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Theorem tgoldbach 47389
Description: The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 47388 and ax-tgoldbachgt 47383. (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 47203 . . . . 5 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
21zred 12718 . . . 4 (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
3 10re 12748 . . . . 5 10 ∈ ℝ
4 2nn0 12541 . . . . . . 7 2 ∈ ℕ0
5 7nn 12356 . . . . . . 7 7 ∈ ℕ
64, 5decnncl 12749 . . . . . 6 27 ∈ ℕ
76nnnn0i 12532 . . . . 5 27 ∈ ℕ0
8 reexpcl 14098 . . . . 5 ((10 ∈ ℝ ∧ 27 ∈ ℕ0) → (10↑27) ∈ ℝ)
93, 7, 8mp2an 690 . . . 4 (10↑27) ∈ ℝ
10 lelttric 11371 . . . 4 ((𝑛 ∈ ℝ ∧ (10↑27) ∈ ℝ) → (𝑛 ≤ (10↑27) ∨ (10↑27) < 𝑛))
112, 9, 10sylancl 584 . . 3 (𝑛 ∈ Odd → (𝑛 ≤ (10↑27) ∨ (10↑27) < 𝑛))
12 tgoldbachlt 47388 . . . . 5 𝑚 ∈ ℕ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ))
13 breq2 5157 . . . . . . . . . . . . 13 (𝑜 = 𝑛 → (7 < 𝑜 ↔ 7 < 𝑛))
14 breq1 5156 . . . . . . . . . . . . 13 (𝑜 = 𝑛 → (𝑜 < 𝑚𝑛 < 𝑚))
1513, 14anbi12d 630 . . . . . . . . . . . 12 (𝑜 = 𝑛 → ((7 < 𝑜𝑜 < 𝑚) ↔ (7 < 𝑛𝑛 < 𝑚)))
16 eleq1w 2809 . . . . . . . . . . . 12 (𝑜 = 𝑛 → (𝑜 ∈ GoldbachOdd ↔ 𝑛 ∈ GoldbachOdd ))
1715, 16imbi12d 343 . . . . . . . . . . 11 (𝑜 = 𝑛 → (((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ) ↔ ((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd )))
1817rspcv 3604 . . . . . . . . . 10 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ) → ((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd )))
199recni 11278 . . . . . . . . . . . . . . . . . . . . . . 23 (10↑27) ∈ ℂ
2019mullidi 11269 . . . . . . . . . . . . . . . . . . . . . 22 (1 · (10↑27)) = (10↑27)
21 1re 11264 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ ℝ
22 8re 12360 . . . . . . . . . . . . . . . . . . . . . . . . 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 11787 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ≤ 1
26 1lt8 12462 . . . . . . . . . . . . . . . . . . . . . . . . 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 12343 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 ∈ ℕ
3029decnncl2 12753 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 30 ∈ ℕ
3130nnnn0i 12532 . . . . . . . . . . . . . . . . . . . . . . . . . 26 30 ∈ ℕ0
32 reexpcl 14098 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((10 ∈ ℝ ∧ 30 ∈ ℕ0) → (10↑30) ∈ ℝ)
333, 31, 32mp2an 690 . . . . . . . . . . . . . . . . . . . . . . . . 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 10nn0 12747 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 10 ∈ ℕ0
3736, 7nn0expcli 14108 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (10↑27) ∈ ℕ0
3837nn0ge0i 12551 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ≤ (10↑27)
396nnzi 12638 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 ∈ ℤ
4030nnzi 12638 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 30 ∈ ℤ
413, 39, 403pm3.2i 1336 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (10 ∈ ℝ ∧ 27 ∈ ℤ ∧ 30 ∈ ℤ)
42 1lt10 12868 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1 < 10
43 3nn0 12542 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 ∈ ℕ0
44 7nn0 12546 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7 ∈ ℕ0
45 0nn0 12539 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ ℕ0
46 7lt10 12862 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7 < 10
47 2lt3 12436 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 < 3
484, 43, 44, 45, 46, 47decltc 12758 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 < 30
4942, 48pm3.2i 469 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (1 < 10 ∧ 27 < 30)
50 ltexp2a 14185 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((10 ∈ ℝ ∧ 27 ∈ ℤ ∧ 30 ∈ ℤ) ∧ (1 < 10 ∧ 27 < 30)) → (10↑27) < (10↑30))
