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Theorem tgoldbachOLD 40239
 Description: Obsolete version of tgoldbach 40232 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 40082 . . . . 5 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
21zred 11358 . . . 4 (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
3 10reOLD 10986 . . . . 5 10 ∈ ℝ
4 2nn0 11186 . . . . . . 7 2 ∈ ℕ0
5 7nn 11067 . . . . . . 7 7 ∈ ℕ
64, 5decnncl 11394 . . . . . 6 27 ∈ ℕ
76nnnn0i 11177 . . . . 5 27 ∈ ℕ0
8 reexpcl 12739 . . . . 5 ((10 ∈ ℝ ∧ 27 ∈ ℕ0) → (10↑27) ∈ ℝ)
93, 7, 8mp2an 704 . . . 4 (10↑27) ∈ ℝ
10 lelttric 10023 . . . 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 tgoldbachltOLD 40237 . . . . 5 𝑚 ∈ ℕ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ))
13 breq2 4587 . . . . . . . . . . . . 13 (𝑜 = 𝑛 → (7 < 𝑜 ↔ 7 < 𝑛))
14 breq1 4586 . . . . . . . . . . . . 13 (𝑜 = 𝑛 → (𝑜 < 𝑚𝑛 < 𝑚))
1513, 14anbi12d 743 . . . . . . . . . . . 12 (𝑜 = 𝑛 → ((7 < 𝑜𝑜 < 𝑚) ↔ (7 < 𝑛𝑛 < 𝑚)))
16 eleq1 2676 . . . . . . . . . . . 12 (𝑜 = 𝑛 → (𝑜 ∈ GoldbachOddALTV ↔ 𝑛 ∈ GoldbachOddALTV ))
1715, 16imbi12d 333 . . . . . . . . . . 11 (𝑜 = 𝑛 → (((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) ↔ ((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV )))
1817rspcv 3278 . . . . . . . . . 10 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) → ((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV )))
199recni 9931 . . . . . . . . . . . . . . . . . . . . . . 23 (10↑27) ∈ ℂ
2019mulid2i 9922 . . . . . . . . . . . . . . . . . . . . . 22 (1 · (10↑27)) = (10↑27)
21 1re 9918 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ ℝ
22 8re 10982 . . . . . . . . . . . . . . . . . . . . . . . . 25 8 ∈ ℝ
2321, 22pm3.2i 470 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 ∈ ℝ ∧ 8 ∈ ℝ)
2423a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (1 ∈ ℝ ∧ 8 ∈ ℝ))
25 0le1 10430 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ≤ 1
26 1lt8 11098 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 < 8
2725, 26pm3.2i 470 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ≤ 1 ∧ 1 < 8)
2827a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (0 ≤ 1 ∧ 1 < 8))
29 3nn 11063 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 ∈ ℕ
3029decnncl2 11401 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 30 ∈ ℕ
3130nnnn0i 11177 . . . . . . . . . . . . . . . . . . . . . . . . . 26 30 ∈ ℕ0
32 reexpcl 12739 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((10 ∈ ℝ ∧ 30 ∈ ℕ0) → (10↑30) ∈ ℝ)
333, 31, 32mp2an 704 . . . . . . . . . . . . . . . . . . . . . . . . 25 (10↑30) ∈ ℝ
349, 33pm3.2i 470 . . . . . . . . . . . . . . . . . . . . . . . 24 ((10↑27) ∈ ℝ ∧ (10↑30) ∈ ℝ)
3534a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((10↑27) ∈ ℝ ∧ (10↑30) ∈ ℝ))
36 10nn0OLD 11194 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 10 ∈ ℕ0
3736, 7nn0expcli 12748 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (10↑27) ∈ ℕ0
3837nn0ge0i 11197 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ≤ (10↑27)
396nnzi 11278 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 ∈ ℤ
4030nnzi 11278 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 30 ∈ ℤ
413, 39, 403pm3.2i 1232 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (10 ∈ ℝ ∧ 27 ∈ ℤ ∧ 30 ∈ ℤ)
42 1lt10OLD 11115 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1 < 10
43 3nn0 11187 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 ∈ ℕ0
44 7nn0 11191 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7 ∈ ℕ0
45 0nn0 11184 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ ℕ0
46 7lt10OLD 11109 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7 < 10
47 2lt3 11072 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 < 3
484, 43, 44, 45, 46, 47decltcOLD 11409 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 < 30
