Theorem tgoldbachOLD 41712

Description: Obsolete version of tgoldbach 41705 as of 9-Sep-2021. (Contributed by AV, 2-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
tgoldbachOLD	⊢ ∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )

Proof of Theorem tgoldbachOLD

Dummy variables 𝑚 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oddz 41544	. . . . 5 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
2	1	zred 11482	. . . 4 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
3		10reOLD 11109	. . . . 5 ⊢ 10 ∈ ℝ
4		2nn0 11309	. . . . . . 7 ⊢ 2 ∈ ℕ0
5		7nn 11190	. . . . . . 7 ⊢ 7 ∈ ℕ
6	4, 5	decnncl 11518	. . . . . 6 ⊢ ;27 ∈ ℕ
7	6	nnnn0i 11300	. . . . 5 ⊢ ;27 ∈ ℕ0
8		reexpcl 12877	. . . . 5 ⊢ ((10 ∈ ℝ ∧ ;27 ∈ ℕ0) → (10↑;27) ∈ ℝ)
9	3, 7, 8	mp2an 708	. . . 4 ⊢ (10↑;27) ∈ ℝ
10		lelttric 10144	. . . 4 ⊢ ((𝑛 ∈ ℝ ∧ (10↑;27) ∈ ℝ) → (𝑛 ≤ (10↑;27) ∨ (10↑;27) < 𝑛))
11	2, 9, 10	sylancl 694	. . 3 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ (10↑;27) ∨ (10↑;27) < 𝑛))
12		tgoldbachltOLD 41710	. . . . 5 ⊢ ∃𝑚 ∈ ℕ ((8 · (10↑;30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜 ∧ 𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ))
13		breq2 4657	. . . . . . . . . . . . 13 ⊢ (𝑜 = 𝑛 → (7 < 𝑜 ↔ 7 < 𝑛))
14		breq1 4656	. . . . . . . . . . . . 13 ⊢ (𝑜 = 𝑛 → (𝑜 < 𝑚 ↔ 𝑛 < 𝑚))
15	13, 14	anbi12d 747	. . . . . . . . . . . 12 ⊢ (𝑜 = 𝑛 → ((7 < 𝑜 ∧ 𝑜 < 𝑚) ↔ (7 < 𝑛 ∧ 𝑛 < 𝑚)))
16		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑜 = 𝑛 → (𝑜 ∈ GoldbachOdd ↔ 𝑛 ∈ GoldbachOdd ))
17	15, 16	imbi12d 334	. . . . . . . . . . 11 ⊢ (𝑜 = 𝑛 → (((7 < 𝑜 ∧ 𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ) ↔ ((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd )))
18	17	rspcv 3305	. . . . . . . . . 10 ⊢ (𝑛 ∈ Odd → (∀𝑜 ∈ Odd ((7 < 𝑜 ∧ 𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ) → ((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd )))
19	9	recni 10052	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (10↑;27) ∈ ℂ
20	19	mulid2i 10043	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 · (10↑;27)) = (10↑;27)
21		1re 10039	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ ℝ
22		8re 11105	. . . . . . . . . . . . . . . . . . . . . . . . 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 10551	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ≤ 1
26		1lt8 11221	. . . . . . . . . . . . . . . . . . . . . . . . 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 11186	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 3 ∈ ℕ
30	29	decnncl2 11525	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ;30 ∈ ℕ
31	30	nnnn0i 11300	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ;30 ∈ ℕ0
32		reexpcl 12877	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((10 ∈ ℝ ∧ ;30 ∈ ℕ0) → (10↑;30) ∈ ℝ)
33	3, 31, 32	mp2an 708	. . . . . . . . . . . . . . . . . . . . . . . . 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		10nn0OLD 11317	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 10 ∈ ℕ0
37	36, 7	nn0expcli 12886	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (10↑;27) ∈ ℕ0
38	37	nn0ge0i 11320	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ≤ (10↑;27)
39	6	nnzi 11401	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ;27 ∈ ℤ
40	30	nnzi 11401	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ;30 ∈ ℤ
41	3, 39, 40	3pm3.2i 1239	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (10 ∈ ℝ ∧ ;27 ∈ ℤ ∧ ;30 ∈ ℤ)
42		1lt10OLD 11238	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 1 < 10
43		3nn0 11310	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 3 ∈ ℕ0
44		7nn0 11314	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 7 ∈ ℕ0
45		0nn0 11307	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 ∈ ℕ0
46		7lt10OLD 11232	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 7 < 10
47		2lt3 11195	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 2 < 3
48	4, 43, 44, 45, 46, 47	decltcOLD 11533	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ;27 < ;30
49	42, 48	pm3.2i 471	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1 < 10 ∧ ;27 < ;30)
50		ltexp2a 12912	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((10 ∈ ℝ ∧ ;27 ∈ ℤ ∧ ;30 ∈ ℤ) ∧ (1 < 10 ∧ ;27 < ;30)) → (10↑;27) < (10↑;30))
