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Theorem sbgoldbalt 43823
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 12002 . . . . . 6 2 ∈ ℤ
2 evenz 43672 . . . . . 6 (𝑛 ∈ Even → 𝑛 ∈ ℤ)
3 zltp1le 12020 . . . . . 6 ((2 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
41, 2, 3sylancr 587 . . . . 5 (𝑛 ∈ Even → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
5 2p1e3 11767 . . . . . . 7 (2 + 1) = 3
65breq1i 5064 . . . . . 6 ((2 + 1) ≤ 𝑛 ↔ 3 ≤ 𝑛)
7 3re 11705 . . . . . . . . 9 3 ∈ ℝ
87a1i 11 . . . . . . . 8 (𝑛 ∈ Even → 3 ∈ ℝ)
92zred 12075 . . . . . . . 8 (𝑛 ∈ Even → 𝑛 ∈ ℝ)
108, 9leloed 10771 . . . . . . 7 (𝑛 ∈ Even → (3 ≤ 𝑛 ↔ (3 < 𝑛 ∨ 3 = 𝑛)))
11 3z 12003 . . . . . . . . . . . 12 3 ∈ ℤ
12 zltp1le 12020 . . . . . . . . . . . 12 ((3 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
1311, 2, 12sylancr 587 . . . . . . . . . . 11 (𝑛 ∈ Even → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
14 3p1e4 11770 . . . . . . . . . . . . 13 (3 + 1) = 4
1514breq1i 5064 . . . . . . . . . . . 12 ((3 + 1) ≤ 𝑛 ↔ 4 ≤ 𝑛)
16 4re 11709 . . . . . . . . . . . . . . 15 4 ∈ ℝ
1716a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ Even → 4 ∈ ℝ)
1817, 9leloed 10771 . . . . . . . . . . . . 13 (𝑛 ∈ Even → (4 ≤ 𝑛 ↔ (4 < 𝑛 ∨ 4 = 𝑛)))
19 pm3.35 799 . . . . . . . . . . . . . . . . . 18 ((4 < 𝑛 ∧ (4 < 𝑛𝑛 ∈ GoldbachEven )) → 𝑛 ∈ GoldbachEven )
20 isgbe 43793 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ GoldbachEven ↔ (𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
21 simp3 1130 . . . . . . . . . . . . . . . . . . . . . . . 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 3271 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Even ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
2423reximdva 3271 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ Even → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
2524imp 407 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))
2620, 25sylbi 218 . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . 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 16024 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℙ
32 2p2e4 11760 . . . . . . . . . . . . . . . . . . . 20 (2 + 2) = 4
3332eqcomi 2827 . . . . . . . . . . . . . . . . . . 19 4 = (2 + 2)
34 rspceov 7192 . . . . . . . . . . . . . . . . . . 19 ((2 ∈ ℙ ∧ 2 ∈ ℙ ∧ 4 = (2 + 2)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞))
3531, 31, 33, 34mp3an 1452 . . . . . . . . . . . . . . . . . 18 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞)
36 eqeq1 2822 . . . . . . . . . . . . . . . . . . 19 (4 = 𝑛 → (4 = (𝑝 + 𝑞) ↔ 𝑛 = (𝑝 + 𝑞)))
37362rexbidv 3297 . . . . . . . . . . . . . . . . . 18 (4 = 𝑛 → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞) ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
3835, 37mpbii 234 . . . . . . . . . . . . . . . . 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 851 . . . . . . . . . . . . . 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 241 . . . . . . . . . . . 12 (𝑛 ∈ Even → (4 ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
4415, 43syl5bi 243 . . . . . . . . . . 11 (𝑛 ∈ Even → ((3 + 1) ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
4513, 44sylbid 241 . . . . . . . . . 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 43750 . . . . . . . . . . . 12 3 ∈ Odd
48 eleq1 2897 . . . . . . . . . . . 12 (3 = 𝑛 → (3 ∈ Odd ↔ 𝑛 ∈ Odd ))
4947, 48mpbii 234 . . . . . . . . . . 11 (3 = 𝑛𝑛 ∈ Odd )
50 oddneven 43686 . . . . . . . . . . 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 851 . . . . . . . 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 241 . . . . . 6 (𝑛 ∈ Even → (3 ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
566, 55syl5bi 243 . . . . 5 (𝑛 ∈ Even → ((2 + 1) ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
574, 56sylbid 241 . . . 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 11800 . . . . . . . 8 2 < 4
60 2re 11699 . . . . . . . . . 10 2 ∈ ℝ
6160a1i 11 . . . . . . . . 9 (𝑛 ∈ Even → 2 ∈ ℝ)
62 lttr 10705 . . . . . . . . 9 ((2 ∈ ℝ ∧ 4 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
6361, 17, 9, 62syl3anc 1363 . . . . . . . 8 (𝑛 ∈ Even → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
6459, 63mpani 692 . . . . . . 7 (𝑛 ∈ Even → (4 < 𝑛 → 2 < 𝑛))
6564imp 407 . . . . . 6 ((𝑛 ∈ Even ∧ 4 < 𝑛) → 2 < 𝑛)
66 simpll 763 . . . . . . . . 9 (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → 𝑛 ∈ Even )
67 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
6867anim1i 614 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
6968adantr 481 . . . . . . . . . . . . . . 15 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
70 simpll 763 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 ∈ Even ∧ 4 < 𝑛))
7170anim1i 614 . . . . . . . . . . . . . . . 16 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
72 df-3an 1081 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Even ∧ 4 < 𝑛𝑛 = (𝑝 + 𝑞)) ↔ ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
7371, 72sylibr 235 . . . . . . . . . . . . . . 15 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑛 ∈ Even ∧ 4 < 𝑛𝑛 = (𝑝 + 𝑞)))
74 sbgoldbaltlem2 43822 . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . 14 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → 𝑛 = (𝑝 + 𝑞))
77 df-3an 1081 . . . . . . . . . . . . . 14 ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) ↔ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ) ∧ 𝑛 = (𝑝 + 𝑞)))
7875, 76, 77sylanbrc 583 . . . . . . . . . . . . 13 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)))
7978ex 413 . . . . . . . . . . . 12 ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 = (𝑝 + 𝑞) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
8079reximdva 3271 . . . . . . . . . . 11 (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
8180reximdva 3271 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ 4 < 𝑛) → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
8281imp 407 . . . . . . . . 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 512 . . . . . . . 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 413 . . . . . . 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 253 . . . . . 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 413 . . . 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 213 . 2 (𝑛 ∈ Even → ((4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
9089ralbiia 3161 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 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136   class class class wbr 5057  (class class class)co 7145  cr 10524  1c1 10526   + caddc 10528   < clt 10663  cle 10664  2c2 11680  3c3 11681  4c4 11682  cz 11969  cprime 16003   Even ceven 43666   Odd codd 43667   GoldbachEven cgbe 43787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-dvds 15596  df-prm 16004  df-even 43668  df-odd 43669  df-gbe 43790
This theorem is referenced by:  sbgoldbb  43824  sbgoldbmb  43828
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