4942, 48pm3.2i 470 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (1 < 10 ∧ 27 < 30)
50 ltexp2a 12774 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((10 ∈ ℝ ∧ 27 ∈ ℤ ∧ 30 ∈ ℤ) ∧ (1 < 10 ∧ 27 < 30)) → (10↑27) < (10↑30))
5141, 49, 50mp2an 704 . . . . . . . . . . . . . . . . . . . . . . . . 25 (10↑27) < (10↑30)
5238, 51pm3.2i 470 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ≤ (10↑27) ∧ (10↑27) < (10↑30))
5352a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (0 ≤ (10↑27) ∧ (10↑27) < (10↑30)))
54 ltmul12a 10758 . . . . . . . . . . . . . . . . . . . . . . 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 1319 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (1 · (10↑27)) < (8 · (10↑30)))
5620, 55syl5eqbrr 4619 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (10↑27) < (8 · (10↑30)))
579a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (10↑27) ∈ ℝ)
5822, 33remulcli 9933 . . . . . . . . . . . . . . . . . . . . . . 23 (8 · (10↑30)) ∈ ℝ
5958a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (8 · (10↑30)) ∈ ℝ)
60 nnre 10904 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
6160adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
62 lttr 9993 . . . . . . . . . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (((10↑27) < (8 · (10↑30)) ∧ (8 · (10↑30)) < 𝑚) → (10↑27) < 𝑚))
6456, 63mpand 707 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((8 · (10↑30)) < 𝑚 → (10↑27) < 𝑚))
6564imp 444 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → (10↑27) < 𝑚)
662adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → 𝑛 ∈ ℝ)
6766, 57, 613jca 1235 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → (𝑛 ∈ ℝ ∧ (10↑27) ∈ ℝ ∧ 𝑚 ∈ ℝ))
6867adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → (𝑛 ∈ ℝ ∧ (10↑27) ∈ ℝ ∧ 𝑚 ∈ ℝ))
69 lelttr 10007 . . . . . . . . . . . . . . . . . . . 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 706 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) → (𝑛 ≤ (10↑27) → 𝑛 < 𝑚))
7271imp 444 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) → 𝑛 < 𝑚)
7372anim1i 590 . . . . . . . . . . . . . . . 16 (((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) ∧ 7 < 𝑛) → (𝑛 < 𝑚 ∧ 7 < 𝑛))
7473ancomd 466 . . . . . . . . . . . . . . 15 (((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) ∧ 7 < 𝑛) → (7 < 𝑛𝑛 < 𝑚))
75 pm2.27 41 . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . 13 ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) → (7 < 𝑛 → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ) → 𝑛 ∈ GoldbachOddALTV )))
7877com23 84 . . . . . . . . . . . 12 ((((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) ∧ (8 · (10↑30)) < 𝑚) ∧ 𝑛 ≤ (10↑27)) → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))
7978exp41 636 . . . . . . . . . . 11 (𝑛 ∈ Odd → (𝑚 ∈ ℕ → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8079com25 97 . . . . . . . . . 10 (𝑛 ∈ Odd → (((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ) → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (𝑚 ∈ ℕ → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8118, 80syld 46 . . . . . . . . 9 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (𝑚 ∈ ℕ → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8281com15 99 . . . . . . . 8 (𝑚 ∈ ℕ → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) → ((8 · (10↑30)) < 𝑚 → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8382com23 84 . . . . . . 7 (𝑚 ∈ ℕ → ((8 · (10↑30)) < 𝑚 → (∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ) → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))))
8483imp32 448 . . . . . 6 ((𝑚 ∈ ℕ ∧ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑜 ∈ Odd ((7 < 𝑜𝑜 < 𝑚) → 𝑜 ∈ GoldbachOddALTV ))) → (𝑛 ≤ (10↑27) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))
