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Theorem tgoldbach 41023
Description: The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 41021 and ax-tgoldbachgt 41022. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.)
Assertion
Ref Expression
tgoldbach 𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOddALTV )

Proof of Theorem tgoldbach
Dummy variables 𝑚 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oddz 40873 . . . . 5 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
21zred 11442 . . . 4 (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
3 10re 11477 . . . . 5 10 ∈ ℝ
4 2nn0 11269 . . . . . . 7 2 ∈ ℕ0
5 7nn 11150 . . . . . . 7 7 ∈ ℕ
64, 5decnncl 11478 . . . . . 6 27 ∈ ℕ
76nnnn0i 11260 . . . . 5 27 ∈ ℕ0
8 reexpcl 12833 . . . . 5 ((10 ∈ ℝ ∧ 27 ∈ ℕ0) → (10↑27) ∈ ℝ)
93, 7, 8mp2an 707 . . . 4 (10↑27) ∈ ℝ
10 lelttric 10104 . . . 4 ((𝑛 ∈ ℝ ∧ (10↑27) ∈ ℝ) → (𝑛 ≤ (10↑27) ∨ (10↑27) < 𝑛))
112, 9, 10sylancl 693 . . 3 (𝑛 ∈ Odd → (𝑛 ≤ (10↑27) ∨ (10↑27) < 𝑛))
12 tgoldbachlt 41021 . . . . 5 𝑚 ∈ ℕ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ))
13 breq2 4627 . . . . . . . . . . . . 13 (𝑜 = 𝑛 → (7 < 𝑜 ↔ 7 < 𝑛))
14 breq1 4626 . . . . . . . . . . . . 13 (𝑜 = 𝑛 → (𝑜 < 𝑚𝑛 < 𝑚))
1513, 14anbi12d 746 . . . . . . . . . . . 12 (𝑜 = 𝑛 → ((7 < 𝑜𝑜 < 𝑚) ↔ (7 < 𝑛𝑛 < 𝑚)))
16 eleq1 2686 . . . . . . . . . . . 12 (𝑜 = 𝑛 → (𝑜 ∈ GoldbachOddALTV ↔ 𝑛 ∈ GoldbachOddALTV ))
1715, 16imbi12d 334 . . . . . . . . . . 11 (𝑜 = 𝑛 → (((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) ↔ ((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV )))
1817rspcv 3295 . . . . . . . . . 10 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) → ((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV )))
199recni 10012 . . . . . . . . . . . . . . . . . . . . . . 23 (10↑27) ∈ ℂ
2019mulid2i 10003 . . . . . . . . . . . . . . . . . . . . . 22 (1 · (10↑27)) = (10↑27)
21 1re 9999 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ ℝ
22 8re 11065 . . . . . . . . . . . . . . . . . . . . . . . . 25 8 ∈ ℝ
2321, 22pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 ∈ ℝ ∧ 8 ∈ ℝ)
2423a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (1 ∈ ℝ ∧ 8 ∈ ℝ))
25 0le1 10511 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ≤ 1
26 1lt8 11181 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 < 8
2725, 26pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ≤ 1 ∧ 1 < 8)
2827a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (0 ≤ 1 ∧ 1 < 8))
29 3nn 11146 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 ∈ ℕ
3029decnncl2 11485 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 30 ∈ ℕ
3130nnnn0i 11260 . . . . . . . . . . . . . . . . . . . . . . . . . 26 30 ∈ ℕ0
32 reexpcl 12833 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((10 ∈ ℝ ∧ 30 ∈ ℕ0) → (10↑30) ∈ ℝ)
333, 31, 32mp2an 707 . . . . . . . . . . . . . . . . . . . . . . . . 25 (10↑30) ∈ ℝ
349, 33pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . . . 24 ((10↑27) ∈ ℝ ∧ (10↑30) ∈ ℝ)
3534a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((10↑27) ∈ ℝ ∧ (10↑30) ∈ ℝ))
36 10nn0 11476 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 10 ∈ ℕ0
3736, 7nn0expcli 12842 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (10↑27) ∈ ℕ0
3837nn0ge0i 11280 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ≤ (10↑27)
396nnzi 11361 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 ∈ ℤ
4030nnzi 11361 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 30 ∈ ℤ
413, 39, 403pm3.2i 1237 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (10 ∈ ℝ ∧ 27 ∈ ℤ ∧ 30 ∈ ℤ)
42 1lt10 11641 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1 < 10
43 3nn0 11270 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 ∈ ℕ0
44 7nn0 11274 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7 ∈ ℕ0
45 0nn0 11267 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ ℕ0
46 7lt10 11635 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7 < 10
47 2lt3 11155 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 < 3
484, 43, 44, 45, 46, 47decltc 11492 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 < 30
4942, 48pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (1 < 10 ∧ 27 < 30)
50 ltexp2a 12868 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((10 ∈ ℝ ∧ 27 ∈ ℤ ∧ 30 ∈ ℤ) ∧ (1 < 10 ∧ 27 < 30)) → (10↑27) < (10↑30))
5141, 49, 50mp2an 707 . . . . . . . . . . . . . . . . . . . . . . . . 25 (10↑27) < (10↑30)
5238, 51pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ≤ (10↑27) ∧ (10↑27) < (10↑30))
5352a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (0 ≤ (10↑27) ∧ (10↑27) < (10↑30)))
54 ltmul12a 10839 . . . . . . . . . . . . . . . . . . . . . . 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 1324 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (1 · (10↑27)) < (8 · (10↑30)))
5620, 55syl5eqbrr 4659 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (10↑27) < (8 · (10↑30)))
579a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (10↑27) ∈ ℝ)
5822, 33remulcli 10014 . . . . . . . . . . . . . . . . . . . . . . 23 (8 · (10↑30)) ∈ ℝ
5958a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (8 · (10↑30)) ∈ ℝ)
60 nnre 10987 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
6160adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
62 lttr 10074 . . . . . . . . . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (((10↑27) < (8 · (10↑30)) ∧ (8 · (10↑30)) < 𝑚) → (10↑27) < 𝑚))
6456, 63mpand 710 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((8 · (10↑30)) < 𝑚 → (10↑27) < 𝑚))
6564imp 445 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → (10↑27) < 𝑚)
662adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → 𝑛 ∈ ℝ)
6766, 57, 613jca 1240 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (𝑛 ∈ ℝ ∧ (10↑27) ∈ ℝ ∧ 𝑚 ∈ ℝ))
6867adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → (𝑛 ∈ ℝ ∧ (10↑27) ∈ ℝ ∧ 𝑚 ∈ ℝ))
69 lelttr 10088 . . . . . . . . . . . . . . . . . . . 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 709 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → (𝑛 ≤ (10↑27) → 𝑛 < 𝑚))
7271imp 445 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) → 𝑛 < 𝑚)
7372anim1i 591 . . . . . . . . . . . . . . . 16 (((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) ∧ 7 < 𝑛) → (𝑛 < 𝑚 ∧ 7 < 𝑛))
7473ancomd 467 . . . . . . . . . . . . . . 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 ∈ 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 450 . . . . . . . . . . . . 13 ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) → (7 < 𝑛 → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ) → 𝑛 ∈ GoldbachOddALTV )))
7877com23 86 . . . . . . . . . . . 12 ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))
7978exp41 637 . . . . . . . . . . 11 (𝑛 ∈ Odd → (𝑚 ∈ ℕ → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8079com25 99 . . . . . . . . . 10 (𝑛 ∈ Odd → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ) → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (𝑚 ∈ ℕ → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8118, 80syld 47 . . . . . . . . 9 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (𝑚 ∈ ℕ → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8281com15 101 . . . . . . . 8 (𝑚 ∈ ℕ → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8382com23 86 . . . . . . 7 (𝑚 ∈ ℕ → ((8 · (10↑30)) < 𝑚 → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8483imp32 449 . . . . . 6 ((𝑚 ∈ ℕ ∧ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ))) → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))
8584rexlimiva 3023 . . . . 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-tgoldbachgt 41022 . . . . 5 𝑚 ∈ ℕ (𝑚 ≤ (10↑27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ))
88 breq2 4627 . . . . . . . . . . 11 (𝑜 = 𝑛 → (𝑚 < 𝑜𝑚 < 𝑛))
8988, 16imbi12d 334 . . . . . . . . . 10 (𝑜 = 𝑛 → ((𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ) ↔ (𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV )))
9089rspcv 3295 . . . . . . . . 9 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ) → (𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV )))
91 lelttr 10088 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℝ ∧ (10↑27) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑚 ≤ (10↑27) ∧ (10↑27) < 𝑛) → 𝑚 < 𝑛))
9261, 57, 66, 91syl3anc 1323 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((𝑚 ≤ (10↑27) ∧ (10↑27) < 𝑛) → 𝑚 < 𝑛))
9392expcomd 454 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((10↑27) < 𝑛 → (𝑚 ≤ (10↑27) → 𝑚 < 𝑛)))
9493ex 450 . . . . . . . . . . . . . . . . 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 620 . . . . . . . . . . . . . . 15 (((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) ∧ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27))) → 𝑚 < 𝑛)
97 pm2.27 42 . . . . . . . . . . . . . . 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 50 . . . . . . . . . . . . 13 (((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) ∧ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27))) → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))
10099ex 450 . . . . . . . . . . . 12 ((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))
101100com23 86 . . . . . . . . . . 11 ((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))
102101ex 450 . . . . . . . . . 10 (𝑛 ∈ Odd → ((10↑27) < 𝑛 → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))))
103102com23 86 . . . . . . . . 9 (𝑛 ∈ Odd → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → ((10↑27) < 𝑛 → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))))
10490, 103syld 47 . . . . . . . 8 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ) → ((10↑27) < 𝑛 → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))))
105104com14 96 . . . . . . 7 ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ) → ((10↑27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))))
106105impr 648 . . . . . 6 ((𝑚 ∈ ℕ ∧ (𝑚 ≤ (10↑27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ))) → ((10↑27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))
107106rexlimiva 3023 . . . . 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 394 . . 3 ((𝑛 ≤ (10↑27) ∨ (10↑27) < 𝑛) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))
11011, 109mpcom 38 . 2 (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))
111110rgen 2918 1 𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOddALTV )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  wrex 2909   class class class wbr 4623  (class class class)co 6615  cr 9895  0cc0 9896  1c1 9897   · cmul 9901   < clt 10034  cle 10035  cn 10980  2c2 11030  3c3 11031  7c7 11035  8c8 11036  0cn0 11252  cz 11337  cdc 11453  cexp 12816   Odd codd 40867   GoldbachOddALTV cgboa 40960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974  ax-bgbltosilva 41017  ax-hgprmladder 41019  ax-tgoldbachgt 41022
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-inf 8309  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-rp 11793  df-ico 12139  df-fz 12285  df-fzo 12423  df-seq 12758  df-exp 12817  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-dvds 14927  df-prm 15329  df-iccp 40678  df-even 40868  df-odd 40869  df-gbe 40961  df-gboa 40963
This theorem is referenced by: (None)
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