51	41, 49, 50	mp2an 708	. . . . . . . . . . . . . . . . . . . . . . . . 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 10879	. . . . . . . . . . . . . . . . . . . . . . 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 1327	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (1 · (10↑;27)) < (8 · (10↑;30)))
56	20, 55	syl5eqbrr 4689	. . . . . . . . . . . . . . . . . . . . 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 10054	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (8 · (10↑;30)) ∈ ℝ
59	58	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (8 · (10↑;30)) ∈ ℝ)
60		nnre 11027	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
61	60	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
62		lttr 10114	. . . . . . . . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (((10↑;27) < (8 · (10↑;30)) ∧ (8 · (10↑;30)) < 𝑚) → (10↑;27) < 𝑚))
64	56, 63	mpand 711	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((8 · (10↑;30)) < 𝑚 → (10↑;27) < 𝑚))
65	64	imp 445	. . . . . . . . . . . . . . . . . . 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 1242	. . . . . . . . . . . . . . . . . . . . 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 10128	. . . . . . . . . . . . . . . . . . . 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 710	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑;30)) < 𝑚) → (𝑛 ≤ (10↑;27) → 𝑛 < 𝑚))
72	71	imp 445	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑;30)) < 𝑚) ∧ 𝑛 ≤ (10↑;27)) → 𝑛 < 𝑚)
73	72	anim1i 592	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑;30)) < 𝑚) ∧ 𝑛 ≤ (10↑;27)) ∧ 7 < 𝑛) → (𝑛 < 𝑚 ∧ 7 < 𝑛))
74	73	ancomd 467	. . . . . . . . . . . . . . 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 450	. . . . . . . . . . . . 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 638	. . . . . . . . . . 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 449	. . . . . 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 3028	. . . . 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		ax-tgoldbachgtOLD 41711	. . . . 5 ⊢ ∃𝑚 ∈ ℕ (𝑚 ≤ (10↑;27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜 → 𝑜 ∈ GoldbachOdd ))
88		breq2 4657	. . . . . . . . . . 11 ⊢ (𝑜 = 𝑛 → (𝑚 < 𝑜 ↔ 𝑚 < 𝑛))
89	88, 16	imbi12d 334	. . . . . . . . . 10 ⊢ (𝑜 = 𝑛 → ((𝑚 < 𝑜 → 𝑜 ∈ GoldbachOdd ) ↔ (𝑚 < 𝑛 → 𝑛 ∈ GoldbachOdd )))
90	89	rspcv 3305	. . . . . . . . 9 ⊢ (𝑛 ∈ Odd → (∀𝑜 ∈ Odd (𝑚 < 𝑜 → 𝑜 ∈ GoldbachOdd ) → (𝑚 < 𝑛 → 𝑛 ∈ GoldbachOdd )))
91		lelttr 10128	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℝ ∧ (10↑;27) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑚 ≤ (10↑;27) ∧ (10↑;27) < 𝑛) → 𝑚 < 𝑛))
92	61, 57, 66, 91	syl3anc 1326	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((𝑚 ≤ (10↑;27) ∧ (10↑;27) < 𝑛) → 𝑚 < 𝑛))
93	92	expcomd 454	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((10↑;27) < 𝑛 → (𝑚 ≤ (10↑;27) → 𝑚 < 𝑛)))
94	93	ex 450	. . . . . . . . . . . . . . . . 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 621	. . . . . . . . . . . . . . 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 450	. . . . . . . . . . . 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 450	. . . . . . . . . 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 649	. . . . . 6 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 ≤ (10↑;27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜 → 𝑜 ∈ GoldbachOdd ))) → ((10↑;27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ))))
107	106	rexlimiva 3028	. . . . 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 394	. . 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 2922	1 ⊢ ∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   class class class wbr 4653  (class class class)co 6650  ℝcr 9935  0cc0 9936  1c1 9937   · cmul 9941   < clt 10074   ≤ cle 10075  ℕcn 11020  2c2 11070  3c3 11071  7c7 11075  8c8 11076  10c10 11078  ℕ₀cn0 11292  ℤcz 11377  ;cdc 11493  ↑cexp 12860   Odd codd 41538   GoldbachOdd cgbo 41635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-bgbltosilvaOLD 41706  ax-hgprmladderOLD 41708  ax-tgoldbachgtOLD 41711

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-10OLD 11087  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386  df-iccp 41350  df-even 41539  df-odd 41540  df-gbe 41636  df-gbo 41638

This theorem is referenced by: (None)