tgoldbach - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > tgoldbach		Structured version Visualization version GIF version

Theorem tgoldbach 46471

Description: The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 46470 and ax-tgoldbachgt 46465. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.)

**Assertion**
Ref	Expression
tgoldbach	⊢ ∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )

Proof of Theorem tgoldbach Dummy variables 𝑚 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oddz 46285	. . . . 5 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
2	1	zred 12662	. . . 4 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
3		10re 12692	. . . . 5 ⊢ ;10 ∈ ℝ
4		2nn0 12485	. . . . . . 7 ⊢ 2 ∈ ℕ₀
5		7nn 12300	. . . . . . 7 ⊢ 7 ∈ ℕ
6	4, 5	decnncl 12693	. . . . . 6 ⊢ ;27 ∈ ℕ
7	6	nnnn0i 12476	. . . . 5 ⊢ ;27 ∈ ℕ₀
8		reexpcl 14040	. . . . 5 ⊢ ((;10 ∈ ℝ ∧ ;27 ∈ ℕ₀) → (;10↑;27) ∈ ℝ)
9	3, 7, 8	mp2an 690	. . . 4 ⊢ (;10↑;27) ∈ ℝ
10		lelttric 11317	. . . 4 ⊢ ((𝑛 ∈ ℝ ∧ (;10↑;27) ∈ ℝ) → (𝑛 ≤ (;10↑;27) ∨ (;10↑;27) < 𝑛))
11	2, 9, 10	sylancl 586	. . 3 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ (;10↑;27) ∨ (;10↑;27) < 𝑛))
12		tgoldbachlt 46470	. . . . 5 ⊢ ∃𝑚 ∈ ℕ ((8 · (;10↑;30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜 ∧ 𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ))
13		breq2 5151	. . . . . . . . . . . . 13 ⊢ (𝑜 = 𝑛 → (7 < 𝑜 ↔ 7 < 𝑛))
14		breq1 5150	. . . . . . . . . . . . 13 ⊢ (𝑜 = 𝑛 → (𝑜 < 𝑚 ↔ 𝑛 < 𝑚))
15	13, 14	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑜 = 𝑛 → ((7 < 𝑜 ∧ 𝑜 < 𝑚) ↔ (7 < 𝑛 ∧ 𝑛 < 𝑚)))
16		eleq1w 2816	. . . . . . . . . . . 12 ⊢ (𝑜 = 𝑛 → (𝑜 ∈ GoldbachOdd ↔ 𝑛 ∈ GoldbachOdd ))
17	15, 16	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑜 = 𝑛 → (((7 < 𝑜 ∧ 𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ) ↔ ((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd )))
18	17	rspcv 3608	. . . . . . . . . 10 ⊢ (𝑛 ∈ Odd → (∀𝑜 ∈ Odd ((7 < 𝑜 ∧ 𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ) → ((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd )))
19	9	recni 11224	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (;10↑;27) ∈ ℂ
20	19	mullidi 11215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 · (;10↑;27)) = (;10↑;27)
21		1re 11210	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ ℝ
22		8re 12304	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 8 ∈ ℝ
23	21, 22	pm3.2i 471	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ∈ ℝ ∧ 8 ∈ ℝ)
24	23	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (1 ∈ ℝ ∧ 8 ∈ ℝ))
25		0le1 11733	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ≤ 1
26		1lt8 12406	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 < 8
27	25, 26	pm3.2i 471	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 ≤ 1 ∧ 1 < 8)
28	27	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (0 ≤ 1 ∧ 1 < 8))
29		3nn 12287	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 3 ∈ ℕ
30	29	decnncl2 12697	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ;30 ∈ ℕ
31	30	nnnn0i 12476	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ;30 ∈ ℕ₀
32		reexpcl 14040	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((;10 ∈ ℝ ∧ ;30 ∈ ℕ₀) → (;10↑;30) ∈ ℝ)
33	3, 31, 32	mp2an 690	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (;10↑;30) ∈ ℝ
34	9, 33	pm3.2i 471	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((;10↑;27) ∈ ℝ ∧ (;10↑;30) ∈ ℝ)
35	34	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((;10↑;27) ∈ ℝ ∧ (;10↑;30) ∈ ℝ))
36		10nn0 12691	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ;10 ∈ ℕ₀
37	36, 7	nn0expcli 14050	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (;10↑;27) ∈ ℕ₀
