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Theorem tgoldbachOLD 40044
Description: Obsolete version of tgoldbach 40037 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 < 𝑛𝑛 ∈ GoldbachOddALTV )

Proof of Theorem tgoldbachOLD
Dummy variables 𝑚 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oddz 39887 . . . . 5 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
21zred 11314 . . . 4 (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
3 10reOLD 10956 . . . . 5 10 ∈ ℝ
4 2nn0 11156 . . . . . . 7 2 ∈ ℕ0
5 7nn 11037 . . . . . . 7 7 ∈ ℕ
64, 5decnncl 11350 . . . . . 6 27 ∈ ℕ
76nnnn0i 11147 . . . . 5 27 ∈ ℕ0
8 reexpcl 12694 . . . . 5 ((10 ∈ ℝ ∧ 27 ∈ ℕ0) → (10↑27) ∈ ℝ)
93, 7, 8mp2an 703 . . . 4 (10↑27) ∈ ℝ
10 lelttric 9995 . . . 4 ((𝑛 ∈ ℝ ∧ (10↑27) ∈ ℝ) → (𝑛 ≤ (10↑27) ∨ (10↑27) < 𝑛))
112, 9, 10sylancl 692 . . 3 (𝑛 ∈ Odd → (𝑛 ≤ (10↑27) ∨ (10↑27) < 𝑛))
12 tgoldbachltOLD 40042 . . . . 5 𝑚 ∈ ℕ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ))
13 breq2 4581 . . . . . . . . . . . . 13 (𝑜 = 𝑛 → (7 < 𝑜 ↔ 7 < 𝑛))
14 breq1 4580 . . . . . . . . . . . . 13 (𝑜 = 𝑛 → (𝑜 < 𝑚𝑛 < 𝑚))
1513, 14anbi12d 742 . . . . . . . . . . . 12 (𝑜 = 𝑛 → ((7 < 𝑜𝑜 < 𝑚) ↔ (7 < 𝑛𝑛 < 𝑚)))
16 eleq1 2675 . . . . . . . . . . . 12 (𝑜 = 𝑛 → (𝑜 ∈ GoldbachOddALTV ↔ 𝑛 ∈ GoldbachOddALTV ))
1715, 16imbi12d 332 . . . . . . . . . . 11 (𝑜 = 𝑛 → (((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) ↔ ((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV )))
1817rspcv 3277 . . . . . . . . . 10 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) → ((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV )))
199recni 9908 . . . . . . . . . . . . . . . . . . . . . . 23 (10↑27) ∈ ℂ
2019mulid2i 9899 . . . . . . . . . . . . . . . . . . . . . 22 (1 · (10↑27)) = (10↑27)
21 1re 9895 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ ℝ
22 8re 10952 . . . . . . . . . . . . . . . . . . . . . . . . 25 8 ∈ ℝ
2321, 22pm3.2i 469 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 ∈ ℝ ∧ 8 ∈ ℝ)
2423a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (1 ∈ ℝ ∧ 8 ∈ ℝ))
25 0le1 10400 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ≤ 1
26 1lt8 11068 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 < 8
2725, 26pm3.2i 469 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ≤ 1 ∧ 1 < 8)
2827a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (0 ≤ 1 ∧ 1 < 8))
29 3nn 11033 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 ∈ ℕ
3029decnncl2 11357 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 30 ∈ ℕ
3130nnnn0i 11147 . . . . . . . . . . . . . . . . . . . . . . . . . 26 30 ∈ ℕ0
32 reexpcl 12694 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((10 ∈ ℝ ∧ 30 ∈ ℕ0) → (10↑30) ∈ ℝ)
333, 31, 32mp2an 703 . . . . . . . . . . . . . . . . . . . . . . . . 25 (10↑30) ∈ ℝ
349, 33pm3.2i 469 . . . . . . . . . . . . . . . . . . . . . . . 24 ((10↑27) ∈ ℝ ∧ (10↑30) ∈ ℝ)
3534a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((10↑27) ∈ ℝ ∧ (10↑30) ∈ ℝ))
36 10nn0OLD 11164 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 10 ∈ ℕ0
3736, 7nn0expcli 12703 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (10↑27) ∈ ℕ0
3837nn0ge0i 11167 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ≤ (10↑27)
396nnzi 11234 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 ∈ ℤ
4030nnzi 11234 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 30 ∈ ℤ
413, 39, 403pm3.2i 1231 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (10 ∈ ℝ ∧ 27 ∈ ℤ ∧ 30 ∈ ℤ)
42 1lt10OLD 11085 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1 < 10
43 3nn0 11157 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 ∈ ℕ0
44 7nn0 11161 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7 ∈ ℕ0
45 0nn0 11154 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ ℕ0
46 7lt10OLD 11079 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7 < 10
47 2lt3 11042 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 < 3
484, 43, 44, 45, 46, 47decltcOLD 11365 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 < 30
4942, 48pm3.2i 469 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (1 < 10 ∧ 27 < 30)
50 ltexp2a 12729 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((10 ∈ ℝ ∧ 27 ∈ ℤ ∧ 30 ∈ ℤ) ∧ (1 < 10 ∧ 27 < 30)) → (10↑27) < (10↑30))
