tgoldbachgt - Mathbox for Thierry Arnoux

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Theorem tgoldbachgt 32543

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 12382	. . 3 ⊢ ;10 ∈ ℕ
2		2nn0 12180	. . . 4 ⊢ 2 ∈ ℕ₀
3		7nn0 12185	. . . 4 ⊢ 7 ∈ ℕ₀
4	2, 3	deccl 12381	. . 3 ⊢ ;27 ∈ ℕ₀
5		nnexpcl 13723	. . 3 ⊢ ((;10 ∈ ℕ ∧ ;27 ∈ ℕ₀) → (;10↑;27) ∈ ℕ)
6	1, 4, 5	mp2an 688	. 2 ⊢ (;10↑;27) ∈ ℕ
7	6	nnrei 11912	. . . 4 ⊢ (;10↑;27) ∈ ℝ
8	7	leidi 11439	. . 3 ⊢ (;10↑;27) ≤ (;10↑;27)
9		simpl 482	. . . . . 6 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 𝑛 ∈ 𝑂)
10		inss2 4160	. . . . . . . . . . . . . 14 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ
11		prmssnn 16309	. . . . . . . . . . . . . 14 ⊢ ℙ ⊆ ℕ
12	10, 11	sstri 3926	. . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑂 ↔ 𝑛 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧})
16		elrabi 3611	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} → 𝑛 ∈ ℤ)
17	15, 16	sylbi 216	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑂 → 𝑛 ∈ ℤ)
18	17	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑛 ∈ ℤ)
19		3nn0 12181	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℕ₀
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 3 ∈ ℕ₀)
21		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛))
22	13, 18, 20, 21	reprf 32492	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑐:(0..^3)⟶(𝑂 ∩ ℙ))
23		c0ex 10900	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
24	23	tpid1 4701	. . . . . . . . . . . . 13 ⊢ 0 ∈ {0, 1, 2}
25		fzo0to3tp 13401	. . . . . . . . . . . . 13 ⊢ (0..^3) = {0, 1, 2}
26	24, 25	eleqtrri 2838	. . . . . . . . . . . 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 6944	. . . . . . . . . 10 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ (𝑂 ∩ ℙ))
29	28	elin2d 4129	. . . . . . . . 9 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ ℙ)
30		1ex 10902	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
31	30	tpid2 4703	. . . . . . . . . . . . 13 ⊢ 1 ∈ {0, 1, 2}
32	31, 25	eleqtrri 2838	. . . . . . . . . . . 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 6944	. . . . . . . . . 10 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ (𝑂 ∩ ℙ))
35	34	elin2d 4129	. . . . . . . . 9 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ ℙ)
36		2ex 11980	. . . . . . . . . . . . . 14 ⊢ 2 ∈ V
37	36	tpid3 4706	. . . . . . . . . . . . 13 ⊢ 2 ∈ {0, 1, 2}
38	37, 25	eleqtrri 2838	. . . . . . . . . . . 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 6944	. . . . . . . . . 10 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ (𝑂 ∩ ℙ))
41	40	elin2d 4129	. . . . . . . . 9 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ ℙ)
42	28	elin1d 4128	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ 𝑂)
43	34	elin1d 4128	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ 𝑂)
44	40	elin1d 4128	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ 𝑂)
45	42, 43, 44	3jca 1126	. . . . . . . . . 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 15341	. . . . . . . . . . 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 32493	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → Σ𝑖 ∈ (0..^3)(𝑐‘𝑖) = 𝑛)
49		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑐‘𝑖) = (𝑐‘0))
50		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝑐‘𝑖) = (𝑐‘1))
51		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑖 = 2 → (𝑐‘𝑖) = (𝑐‘2))
52	12, 28	sselid 3915	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ ℕ)
53	52	nncnd 11919	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ ℂ)
54	12, 34	sselid 3915	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ ℕ)
55	54	nncnd 11919	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ ℂ)
56	12, 40	sselid 3915	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ ℕ)
57	56	nncnd 11919	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ ℂ)
58	53, 55, 57	3jca 1126	. . . . . . . . . . . 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 1126	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ V))
63		0ne1 11974	. . . . . . . . . . . . 13 ⊢ 0 ≠ 1
64	63	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ≠ 1)
65		0ne2 12110	. . . . . . . . . . . . 13 ⊢ 0 ≠ 2
66	65	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ≠ 2)
