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Theorem sbgoldbalt 42197
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 11621 . . . . . 6 2 ∈ ℤ
2 evenz 42071 . . . . . 6 (𝑛 ∈ Even → 𝑛 ∈ ℤ)
3 zltp1le 11639 . . . . . 6 ((2 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
41, 2, 3sylancr 698 . . . . 5 (𝑛 ∈ Even → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
5 2p1e3 11363 . . . . . . 7 (2 + 1) = 3
65breq1i 4811 . . . . . 6 ((2 + 1) ≤ 𝑛 ↔ 3 ≤ 𝑛)
7 3re 11306 . . . . . . . . 9 3 ∈ ℝ
87a1i 11 . . . . . . . 8 (𝑛 ∈ Even → 3 ∈ ℝ)
92zred 11694 . . . . . . . 8 (𝑛 ∈ Even → 𝑛 ∈ ℝ)
108, 9leloed 10392 . . . . . . 7 (𝑛 ∈ Even → (3 ≤ 𝑛 ↔ (3 < 𝑛 ∨ 3 = 𝑛)))
11 3z 11622 . . . . . . . . . . . 12 3 ∈ ℤ
12 zltp1le 11639 . . . . . . . . . . . 12 ((3 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
1311, 2, 12sylancr 698 . . . . . . . . . . 11 (𝑛 ∈ Even → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
14 3p1e4 11365 . . . . . . . . . . . . 13 (3 + 1) = 4
1514breq1i 4811 . . . . . . . . . . . 12 ((3 + 1) ≤ 𝑛 ↔ 4 ≤ 𝑛)
16 4re 11309 . . . . . . . . . . . . . . 15 4 ∈ ℝ
1716a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ Even → 4 ∈ ℝ)
1817, 9leloed 10392 . . . . . . . . . . . . 13 (𝑛 ∈ Even → (4 ≤ 𝑛 ↔ (4 < 𝑛 ∨ 4 = 𝑛)))
19 pm3.35 612 . . . . . . . . . . . . . . . . . 18 ((4 < 𝑛 ∧ (4 < 𝑛𝑛 ∈ GoldbachEven )) → 𝑛 ∈ GoldbachEven )
20 isgbe 42167 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ GoldbachEven ↔ (𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
21 simp3 1133 . . . . . . . . . . . . . . . . . . . . . . . 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 3155 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Even ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
2423reximdva 3155 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ Even → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
2524imp 444 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))
2620, 25sylbi 207 . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . 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 15627 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℙ
32 2p2e4 11356 . . . . . . . . . . . . . . . . . . . 20 (2 + 2) = 4
3332eqcomi 2769 . . . . . . . . . . . . . . . . . . 19 4 = (2 + 2)
34 rspceov 6856 . . . . . . . . . . . . . . . . . . 19 ((2 ∈ ℙ ∧ 2 ∈ ℙ ∧ 4 = (2 + 2)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞))
3531, 31, 33, 34mp3an 1573 . . . . . . . . . . . . . . . . . 18 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞)
36 eqeq1 2764 . . . . . . . . . . . . . . . . . . 19 (4 = 𝑛 → (4 = (𝑝 + 𝑞) ↔ 𝑛 = (𝑝 + 𝑞)))
37362rexbidv 3195 . . . . . . . . . . . . . . . . . 18 (4 = 𝑛 → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞) ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
3835, 37mpbii 223 . . . . . . . . . . . . . . . . 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 393 . . . . . . . . . . . . . 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 230 . . . . . . . . . . . 12 (𝑛 ∈ Even → (4 ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
4415, 43syl5bi 232 . . . . . . . . . . 11 (𝑛 ∈ Even → ((3 + 1) ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
4513, 44sylbid 230 . . . . . . . . . 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 42145 . . . . . . . . . . . 12 3 ∈ Odd
48 eleq1 2827 . . . . . . . . . . . 12 (3 = 𝑛 → (3 ∈ Odd ↔ 𝑛 ∈ Odd ))
4947, 48mpbii 223 . . . . . . . . . . 11 (3 = 𝑛𝑛 ∈ Odd )
50 oddneven 42085 . . . . . . . . . . 11 (𝑛 ∈ Odd → ¬ 𝑛 ∈ Even )
5149, 50syl 17 . . . . . . . . . 10 (3 = 𝑛 → ¬ 𝑛 ∈ Even )
5251pm2.21d 118 . . . . . . . . 9 (3 = 𝑛 → (𝑛 ∈ Even → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
5346, 52jaoi 393 . . . . . . . 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 230 . . . . . 6 (𝑛 ∈ Even → (3 ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
566, 55syl5bi 232 . . . . 5 (𝑛 ∈ Even → ((2 + 1) ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
574, 56sylbid 230 . . . 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 11410 . . . . . . . 8 2 < 4
60 2re 11302 . . . . . . . . . 10 2 ∈ ℝ
6160a1i 11 . . . . . . . . 9 (𝑛 ∈ Even → 2 ∈ ℝ)
62 lttr 10326 . . . . . . . . 9 ((2 ∈ ℝ ∧ 4 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
6361, 17, 9, 62syl3anc 1477 . . . . . . . 8 (𝑛 ∈ Even → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
6459, 63mpani 714 . . . . . . 7 (𝑛 ∈ Even → (4 < 𝑛 → 2 < 𝑛))
6564imp 444 . . . . . 6 ((𝑛 ∈ Even ∧ 4 < 𝑛) → 2 < 𝑛)
66 simpll 807 . . . . . . . . 9 (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → 𝑛 ∈ Even )
67 simpr 479 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
6867anim1i 593 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
6968adantr 472 . . . . . . . . . . . . . . 15 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
70 simpll 807 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 ∈ Even ∧ 4 < 𝑛))
7170anim1i 593 . . . . . . . . . . . . . . . 16 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
72 df-3an 1074 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Even ∧ 4 < 𝑛𝑛 = (𝑝 + 𝑞)) ↔ ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
7371, 72sylibr 224 . . . . . . . . . . . . . . 15 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑛 ∈ Even ∧ 4 < 𝑛𝑛 = (𝑝 + 𝑞)))
74 sbgoldbaltlem2 42196 . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . 14 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → 𝑛 = (𝑝 + 𝑞))
77 df-3an 1074 . . . . . . . . . . . . . 14 ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) ↔ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ) ∧ 𝑛 = (𝑝 + 𝑞)))
7875, 76, 77sylanbrc 701 . . . . . . . . . . . . 13 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)))
7978ex 449 . . . . . . . . . . . 12 ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 = (𝑝 + 𝑞) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
8079reximdva 3155 . . . . . . . . . . 11 (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
8180reximdva 3155 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ 4 < 𝑛) → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
8281imp 444 . . . . . . . . 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 555 . . . . . . . 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 449 . . . . . . 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 242 . . . . . 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 449 . . . 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 202 . 2 (𝑛 ∈ Even → ((4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
9089ralbiia 3117 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 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051   class class class wbr 4804  (class class class)co 6814  cr 10147  1c1 10149   + caddc 10151   < clt 10286  cle 10287  2c2 11282  3c3 11283  4c4 11284  cz 11589  cprime 15607   Even ceven 42065   Odd codd 42066   GoldbachEven cgbe 42161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-n0 11505  df-z 11590  df-uz 11900  df-rp 12046  df-fz 12540  df-seq 13016  df-exp 13075  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-dvds 15203  df-prm 15608  df-even 42067  df-odd 42068  df-gbe 42164
This theorem is referenced by:  sbgoldbb  42198  sbgoldbmb  42202
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