5141, 49, 50mp2an 703 . . . . . . . . . . . . . . . . . . . . . . . . 25 (10↑27) < (10↑30)
5238, 51pm3.2i 469 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ≤ (10↑27) ∧ (10↑27) < (10↑30))
5352a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (0 ≤ (10↑27) ∧ (10↑27) < (10↑30)))
54 ltmul12a 10728 . . . . . . . . . . . . . . . . . . . . . . 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)))
5524, 28, 35, 53, 54syl22anc 1318 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (1 · (10↑27)) < (8 · (10↑30)))
5620, 55syl5eqbrr 4613 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (10↑27) < (8 · (10↑30)))
579a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (10↑27) ∈ ℝ)
5822, 33remulcli 9910 . . . . . . . . . . . . . . . . . . . . . . 23 (8 · (10↑30)) ∈ ℝ
5958a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (8 · (10↑30)) ∈ ℝ)
60 nnre 10874 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
6160adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
62 lttr 9965 . . . . . . . . . . . . . . . . . . . . . 22 (((10↑27) ∈ ℝ ∧ (8 · (10↑30)) ∈ ℝ ∧ 𝑚 ∈ ℝ) → (((10↑27) < (8 · (10↑30)) ∧ (8 · (10↑30)) < 𝑚) → (10↑27) < 𝑚))
6357, 59, 61, 62syl3anc 1317 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (((10↑27) < (8 · (10↑30)) ∧ (8 · (10↑30)) < 𝑚) → (10↑27) < 𝑚))
6456, 63mpand 706 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((8 · (10↑30)) < 𝑚 → (10↑27) < 𝑚))
6564imp 443 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → (10↑27) < 𝑚)
662adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → 𝑛 ∈ ℝ)
6766, 57, 613jca 1234 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (𝑛 ∈ ℝ ∧ (10↑27) ∈ ℝ ∧ 𝑚 ∈ ℝ))
6867adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → (𝑛 ∈ ℝ ∧ (10↑27) ∈ ℝ ∧ 𝑚 ∈ ℝ))
69 lelttr 9979 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℝ ∧ (10↑27) ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝑛 ≤ (10↑27) ∧ (10↑27) < 𝑚) → 𝑛 < 𝑚))
7068, 69syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → ((𝑛 ≤ (10↑27) ∧ (10↑27) < 𝑚) → 𝑛 < 𝑚))
7165, 70mpan2d 705 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → (𝑛 ≤ (10↑27) → 𝑛 < 𝑚))
7271imp 443 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) → 𝑛 < 𝑚)
7372anim1i 589 . . . . . . . . . . . . . . . 16 (((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) ∧ 7 < 𝑛) → (𝑛 < 𝑚 ∧ 7 < 𝑛))
7473ancomd 465 . . . . . . . . . . . . . . 15 (((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) ∧ 7 < 𝑛) → (7 < 𝑛𝑛 < 𝑚))
75 pm2.27 40 . . . . . . . . . . . . . . 15 ((7 < 𝑛𝑛 < 𝑚) → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ) → 𝑛 ∈ GoldbachOddALTV ))
7674, 75syl 17 . . . . . . . . . . . . . 14 (((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) ∧ 7 < 𝑛) → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ) → 𝑛 ∈ GoldbachOddALTV ))
7776ex 448 . . . . . . . . . . . . 13 ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) → (7 < 𝑛 → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ) → 𝑛 ∈ GoldbachOddALTV )))
7877com23 83 . . . . . . . . . . . 12 ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))
7978exp41 635 . . . . . . . . . . 11 (𝑛 ∈ Odd → (𝑚 ∈ ℕ → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8079com25 96 . . . . . . . . . 10 (𝑛 ∈ Odd → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ) → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (𝑚 ∈ ℕ → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8118, 80syld 45 . . . . . . . . 9 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (𝑚 ∈ ℕ → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8281com15 98 . . . . . . . 8 (𝑚 ∈ ℕ → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8382com23 83 . . . . . . 7 (𝑚 ∈ ℕ → ((8 · (10↑30)) < 𝑚 → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8483imp32 447 . . . . . 6 ((𝑚 ∈ ℕ ∧ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ))) → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))
8584rexlimiva 3009 . . . . 5 (∃𝑚 ∈ ℕ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV )) → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))
8612, 85ax-mp 5 . . . 4 (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))
87 ax-tgoldbachgtOLD 40043 . . . . 5 𝑚 ∈ ℕ (𝑚 ≤ (10↑27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ))
88 breq2 4581 . . . . . . . . . . 11 (𝑜 = 𝑛 → (𝑚 < 𝑜𝑚 < 𝑛))