67		1ne2 12111	. . . . . . . . . . . . 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 15389	. . . . . . . . . . 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 2786	. . . . . . . . . 10 ⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑛 = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)))
71	45, 70	jca 511	. . . . . . . . 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 2826	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝑐‘0) → (𝑝 ∈ 𝑂 ↔ (𝑐‘0) ∈ 𝑂))
73	72	3anbi1d 1438	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑐‘0) → ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ↔ ((𝑐‘0) ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂)))
74		oveq1 7262	. . . . . . . . . . . . 13 ⊢ (𝑝 = (𝑐‘0) → (𝑝 + 𝑞) = ((𝑐‘0) + 𝑞))
75	74	oveq1d 7270	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝑐‘0) → ((𝑝 + 𝑞) + 𝑟) = (((𝑐‘0) + 𝑞) + 𝑟))
76	75	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑐‘0) → (𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑛 = (((𝑐‘0) + 𝑞) + 𝑟)))
77	73, 76	anbi12d 630	. . . . . . . . . 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 2826	. . . . . . . . . . . 12 ⊢ (𝑞 = (𝑐‘1) → (𝑞 ∈ 𝑂 ↔ (𝑐‘1) ∈ 𝑂))
79	78	3anbi2d 1439	. . . . . . . . . . 11 ⊢ (𝑞 = (𝑐‘1) → (((𝑐‘0) ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ↔ ((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ 𝑟 ∈ 𝑂)))
80		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑞 = (𝑐‘1) → ((𝑐‘0) + 𝑞) = ((𝑐‘0) + (𝑐‘1)))
81	80	oveq1d 7270	. . . . . . . . . . . 12 ⊢ (𝑞 = (𝑐‘1) → (((𝑐‘0) + 𝑞) + 𝑟) = (((𝑐‘0) + (𝑐‘1)) + 𝑟))
82	81	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑞 = (𝑐‘1) → (𝑛 = (((𝑐‘0) + 𝑞) + 𝑟) ↔ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + 𝑟)))
83	79, 82	anbi12d 630	. . . . . . . . . 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 2826	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑐‘2) → (𝑟 ∈ 𝑂 ↔ (𝑐‘2) ∈ 𝑂))
85	84	3anbi3d 1440	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑐‘2) → (((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ↔ ((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ (𝑐‘2) ∈ 𝑂)))
86		oveq2 7263	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑐‘2) → (((𝑐‘0) + (𝑐‘1)) + 𝑟) = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)))
87	86	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑐‘2) → (𝑛 = (((𝑐‘0) + (𝑐‘1)) + 𝑟) ↔ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2))))
88	85, 87	anbi12d 630	. . . . . . . . . 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 3566	. . . . . . . . 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 1371	. . . . . . . 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 480	. . . . . . 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 11918	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) ∈ ℝ)
94	17	zred 12355	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑂 → 𝑛 ∈ ℝ)
95	94	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 𝑛 ∈ ℝ)
96		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) < 𝑛)
97	93, 95, 96	ltled 11053	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) ≤ 𝑛)
98	14, 9, 97	tgoldbachgtd 32542	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 0 < (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑛)))
99		ovex 7288	. . . . . . . . . . . . . . 15 ⊢ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ∈ V
100		hashneq0 14007	. . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ((𝑂 ∩ ℙ)(repr‘3)𝑛) ≠ ∅)
103	102	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ¬ ((𝑂 ∩ ℙ)(repr‘3)𝑛) = ∅)
104		neq0 4276	. . . . . . . . . . . 12 ⊢ (¬ ((𝑂 ∩ ℙ)(repr‘3)𝑛) = ∅ ↔ ∃𝑐 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛))
105	103, 104	sylib 217	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ∃𝑐 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛))
106		tru 1543	. . . . . . . . . . 11 ⊢ ⊤
107	105, 106	jctil 519	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (⊤ ∧ ∃𝑐 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)))
108		19.42v 1958	. . . . . . . . . 10 ⊢ (∃𝑐(⊤ ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) ↔ (⊤ ∧ ∃𝑐 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)))
