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Theorem tgoldbach 44330
Description: The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 44329 and ax-tgoldbachgt 44324. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.)
Assertion
Ref Expression
tgoldbach 𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOdd )

Proof of Theorem tgoldbach
Dummy variables 𝑚 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oddz 44144 . . . . 5 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
21zred 12075 . . . 4 (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
3 10re 12105 . . . . 5 10 ∈ ℝ
4 2nn0 11902 . . . . . . 7 2 ∈ ℕ0
5 7nn 11717 . . . . . . 7 7 ∈ ℕ
64, 5decnncl 12106 . . . . . 6 27 ∈ ℕ
76nnnn0i 11893 . . . . 5 27 ∈ ℕ0
8 reexpcl 13442 . . . . 5 ((10 ∈ ℝ ∧ 27 ∈ ℕ0) → (10↑27) ∈ ℝ)
93, 7, 8mp2an 691 . . . 4 (10↑27) ∈ ℝ
10 lelttric 10736 . . . 4 ((𝑛 ∈ ℝ ∧ (10↑27) ∈ ℝ) → (𝑛 ≤ (10↑27) ∨ (10↑27) < 𝑛))
112, 9, 10sylancl 589 . . 3 (𝑛 ∈ Odd → (𝑛 ≤ (10↑27) ∨ (10↑27) < 𝑛))
12 tgoldbachlt 44329 . . . . 5 𝑚 ∈ ℕ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ))
13 breq2 5034 . . . . . . . . . . . . 13 (𝑜 = 𝑛 → (7 < 𝑜 ↔ 7 < 𝑛))
14 breq1 5033 . . . . . . . . . . . . 13 (𝑜 = 𝑛 → (𝑜 < 𝑚𝑛 < 𝑚))
1513, 14anbi12d 633 . . . . . . . . . . . 12 (𝑜 = 𝑛 → ((7 < 𝑜𝑜 < 𝑚) ↔ (7 < 𝑛𝑛 < 𝑚)))
16 eleq1w 2872 . . . . . . . . . . . 12 (𝑜 = 𝑛 → (𝑜 ∈ GoldbachOdd ↔ 𝑛 ∈ GoldbachOdd ))
1715, 16imbi12d 348 . . . . . . . . . . 11 (𝑜 = 𝑛 → (((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ) ↔ ((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd )))
1817rspcv 3566 . . . . . . . . . 10 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ) → ((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd )))
199recni 10644 . . . . . . . . . . . . . . . . . . . . . . 23 (10↑27) ∈ ℂ
2019mulid2i 10635 . . . . . . . . . . . . . . . . . . . . . 22 (1 · (10↑27)) = (10↑27)
21 1re 10630 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ ℝ
22 8re 11721 . . . . . . . . . . . . . . . . . . . . . . . . 25 8 ∈ ℝ
2321, 22pm3.2i 474 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 ∈ ℝ ∧ 8 ∈ ℝ)
2423a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (1 ∈ ℝ ∧ 8 ∈ ℝ))
25 0le1 11152 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ≤ 1
26 1lt8 11823 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 < 8
2725, 26pm3.2i 474 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ≤ 1 ∧ 1 < 8)
2827a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (0 ≤ 1 ∧ 1 < 8))
29 3nn 11704 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 ∈ ℕ
3029decnncl2 12110 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 30 ∈ ℕ
3130nnnn0i 11893 . . . . . . . . . . . . . . . . . . . . . . . . . 26 30 ∈ ℕ0
32 reexpcl 13442 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((10 ∈ ℝ ∧ 30 ∈ ℕ0) → (10↑30) ∈ ℝ)
333, 31, 32mp2an 691 . . . . . . . . . . . . . . . . . . . . . . . . 25 (10↑30) ∈ ℝ
349, 33pm3.2i 474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((10↑27) ∈ ℝ ∧ (10↑30) ∈ ℝ)
3534a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((10↑27) ∈ ℝ ∧ (10↑30) ∈ ℝ))
36 10nn0 12104 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 10 ∈ ℕ0
3736, 7nn0expcli 13451 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (10↑27) ∈ ℕ0
3837nn0ge0i 11912 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ≤ (10↑27)
396nnzi 11994 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 ∈ ℤ
4030nnzi 11994 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 30 ∈ ℤ
413, 39, 403pm3.2i 1336 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (10 ∈ ℝ ∧ 27 ∈ ℤ ∧ 30 ∈ ℤ)
42 1lt10 12225 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1 < 10
43 3nn0 11903 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 ∈ ℕ0
44 7nn0 11907 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7 ∈ ℕ0
45 0nn0 11900 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ ℕ0
46 7lt10 12219 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7 < 10
47 2lt3 11797 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 < 3
484, 43, 44, 45, 46, 47decltc 12115 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 < 30
4942, 48pm3.2i 474 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (1 < 10 ∧ 27 < 30)
