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Theorem sbgoldbalt 44229
Description: An alternate (related to the original) formulation of the binary Goldbach conjecture: Every even integer greater than 2 can be expressed as the sum of two primes. (Contributed by AV, 22-Jul-2020.)
Assertion
Ref Expression
sbgoldbalt (∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
Distinct variable group:   𝑛,𝑝,𝑞

Proof of Theorem sbgoldbalt
StepHypRef Expression
1 2z 12011 . . . . . 6 2 ∈ ℤ
2 evenz 44078 . . . . . 6 (𝑛 ∈ Even → 𝑛 ∈ ℤ)
3 zltp1le 12029 . . . . . 6 ((2 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
41, 2, 3sylancr 590 . . . . 5 (𝑛 ∈ Even → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
5 2p1e3 11776 . . . . . . 7 (2 + 1) = 3
65breq1i 5059 . . . . . 6 ((2 + 1) ≤ 𝑛 ↔ 3 ≤ 𝑛)
7 3re 11714 . . . . . . . . 9 3 ∈ ℝ
87a1i 11 . . . . . . . 8 (𝑛 ∈ Even → 3 ∈ ℝ)
92zred 12084 . . . . . . . 8 (𝑛 ∈ Even → 𝑛 ∈ ℝ)
108, 9leloed 10781 . . . . . . 7 (𝑛 ∈ Even → (3 ≤ 𝑛 ↔ (3 < 𝑛 ∨ 3 = 𝑛)))
11 3z 12012 . . . . . . . . . . . 12 3 ∈ ℤ
12 zltp1le 12029 . . . . . . . . . . . 12 ((3 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
1311, 2, 12sylancr 590 . . . . . . . . . . 11 (𝑛 ∈ Even → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
14 3p1e4 11779 . . . . . . . . . . . . 13 (3 + 1) = 4
1514breq1i 5059 . . . . . . . . . . . 12 ((3 + 1) ≤ 𝑛 ↔ 4 ≤ 𝑛)
16 4re 11718 . . . . . . . . . . . . . . 15 4 ∈ ℝ
1716a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ Even → 4 ∈ ℝ)
1817, 9leloed 10781 . . . . . . . . . . . . 13 (𝑛 ∈ Even → (4 ≤ 𝑛 ↔ (4 < 𝑛 ∨ 4 = 𝑛)))
19 pm3.35 802 . . . . . . . . . . . . . . . . . 18 ((4 < 𝑛 ∧ (4 < 𝑛𝑛 ∈ GoldbachEven )) → 𝑛 ∈ GoldbachEven )
20 isgbe 44199 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ GoldbachEven ↔ (𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
21 simp3 1135 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → 𝑛 = (𝑝 + 𝑞))
2221a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ Even ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → 𝑛 = (𝑝 + 𝑞)))
2322reximdva 3266 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Even ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
2423reximdva 3266 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ Even → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
2524imp 410 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))
2620, 25sylbi 220 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ GoldbachEven → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))
2726a1d 25 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ GoldbachEven → (𝑛 ∈ Even → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
2819, 27syl 17 . . . . . . . . . . . . . . . . 17 ((4 < 𝑛 ∧ (4 < 𝑛𝑛 ∈ GoldbachEven )) → (𝑛 ∈ Even → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
2928ex 416 . . . . . . . . . . . . . . . 16 (4 < 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → (𝑛 ∈ Even → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
3029com23 86 . . . . . . . . . . . . . . 15 (4 < 𝑛 → (𝑛 ∈ Even → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
31 2prm 16034 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℙ
32 2p2e4 11769 . . . . . . . . . . . . . . . . . . . 20 (2 + 2) = 4
3332eqcomi 2833 . . . . . . . . . . . . . . . . . . 19 4 = (2 + 2)
34 rspceov 7196 . . . . . . . . . . . . . . . . . . 19 ((2 ∈ ℙ ∧ 2 ∈ ℙ ∧ 4 = (2 + 2)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞))
3531, 31, 33, 34mp3an 1458 . . . . . . . . . . . . . . . . . 18 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞)
36 eqeq1 2828 . . . . . . . . . . . . . . . . . . 19 (4 = 𝑛 → (4 = (𝑝 + 𝑞) ↔ 𝑛 = (𝑝 + 𝑞)))
37362rexbidv 3292 . . . . . . . . . . . . . . . . . 18 (4 = 𝑛 → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞) ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
3835, 37mpbii 236 . . . . . . . . . . . . . . . . 17 (4 = 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))
3938a1d 25 . . . . . . . . . . . . . . . 16 (4 = 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
4039a1d 25 . . . . . . . . . . . . . . 15 (4 = 𝑛 → (𝑛 ∈ Even → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
4130, 40jaoi 854 . . . . . . . . . . . . . 14 ((4 < 𝑛 ∨ 4 = 𝑛) → (𝑛 ∈ Even → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
4241com12 32 . . . . . . . . . . . . 13 (𝑛 ∈ Even → ((4 < 𝑛 ∨ 4 = 𝑛) → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
4318, 42sylbid 243 . . . . . . . . . . . 12 (𝑛 ∈ Even → (4 ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
4415, 43syl5bi 245 . . . . . . . . . . 11 (𝑛 ∈ Even → ((3 + 1) ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
4513, 44sylbid 243 . . . . . . . . . 10 (𝑛 ∈ Even → (3 < 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
4645com12 32 . . . . . . . . 9 (3 < 𝑛 → (𝑛 ∈ Even → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
47 3odd 44156 . . . . . . . . . . . 12 3 ∈ Odd
48 eleq1 2903 . . . . . . . . . . . 12 (3 = 𝑛 → (3 ∈ Odd ↔ 𝑛 ∈ Odd ))