109	107, 108	sylibr 233	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ∃𝑐(⊤ ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)))
110		exancom 1865	. . . . . . . . 9 ⊢ (∃𝑐(⊤ ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) ↔ ∃𝑐(𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ∧ ⊤))
111	109, 110	sylib 217	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ∃𝑐(𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ∧ ⊤))
112		df-rex 3069	. . . . . . . 8 ⊢ (∃𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)⊤ ↔ ∃𝑐(𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ∧ ⊤))
113	111, 112	sylibr 233	. . . . . . 7 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ∃𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)⊤)
114	91, 113	r19.29a 3217	. . . . . 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 2830	. . . . . . . 8 ⊢ (𝑛 ∈ 𝐺 ↔ 𝑛 ∈ {𝑧 ∈ 𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))})
117		eqeq1 2742	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑛 → (𝑧 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
118	117	anbi2d 628	. . . . . . . . . . . 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 3225	. . . . . . . . . . 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 3225	. . . . . . . . . 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 3225	. . . . . . . . 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 3618	. . . . . . . 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 282	. . . . . . 7 ⊢ (𝑛 ∈ 𝑂 → (𝑛 ∈ 𝐺 ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
124	123	biimpar 477	. . . . . 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 583	. . . . 5 ⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 𝑛 ∈ 𝐺)
126	125	ex 412	. . . 4 ⊢ (𝑛 ∈ 𝑂 → ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺))
127	126	rgen 3073	. . 3 ⊢ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺)
128	8, 127	pm3.2i 470	. 2 ⊢ ((;10↑;27) ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺))
129		breq1 5073	. . . 4 ⊢ (𝑚 = (;10↑;27) → (𝑚 ≤ (;10↑;27) ↔ (;10↑;27) ≤ (;10↑;27)))
130		breq1 5073	. . . . . 6 ⊢ (𝑚 = (;10↑;27) → (𝑚 < 𝑛 ↔ (;10↑;27) < 𝑛))
131	130	imbi1d 341	. . . . 5 ⊢ (𝑚 = (;10↑;27) → ((𝑚 < 𝑛 → 𝑛 ∈ 𝐺) ↔ ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺)))
132	131	ralbidv 3120	. . . 4 ⊢ (𝑚 = (;10↑;27) → (∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺) ↔ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺)))
133	129, 132	anbi12d 630	. . 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 3552	. 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 688	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 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ⊤wtru 1540 ∃wex 1783 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {crab 3067 Vcvv 3422 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 {ctp 4562 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 < clt 10940 ≤ cle 10941 ℕcn 11903 2c2 11958 3c3 11959 7c7 11963 ℕ₀cn0 12163 ℤcz 12249 cdc 12366 ..^cfzo 13311 ↑cexp 13710 ♯chash 13972 Σcsu 15325 ∥ cdvds 15891 ℙcprime 16304 reprcrepr 32488

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-reg 9281 ax-inf2 9329 ax-cc 10122 ax-ac2 10150 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882 ax-hgt749 32524 ax-ros335 32525 ax-ros336 32526

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-symdif 4173 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-omul 8272 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-r1 9453 df-rank 9454 df-dju 9590 df-card 9628 df-acn 9631 df-ac 9803 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-word 14146 df-concat 14202 df-s1 14229 df-s2 14489 df-s3 14490 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-prod 15544 df-ef 15705 df-e 15706 df-sin 15707 df-cos 15708 df-tan 15709 df-pi 15710 df-dvds 15892 df-gcd 16130 df-prm 16305 df-pc 16466 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-pmtr 18965 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-ovol 24533 df-vol 24534 df-mbf 24688 df-itg1 24689 df-itg2 24690 df-ibl 24691 df-itg 24692 df-0p 24739 df-limc 24935 df-dv 24936 df-ulm 25441 df-log 25617 df-cxp 25618 df-atan 25922 df-cht 26151 df-vma 26152 df-chp 26153 df-dp2 31048 df-dp 31065 df-repr 32489 df-vts 32516

This theorem is referenced by: (None)

W3C validator