8584rexlimiva 3010 . . . . 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 40238 . . . . 5 𝑚 ∈ ℕ (𝑚 ≤ (10↑27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ))
88 breq2 4587 . . . . . . . . . . 11 (𝑜 = 𝑛 → (𝑚 < 𝑜𝑚 < 𝑛))
8988, 16imbi12d 333 . . . . . . . . . 10 (𝑜 = 𝑛 → ((𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ) ↔ (𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV )))
9089rspcv 3278 . . . . . . . . 9 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ) → (𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV )))
91 lelttr 10007 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℝ ∧ (10↑27) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑚 ≤ (10↑27) ∧ (10↑27) < 𝑛) → 𝑚 < 𝑛))
9261, 57, 66, 91syl3anc 1318 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((𝑚 ≤ (10↑27) ∧ (10↑27) < 𝑛) → 𝑚 < 𝑛))
9392expcomd 453 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ Odd ∧ 𝑚 ∈ ℕ) → ((10↑27) < 𝑛 → (𝑚 ≤ (10↑27) → 𝑚 < 𝑛)))
9493ex 449 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ Odd → (𝑚 ∈ ℕ → ((10↑27) < 𝑛 → (𝑚 ≤ (10↑27) → 𝑚 < 𝑛))))
9594com23 84 . . . . . . . . . . . . . . . 16 (𝑛 ∈ Odd → ((10↑27) < 𝑛 → (𝑚 ∈ ℕ → (𝑚 ≤ (10↑27) → 𝑚 < 𝑛))))
9695imp43 619 . . . . . . . . . . . . . . 15 (((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) ∧ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27))) → 𝑚 < 𝑛)
97 pm2.27 41 . . . . . . . . . . . . . . 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 48 . . . . . . . . . . . . 13 (((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) ∧ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27))) → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))
10099ex 449 . . . . . . . . . . . 12 ((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))
101100com23 84 . . . . . . . . . . 11 ((𝑛 ∈ Odd ∧ (10↑27) < 𝑛) → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))
102101ex 449 . . . . . . . . . 10 (𝑛 ∈ Odd → ((10↑27) < 𝑛 → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))))
103102com23 84 . . . . . . . . 9 (𝑛 ∈ Odd → ((𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ) → ((10↑27) < 𝑛 → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))))
10490, 103syld 46 . . . . . . . 8 (𝑛 ∈ Odd → (∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ) → ((10↑27) < 𝑛 → ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))))
105104com14 94 . . . . . . 7 ((𝑚 ∈ ℕ ∧ 𝑚 ≤ (10↑27)) → (∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ) → ((10↑27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))))
106105impr 647 . . . . . 6 ((𝑚 ∈ ℕ ∧ (𝑚 ≤ (10↑27) ∧ ∀𝑜 ∈ Odd (𝑚 < 𝑜𝑜 ∈ GoldbachOddALTV ))) → ((10↑27) < 𝑛 → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))))
107106rexlimiva 3010 . . . . 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 393 . . 3 ((𝑛 ≤ (10↑27) ∨ (10↑27) < 𝑛) → (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV )))
11011, 109mpcom 37 . 2 (𝑛 ∈ Odd → (7 < 𝑛𝑛 ∈ GoldbachOddALTV ))
111110rgen 2906 1 𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOddALTV )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   class class class wbr 4583  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   · cmul 9820   < clt 9953   ≤ cle 9954  ℕcn 10897  2c2 10947  3c3 10948  7c7 10952  8c8 10953  10c10 10955  ℕ0cn0 11169  ℤcz 11254  ;cdc 11369  ↑cexp 12722   Odd codd 40076   GoldbachOddALTV cgboa 40169 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-bgbltosilvaOLD 40233  ax-hgprmladderOLD 40235  ax-tgoldbachgtOLD 40238 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-10OLD 10964  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-rp 11709  df-ico 12052  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-prm 15224  df-iccp 39952  df-even 40077  df-odd 40078  df-gbe 40170  df-gboa 40172 This theorem is referenced by: (None)
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