ∈ 𝐺)
126	125	ex 416	. . . 4 ⊢ (𝑛 ∈ 𝑂 → ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺))
127	126	rgen 3116	. . 3 ⊢ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺)
128	8, 127	pm3.2i 474	. 2 ⊢ ((;10↑;27) ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺))
129		breq1 5033	. . . 4 ⊢ (𝑚 = (;10↑;27) → (𝑚 ≤ (;10↑;27) ↔ (;10↑;27) ≤ (;10↑;27)))
130		breq1 5033	. . . . . 6 ⊢ (𝑚 = (;10↑;27) → (𝑚 < 𝑛 ↔ (;10↑;27) < 𝑛))
131	130	imbi1d 345	. . . . 5 ⊢ (𝑚 = (;10↑;27) → ((𝑚 < 𝑛 → 𝑛 ∈ 𝐺) ↔ ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺)))
132	131	ralbidv 3162	. . . 4 ⊢ (𝑚 = (;10↑;27) → (∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺) ↔ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺)))
133	129, 132	anbi12d 633	. . 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 3571	. 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 691	1 ⊢ ∃𝑚 ∈ ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ⊤wtru 1539 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 {crab 3110 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {ctp 4529 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 ℕcn 11625 2c2 11680 3c3 11681 7c7 11685 ℕ₀cn0 11885 ℤcz 11969 cdc 12086 ..^cfzo 13028 ↑cexp 13425 ♯chash 13686 Σcsu 15034 ∥ cdvds 15599 ℙcprime 16005 reprcrepr 31989

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 ax-cc 9846 ax-ac2 9874 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 ax-hgt749 32025 ax-ros335 32026 ax-ros336 32027

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-symdif 4169 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-r1 9177 df-rank 9178 df-dju 9314 df-card 9352 df-acn 9355 df-ac 9527 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-prod 15252 df-ef 15413 df-e 15414 df-sin 15415 df-cos 15416 df-tan 15417 df-pi 15418 df-dvds 15600 df-gcd 15834 df-prm 16006 df-pc 16164 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-pmtr 18562 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-ovol 24068 df-vol 24069 df-mbf 24223 df-itg1 24224 df-itg2 24225 df-ibl 24226 df-itg 24227 df-0p 24274 df-limc 24469 df-dv 24470 df-ulm 24972 df-log 25148 df-cxp 25149 df-atan 25453 df-cht 25682 df-vma 25683 df-chp 25684 df-dp2 30574 df-dp 30591 df-repr 31990 df-vts 32017

This theorem is referenced by: (None)

W3C validator