38	37	nn0ge0i 12495	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ≤ (;10↑;27)
39	6	nnzi 12582	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ;27 ∈ ℤ
40	30	nnzi 12582	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ;30 ∈ ℤ
41	3, 39, 40	3pm3.2i 1339	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (;10 ∈ ℝ ∧ ;27 ∈ ℤ ∧ ;30 ∈ ℤ)
42		1lt10 12812	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 1 < ;10
43		3nn0 12486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 3 ∈ ℕ₀
44		7nn0 12490	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 7 ∈ ℕ₀
45		0nn0 12483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 ∈ ℕ₀
46		7lt10 12806	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 7 < ;10
47		2lt3 12380	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 2 < 3
48	4, 43, 44, 45, 46, 47	decltc 12702	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ;27 < ;30
49	42, 48	pm3.2i 471	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1 < ;10 ∧ ;27 < ;30)
50		ltexp2a 14127	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((;10 ∈ ℝ ∧ ;27 ∈ ℤ ∧ ;30 ∈ ℤ) ∧ (1 < ;10 ∧ ;27 < ;30)) → (;10↑;27) < (;10↑;30))
51	41, 49, 50	mp2an 690	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (;10↑;27) < (;10↑;30)
52	38, 51	pm3.2i 471	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 ≤ (;10↑;27) ∧ (;10↑;27) < (;10↑;30))
53	52	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (0 ≤ (;10↑;27) ∧ (;10↑;27) < (;10↑;30)))
54		ltmul12a 12066	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((1 ∈ ℝ ∧ 8 ∈ ℝ) ∧ (0 ≤ 1 ∧ 1 < 8)) ∧ (((;10↑;27) ∈ ℝ ∧ (;10↑;30) ∈ ℝ) ∧ (0 ≤ (;10↑;27) ∧ (;10↑;27) < (;10↑;30)))) → (1 · (;10↑;27)) < (8 · (;10↑;30)))
55	24, 28, 35, 53, 54	syl22anc 837	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (1 · (;10↑;27)) < (8 · (;10↑;30)))
56	20, 55	eqbrtrrid 5183	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (;10↑;27) < (8 · (;10↑;30)))
57	9	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (;10↑;27) ∈ ℝ)
58	22, 33	remulcli 11226	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (8 · (;10↑;30)) ∈ ℝ
59	58	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (8 · (;10↑;30)) ∈ ℝ)
60		nnre 12215	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
61	60	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
62		lttr 11286	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((;10↑;27) ∈ ℝ ∧ (8 · (;10↑;30)) ∈ ℝ ∧ 𝑚 ∈ ℝ) → (((;10↑;27) < (8 · (;10↑;30)) ∧ (8 · (;10↑;30)) < 𝑚) → (;10↑;27) < 𝑚))
63	57, 59, 61, 62	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (((;10↑;27) < (8 · (;10↑;30)) ∧ (8 · (;10↑;30)) < 𝑚) → (;10↑;27) < 𝑚))
64	56, 63	mpand 693	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((8 · (;10↑;30)) < 𝑚 → (;10↑;27) < 𝑚))
65	64	imp 407	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (;10↑;30)) < 𝑚) → (;10↑;27) < 𝑚)
66	2	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → 𝑛 ∈ ℝ)
67	66, 57, 61	3jca 1128	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (𝑛 ∈ ℝ ∧ (;10↑;27) ∈ ℝ ∧ 𝑚 ∈ ℝ))
68	67	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (;10↑;30)) < 𝑚) → (𝑛 ∈ ℝ ∧ (;10↑;27) ∈ ℝ ∧ 𝑚 ∈ ℝ))
69		lelttr 11300	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℝ ∧ (;10↑;27) ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝑛 ≤ (;10↑;27) ∧ (;10↑;27) < 𝑚) → 𝑛 < 𝑚))
70	68, 69	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (;10↑;30)) < 𝑚) → ((𝑛 ≤ (;10↑;27) ∧ (;10↑;27) < 𝑚) → 𝑛 < 𝑚))
71	65, 70	mpan2d 692	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (;10↑;30)) < 𝑚) → (𝑛 ≤ (;10↑;27) → 𝑛 < 𝑚))
72	71	imp 407	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (;10↑;30)) < 𝑚) ∧ 𝑛 ≤ (;10↑;27)) → 𝑛 < 𝑚)