5141, 49, 50mp2an 690 . . . . . . . . . . . . . . . . . . . . . . . . 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 12121 . . . . . . . . . . . . . . . . . . . . . . 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 5189 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (10↑27) < (8 · (10↑30)))
579a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (10↑27) ∈ ℝ)
5822, 33remulcli 11280 . . . . . . . . . . . . . . . . . . . . . . 23 (8 · (10↑30)) ∈ ℝ
5958a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (8 · (10↑30)) ∈ ℝ)
60 nnre 12271 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
6160adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
62 lttr 11340 . . . . . . . . . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((8 · (10↑30)) < 𝑚 → (10↑27) < 𝑚))
6564imp 405 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → (10↑27) < 𝑚)
662adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → 𝑛 ∈ ℝ)
6766, 57, 613jca 1125 . . . . . . . . . . . . . . . . . . . . 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 11354 . . . . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → (𝑛 ≤ (10↑27) → 𝑛 < 𝑚))
7271imp 405 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) → 𝑛 < 𝑚)
7372anim1i 613 . . . . . . . . . . . . . . . 16 (((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) ∧ 7 < 𝑛) → (𝑛 < 𝑚 ∧ 7 < 𝑛))
7473ancomd 460 . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . 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 433 . . . . . . . . . . 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 417 . . . . . 6 ((𝑚 ∈ ℕ ∧ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOdd ))) → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd ))))
8584rexlimiva 3137 . . . . 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 47384 . . . . 5 𝑚 ∈ ℕ (𝑚 ≤ (10↑27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOdd ))
88 breq2 5157 . . . . . . . . . . 11 (𝑜 = 𝑛 → (𝑚 < 𝑜𝑚 < 𝑛))
8988, 16imbi12d 343 . . . . . . . . . 10 (𝑜 = 𝑛 → ((𝑚 < 𝑜𝑜 ∈ GoldbachOdd ) ↔ (𝑚 < 𝑛𝑛 ∈ GoldbachOdd )))
9089rspcv 3604 . . . . . . . . 9 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOdd ) → (𝑚 < 𝑛𝑛 ∈ GoldbachOdd )))
91 lelttr 11354 . . . . . . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((10↑27) < 𝑛 → (𝑚 ≤ (10↑27) → 𝑚 < 𝑛)))
9493ex 411 . . . . . . . . . . . . . . . . 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 426 . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 411 . . . . . . . . . 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 453 . . . . . 6 ((𝑚 ∈ ℕ ∧ (𝑚 ≤ (10↑27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOdd ))) → ((10↑27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd ))))
107106rexlimiva 3137 . . . . 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 855 . . 3 ((𝑛 ≤ (10↑27) ∨ (10↑27) < 𝑛) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd )))
11011, 109mpcom 38 . 2 (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOdd ))
111110rgen 3053 1 𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOdd )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845  w3a 1084   = wceq 1534  wcel 2099  wral 3051  wrex 3060   class class class wbr 5153  (class class class)co 7424  cr 11157  0cc0 11158  1c1 11159   · cmul 11163   < clt 11298  cle 11299  cn 12264  2c2 12319  3c3 12320  7c7 12324  8c8 12325  0cn0 12524  cz 12610  cdc 12729  cexp 14081   Odd codd 47197   GoldbachOdd cgbo 47319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236  ax-bgbltosilva 47382  ax-tgoldbachgt 47383  ax-hgprmladder 47386
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-inf 9486  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12611  df-dec 12730  df-uz 12875  df-rp 13029  df-ico 13384  df-fz 13539  df-fzo 13682  df-seq 14022  df-exp 14082  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-dvds 16257  df-prm 16673  df-iccp 46986  df-even 47198  df-odd 47199  df-gbe 47320  df-gbo 47322
This theorem is referenced by: (None)
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