4947, 48mpbii 236 . . . . . . . . . . 11 (3 = 𝑛𝑛 ∈ Odd )
50 oddneven 44092 . . . . . . . . . . 11 (𝑛 ∈ Odd → ¬ 𝑛 ∈ Even )
5149, 50syl 17 . . . . . . . . . 10 (3 = 𝑛 → ¬ 𝑛 ∈ Even )
5251pm2.21d 121 . . . . . . . . 9 (3 = 𝑛 → (𝑛 ∈ Even → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
5346, 52jaoi 854 . . . . . . . 8 ((3 < 𝑛 ∨ 3 = 𝑛) → (𝑛 ∈ Even → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
5453com12 32 . . . . . . 7 (𝑛 ∈ Even → ((3 < 𝑛 ∨ 3 = 𝑛) → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
5510, 54sylbid 243 . . . . . 6 (𝑛 ∈ Even → (3 ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
566, 55syl5bi 245 . . . . 5 (𝑛 ∈ Even → ((2 + 1) ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
574, 56sylbid 243 . . . 4 (𝑛 ∈ Even → (2 < 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
5857com23 86 . . 3 (𝑛 ∈ Even → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
59 2lt4 11809 . . . . . . . 8 2 < 4
60 2re 11708 . . . . . . . . . 10 2 ∈ ℝ
6160a1i 11 . . . . . . . . 9 (𝑛 ∈ Even → 2 ∈ ℝ)
62 lttr 10715 . . . . . . . . 9 ((2 ∈ ℝ ∧ 4 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
6361, 17, 9, 62syl3anc 1368 . . . . . . . 8 (𝑛 ∈ Even → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
6459, 63mpani 695 . . . . . . 7 (𝑛 ∈ Even → (4 < 𝑛 → 2 < 𝑛))
6564imp 410 . . . . . 6 ((𝑛 ∈ Even ∧ 4 < 𝑛) → 2 < 𝑛)
66 simpll 766 . . . . . . . . 9 (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → 𝑛 ∈ Even )
67 simpr 488 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
6867anim1i 617 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
6968adantr 484 . . . . . . . . . . . . . . 15 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
70 simpll 766 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 ∈ Even ∧ 4 < 𝑛))
7170anim1i 617 . . . . . . . . . . . . . . . 16 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
72 df-3an 1086 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Even ∧ 4 < 𝑛𝑛 = (𝑝 + 𝑞)) ↔ ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
7371, 72sylibr 237 . . . . . . . . . . . . . . 15 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑛 ∈ Even ∧ 4 < 𝑛𝑛 = (𝑝 + 𝑞)))
74 sbgoldbaltlem2 44228 . . . . . . . . . . . . . . 15 ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → ((𝑛 ∈ Even ∧ 4 < 𝑛𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd )))
7569, 73, 74sylc 65 . . . . . . . . . . . . . 14 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ))
76 simpr 488 . . . . . . . . . . . . . 14 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → 𝑛 = (𝑝 + 𝑞))
77 df-3an 1086 . . . . . . . . . . . . . 14 ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) ↔ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ) ∧ 𝑛 = (𝑝 + 𝑞)))
7875, 76, 77sylanbrc 586 . . . . . . . . . . . . 13 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)))
7978ex 416 . . . . . . . . . . . 12 ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 = (𝑝 + 𝑞) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
8079reximdva 3266 . . . . . . . . . . 11 (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
8180reximdva 3266 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ 4 < 𝑛) → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
8281imp 410 . . . . . . . . 9 (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)))
8366, 82jca 515 . . . . . . . 8 (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → (𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
8483ex 416 . . . . . . 7 ((𝑛 ∈ Even ∧ 4 < 𝑛) → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → (𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)))))
8584, 20syl6ibr 255 . . . . . 6 ((𝑛 ∈ Even ∧ 4 < 𝑛) → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → 𝑛 ∈ GoldbachEven ))
8665, 85embantd 59 . . . . 5 ((𝑛 ∈ Even ∧ 4 < 𝑛) → ((2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → 𝑛 ∈ GoldbachEven ))
8786ex 416 . . . 4 (𝑛 ∈ Even → (4 < 𝑛 → ((2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → 𝑛 ∈ GoldbachEven )))
8887com23 86 . . 3 (𝑛 ∈ Even → ((2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → (4 < 𝑛𝑛 ∈ GoldbachEven )))
8958, 88impbid 215 . 2 (𝑛 ∈ Even → ((4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
9089ralbiia 3159 1 (∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2115  wral 3133  wrex 3134   class class class wbr 5052  (class class class)co 7149  cr 10534  1c1 10536   + caddc 10538   < clt 10673  cle 10674  2c2 11689  3c3 11690  4c4 11691  cz 11978  cprime 16013   Even ceven 44072   Odd codd 44073   GoldbachEven cgbe 44193
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-n0 11895  df-z 11979  df-uz 12241  df-rp 12387  df-fz 12895  df-seq 13374  df-exp 13435  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-dvds 15608  df-prm 16014  df-even 44074  df-odd 44075  df-gbe 44196
This theorem is referenced by:  sbgoldbb  44230  sbgoldbmb  44234
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