73	72	anim1i 615	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (;10↑;30)) < 𝑚) ∧ 𝑛 ≤ (;10↑;27)) ∧ 7 < 𝑛) → (𝑛 < 𝑚 ∧ 7 < 𝑛))
74	73	ancomd 462	. . . . . . . . . . . . . . 15 ⊢ (((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (;10↑;30)) < 𝑚) ∧ 𝑛 ≤ (;10↑;27)) ∧ 7 < 𝑛) → (7 < 𝑛 ∧ 𝑛 < 𝑚))
75		pm2.27 42	. . . . . . . . . . . . . . 15 ⊢ ((7 < 𝑛 ∧ 𝑛 < 𝑚) → (((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) → 𝑛 ∈ GoldbachOdd ))
76	74, 75	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (;10↑;30)) < 𝑚) ∧ 𝑛 ≤ (;10↑;27)) ∧ 7 < 𝑛) → (((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) → 𝑛 ∈ GoldbachOdd ))
77	76	ex 413	. . . . . . . . . . . . 13 ⊢ ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (;10↑;30)) < 𝑚) ∧ 𝑛 ≤ (;10↑;27)) → (7 < 𝑛 → (((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) → 𝑛 ∈ GoldbachOdd )))
78	77	com23 86	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (;10↑;30)) < 𝑚) ∧ 𝑛 ≤ (;10↑;27)) → (((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )))
79	78	exp41 435	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Odd → (𝑚 ∈ ℕ → ((8 · (;10↑;30)) < 𝑚 → (𝑛 ≤ (;10↑;27) → (((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ))))))
80	79	com25 99	. . . . . . . . . 10 ⊢ (𝑛 ∈ Odd → (((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) → ((8 · (;10↑;30)) < 𝑚 → (𝑛 ≤ (;10↑;27) → (𝑚 ∈ ℕ → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ))))))
81	18, 80	syld 47	. . . . . . . . 9 ⊢ (𝑛 ∈ Odd → (∀𝑜 ∈ Odd ((7 < 𝑜 ∧ 𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ) → ((8 · (;10↑;30)) < 𝑚 → (𝑛 ≤ (;10↑;27) → (𝑚 ∈ ℕ → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ))))))
82	81	com15 101	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → (∀𝑜 ∈ Odd ((7 < 𝑜 ∧ 𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ) → ((8 · (;10↑;30)) < 𝑚 → (𝑛 ≤ (;10↑;27) → (𝑛 ∈ Odd → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ))))))
83	82	com23 86	. . . . . . 7 ⊢ (𝑚 ∈ ℕ → ((8 · (;10↑;30)) < 𝑚 → (∀𝑜 ∈ Odd ((7 < 𝑜 ∧ 𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ) → (𝑛 ≤ (;10↑;27) → (𝑛 ∈ Odd → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ))))))
84	83	imp32 419	. . . . . 6 ⊢ ((𝑚 ∈ ℕ ∧ ((8 · (;10↑;30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜 ∧ 𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ))) → (𝑛 ≤ (;10↑;27) → (𝑛 ∈ Odd → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ))))
85	84	rexlimiva 3147	. . . . 5 ⊢ (∃𝑚 ∈ ℕ ((8 · (;10↑;30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜 ∧ 𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd )) → (𝑛 ≤ (;10↑;27) → (𝑛 ∈ Odd → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ))))
86	12, 85	ax-mp 5	. . . 4 ⊢ (𝑛 ≤ (;10↑;27) → (𝑛 ∈ Odd → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )))
87		tgoldbachgtALTV 46466	. . . . 5 ⊢ ∃𝑚 ∈ ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜 → 𝑜 ∈ GoldbachOdd ))
88		breq2 5151	. . . . . . . . . . 11 ⊢ (𝑜 = 𝑛 → (𝑚 < 𝑜 ↔ 𝑚 < 𝑛))
89	88, 16	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑜 = 𝑛 → ((𝑚 < 𝑜 → 𝑜 ∈ GoldbachOdd ) ↔ (𝑚 < 𝑛 → 𝑛 ∈ GoldbachOdd )))
90	89	rspcv 3608	. . . . . . . . 9 ⊢ (𝑛 ∈ Odd → (∀𝑜 ∈ Odd (𝑚 < 𝑜 → 𝑜 ∈ GoldbachOdd ) → (𝑚 < 𝑛 → 𝑛 ∈ GoldbachOdd )))
91		lelttr 11300	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℝ ∧ (;10↑;27) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑚 ≤ (;10↑;27) ∧ (;10↑;27) < 𝑛) → 𝑚 < 𝑛))
92	61, 57, 66, 91	syl3anc 1371	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((𝑚 ≤ (;10↑;27) ∧ (;10↑;27) < 𝑛) → 𝑚 < 𝑛))
93	92	expcomd 417	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((;10↑;27) < 𝑛 → (𝑚 ≤ (;10↑;27) → 𝑚 < 𝑛)))
94	93	ex 413	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ Odd → (𝑚 ∈ ℕ → ((;10↑;27) < 𝑛 → (𝑚 ≤ (;10↑;27) → 𝑚 < 𝑛))))