50 ltexp2a 13526 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((10 ∈ ℝ ∧ 27 ∈ ℤ ∧ 30 ∈ ℤ) ∧ (1 < 10 ∧ 27 < 30)) → (10↑27) < (10↑30))
5141, 49, 50mp2an 691 . . . . . . . . . . . . . . . . . . . . . . . . 25 (10↑27) < (10↑30)
5238, 51pm3.2i 474 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ≤ (10↑27) ∧ (10↑27) < (10↑30))
5352a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (0 ≤ (10↑27) ∧ (10↑27) < (10↑30)))
54 ltmul12a 11485 . . . . . . . . . . . . . . . . . . . . . . 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 837 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (1 · (10↑27)) < (8 · (10↑30)))
5620, 55eqbrtrrid 5066 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (10↑27) < (8 · (10↑30)))
579a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (10↑27) ∈ ℝ)
5822, 33remulcli 10646 . . . . . . . . . . . . . . . . . . . . . . 23 (8 · (10↑30)) ∈ ℝ
5958a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (8 · (10↑30)) ∈ ℝ)
60 nnre 11632 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
6160adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
62 lttr 10706 . . . . . . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (((10↑27) < (8 · (10↑30)) ∧ (8 · (10↑30)) < 𝑚) → (10↑27) < 𝑚))
6456, 63mpand 694 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((8 · (10↑30)) < 𝑚 → (10↑27) < 𝑚))
6564imp 410 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → (10↑27) < 𝑚)
662adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → 𝑛 ∈ ℝ)
6766, 57, 613jca 1125 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (𝑛 ∈ ℝ ∧ (10↑27) ∈ ℝ ∧ 𝑚 ∈ ℝ))
6867adantr 484 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → (𝑛 ∈ ℝ ∧ (10↑27) ∈ ℝ ∧ 𝑚 ∈ ℝ))
69 lelttr 10720 . . . . . . . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → (𝑛 ≤ (10↑27) → 𝑛 < 𝑚))
7271imp 410 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) → 𝑛 < 𝑚)
7372anim1i 617 . . . . . . . . . . . . . . . 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 42 . . . . . . . . . . . . . . 15 ((7 < 𝑛𝑛 < 𝑚) → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) → 𝑛 ∈ GoldbachOdd ))
7674, 75syl 17 . . . . . . . . . . . . . 14 (((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) ∧ 7 < 𝑛) → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) → 𝑛 ∈ GoldbachOdd ))
7776ex 416 . . . . . . . . . . . . 13 ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) → (7 < 𝑛 → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) → 𝑛 ∈ GoldbachOdd )))
7877com23 86 . . . . . . . . . . . 12 ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) → (7 < 𝑛𝑛 ∈ GoldbachOdd )))
7978exp41 438 . . . . . . . . . . 11 (𝑛 ∈ Odd → (𝑚 ∈ ℕ → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) → (7 < 𝑛𝑛 ∈ GoldbachOdd ))))))
8079com25 99 . . . . . . . . . 10 (𝑛 ∈ Odd → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (𝑚 ∈ ℕ → (7 < 𝑛𝑛 ∈ GoldbachOdd ))))))
8118, 80syld 47 . . . . . . . . 9 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ) → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (𝑚 ∈ ℕ → (7 < 𝑛𝑛 ∈ GoldbachOdd ))))))
8281com15 101 . . . . . . . 8 (𝑚 ∈ ℕ → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ) → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd ))))))
8382com23 86 . . . . . . 7 (𝑚 ∈ ℕ → ((8 · (10↑30)) < 𝑚 → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ) → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd ))))))
8483imp32 422 . . . . . 6 ((𝑚 ∈ ℕ ∧ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ))) → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd ))))
8584rexlimiva 3240 . . . . 5 (∃𝑚 ∈ ℕ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd )) → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd ))))
8612, 85ax-mp 5 . . . 4 (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd )))
87 tgoldbachgtALTV 44325 . . . . 5 𝑚 ∈ ℕ (𝑚 ≤ (10↑27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOdd ))
88 breq2 5034 . . . . . . . . . . 11 (𝑜 = 𝑛 → (𝑚 < 𝑜𝑚 < 𝑛))
8988, 16imbi12d 348 . . . . . . . . . 10 (𝑜 = 𝑛 → ((𝑚 < 𝑜𝑜 ∈ GoldbachOdd ) ↔ (𝑚 < 𝑛𝑛 ∈ GoldbachOdd )))
9089rspcv 3566 . . . . . . . . 9 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOdd ) → (𝑚 < 𝑛𝑛 ∈ GoldbachOdd )))