8988, 16imbi12d 332 . . . . . . . . . 10 (𝑜 = 𝑛 → ((𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ) ↔ (𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV )))
9089rspcv 3277 . . . . . . . . 9 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ) → (𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV )))
91 lelttr 9979 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℝ ∧ (10↑27) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑚 ≤ (10↑27) ∧ (10↑27) < 𝑛) → 𝑚 < 𝑛))
9261, 57, 66, 91syl3anc 1317 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((𝑚 ≤ (10↑27) ∧ (10↑27) < 𝑛) → 𝑚 < 𝑛))
9392expcomd 452 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((10↑27) < 𝑛 → (𝑚 ≤ (10↑27) → 𝑚 < 𝑛)))
9493ex 448 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ Odd → (𝑚 ∈ ℕ → ((10↑27) < 𝑛 → (𝑚 ≤ (10↑27) → 𝑚 < 𝑛))))
9594com23 83 . . . . . . . . . . . . . . . 16 (𝑛 ∈ Odd → ((10↑27) < 𝑛 → (𝑚 ∈ ℕ → (𝑚 ≤ (10↑27) → 𝑚 < 𝑛))))
9695imp43 618 . . . . . . . . . . . . . . 15 (((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) ∧ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27))) → 𝑚 < 𝑛)
97 pm2.27 40 . . . . . . . . . . . . . . 15 (𝑚 < 𝑛 → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → 𝑛 ∈ GoldbachOddALTV ))
9896, 97syl 17 . . . . . . . . . . . . . 14 (((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) ∧ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27))) → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → 𝑛 ∈ GoldbachOddALTV ))
9998a1dd 47 . . . . . . . . . . . . 13 (((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) ∧ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27))) → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))
10099ex 448 . . . . . . . . . . . 12 ((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))
101100com23 83 . . . . . . . . . . 11 ((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))
102101ex 448 . . . . . . . . . 10 (𝑛 ∈ Odd → ((10↑27) < 𝑛 → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))))
103102com23 83 . . . . . . . . 9 (𝑛 ∈ Odd → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → ((10↑27) < 𝑛 → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))))
10490, 103syld 45 . . . . . . . 8 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ) → ((10↑27) < 𝑛 → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))))
105104com14 93 . . . . . . 7 ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ) → ((10↑27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))))
106105impr 646 . . . . . 6 ((𝑚 ∈ ℕ ∧ (𝑚 ≤ (10↑27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ))) → ((10↑27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))
107106rexlimiva 3009 . . . . 5 (∃𝑚 ∈ ℕ (𝑚 ≤ (10↑27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV )) → ((10↑27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))
10887, 107ax-mp 5 . . . 4 ((10↑27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))
10986, 108jaoi 392 . . 3 ((𝑛 ≤ (10↑27) ∨ (10↑27) < 𝑛) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))
11011, 109mpcom 37 . 2 (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))
111110rgen 2905 1 𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOddALTV )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 381  wa 382  w3a 1030  wcel 1976  wral 2895  wrex 2896   class class class wbr 4577  (class class class)co 6527  cr 9791  0cc0 9792  1c1 9793   · cmul 9797   < clt 9930  cle 9931  cn 10867  2c2 10917  3c3 10918  7c7 10922  8c8 10923  10c10 10925  0cn0 11139  cz 11210  cdc 11325  cexp 12677   Odd codd 39881   GoldbachOddALTV cgboa 39974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870  ax-bgbltosilvaOLD 40038  ax-hgprmladderOLD 40040  ax-tgoldbachgtOLD 40043
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-inf 8209  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-10OLD 10934  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-rp 11665  df-ico 12008  df-fz 12153  df-fzo 12290  df-seq 12619  df-exp 12678  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-dvds 14768  df-prm 15170  df-iccp 39757  df-even 39882  df-odd 39883  df-gbe 39975  df-gboa 39977
This theorem is referenced by: (None)
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