95	94	com23 86	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ Odd → ((;10↑;27) < 𝑛 → (𝑚 ∈ ℕ → (𝑚 ≤ (;10↑;27) → 𝑚 < 𝑛))))
96	95	imp43 428	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ Odd ∧ (;10↑;27) < 𝑛) ∧ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (;10↑;27))) → 𝑚 < 𝑛)
97		pm2.27 42	. . . . . . . . . . . . . . 15 ⊢ (𝑚 < 𝑛 → ((𝑚 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → 𝑛 ∈ GoldbachOdd ))
98	96, 97	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ Odd ∧ (;10↑;27) < 𝑛) ∧ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (;10↑;27))) → ((𝑚 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → 𝑛 ∈ GoldbachOdd ))
99	98	a1dd 50	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ Odd ∧ (;10↑;27) < 𝑛) ∧ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (;10↑;27))) → ((𝑚 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )))
100	99	ex 413	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ Odd ∧ (;10↑;27) < 𝑛) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (;10↑;27)) → ((𝑚 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ))))
101	100	com23 86	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ Odd ∧ (;10↑;27) < 𝑛) → ((𝑚 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (;10↑;27)) → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ))))
102	101	ex 413	. . . . . . . . . 10 ⊢ (𝑛 ∈ Odd → ((;10↑;27) < 𝑛 → ((𝑚 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (;10↑;27)) → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )))))
103	102	com23 86	. . . . . . . . 9 ⊢ (𝑛 ∈ Odd → ((𝑚 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → ((;10↑;27) < 𝑛 → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (;10↑;27)) → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )))))
104	90, 103	syld 47	. . . . . . . 8 ⊢ (𝑛 ∈ Odd → (∀𝑜 ∈ Odd (𝑚 < 𝑜 → 𝑜 ∈ GoldbachOdd ) → ((;10↑;27) < 𝑛 → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (;10↑;27)) → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )))))
105	104	com14 96	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (;10↑;27)) → (∀𝑜 ∈ Odd (𝑚 < 𝑜 → 𝑜 ∈ GoldbachOdd ) → ((;10↑;27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )))))
106	105	impr 455	. . . . . 6 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 ≤ (;10↑;27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜 → 𝑜 ∈ GoldbachOdd ))) → ((;10↑;27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ))))
107	106	rexlimiva 3147	. . . . 5 ⊢ (∃𝑚 ∈ ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜 → 𝑜 ∈ GoldbachOdd )) → ((;10↑;27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ))))
108	87, 107	ax-mp 5	. . . 4 ⊢ ((;10↑;27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )))
109	86, 108	jaoi 855	. . 3 ⊢ ((𝑛 ≤ (;10↑;27) ∨ (;10↑;27) < 𝑛) → (𝑛 ∈ Odd → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )))
110	11, 109	mpcom 38	. 2 ⊢ (𝑛 ∈ Odd → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ))
111	110	rgen 3063	1 ⊢ ∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 class class class wbr 5147 (class class class)co 7405 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 < clt 11244 ≤ cle 11245 ℕcn 12208 2c2 12263 3c3 12264 7c7 12268 8c8 12269 ℕ₀cn0 12468 ℤcz 12554 cdc 12673 ↑cexp 14023 Odd codd 46279 GoldbachOdd cgbo 46401

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-bgbltosilva 46464 ax-tgoldbachgt 46465 ax-hgprmladder 46468

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-rp 12971 df-ico 13326 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-prm 16605 df-iccp 46068 df-even 46280 df-odd 46281 df-gbe 46402 df-gbo 46404

This theorem is referenced by: (None)

W3C validator