91 lelttr 10720 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℝ ∧ (10↑27) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑚 ≤ (10↑27) ∧ (10↑27) < 𝑛) → 𝑚 < 𝑛))
9261, 57, 66, 91syl3anc 1368 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((𝑚 ≤ (10↑27) ∧ (10↑27) < 𝑛) → 𝑚 < 𝑛))
9392expcomd 420 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((10↑27) < 𝑛 → (𝑚 ≤ (10↑27) → 𝑚 < 𝑛)))
9493ex 416 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ Odd → (𝑚 ∈ ℕ → ((10↑27) < 𝑛 → (𝑚 ≤ (10↑27) → 𝑚 < 𝑛))))
9594com23 86 . . . . . . . . . . . . . . . 16 (𝑛 ∈ Odd → ((10↑27) < 𝑛 → (𝑚 ∈ ℕ → (𝑚 ≤ (10↑27) → 𝑚 < 𝑛))))
9695imp43 431 . . . . . . . . . . . . . . 15 (((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) ∧ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27))) → 𝑚 < 𝑛)
97 pm2.27 42 . . . . . . . . . . . . . . 15 (𝑚 < 𝑛 → ((𝑚 < 𝑛𝑛 ∈ GoldbachOdd ) → 𝑛 ∈ GoldbachOdd ))
9896, 97syl 17 . . . . . . . . . . . . . 14 (((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) ∧ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27))) → ((𝑚 < 𝑛𝑛 ∈ GoldbachOdd ) → 𝑛 ∈ GoldbachOdd ))
9998a1dd 50 . . . . . . . . . . . . 13 (((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) ∧ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27))) → ((𝑚 < 𝑛𝑛 ∈ GoldbachOdd ) → (7 < 𝑛𝑛 ∈ GoldbachOdd )))
10099ex 416 . . . . . . . . . . . 12 ((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → ((𝑚 < 𝑛𝑛 ∈ GoldbachOdd ) → (7 < 𝑛𝑛 ∈ GoldbachOdd ))))
101100com23 86 . . . . . . . . . . 11 ((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) → ((𝑚 < 𝑛𝑛 ∈ GoldbachOdd ) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOdd ))))
102101ex 416 . . . . . . . . . 10 (𝑛 ∈ Odd → ((10↑27) < 𝑛 → ((𝑚 < 𝑛𝑛 ∈ GoldbachOdd ) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOdd )))))
103102com23 86 . . . . . . . . 9 (𝑛 ∈ Odd → ((𝑚 < 𝑛𝑛 ∈ GoldbachOdd ) → ((10↑27) < 𝑛 → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOdd )))))
10490, 103syld 47 . . . . . . . 8 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOdd ) → ((10↑27) < 𝑛 → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOdd )))))
105104com14 96 . . . . . . 7 ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOdd ) → ((10↑27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd )))))
106105impr 458 . . . . . 6 ((𝑚 ∈ ℕ ∧ (𝑚 ≤ (10↑27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOdd ))) → ((10↑27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd ))))
107106rexlimiva 3240 . . . . 5 (∃𝑚 ∈ ℕ (𝑚 ≤ (10↑27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOdd )) → ((10↑27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd ))))
10887, 107ax-mp 5 . . . 4 ((10↑27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd )))
10986, 108jaoi 854 . . 3 ((𝑛 ≤ (10↑27) ∨ (10↑27) < 𝑛) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd )))
11011, 109mpcom 38 . 2 (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd ))
111110rgen 3116 1 𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOdd )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2111  wral 3106  wrex 3107   class class class wbr 5030  (class class class)co 7135  cr 10525  0cc0 10526  1c1 10527   · cmul 10531   < clt 10664  cle 10665  cn 11625  2c2 11680  3c3 11681  7c7 11685  8c8 11686  0cn0 11885  cz 11969  cdc 12086  cexp 13425   Odd codd 44138   GoldbachOdd cgbo 44260
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-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  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-bgbltosilva 44323  ax-tgoldbachgt 44324  ax-hgprmladder 44327
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-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  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-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-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-inf 8891  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-z 11970  df-dec 12087  df-uz 12232  df-rp 12378  df-ico 12732  df-fz 12886  df-fzo 13029  df-seq 13365  df-exp 13426  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-dvds 15600  df-prm 16006  df-iccp 43926  df-even 44139  df-odd 44140  df-gbe 44261  df-gbo 44263
This theorem is referenced by: (None)
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