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Theorem sbgoldbalt 45963
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 12535 . . . . . 6 2 ∈ ℤ
2 evenz 45812 . . . . . 6 (𝑛 ∈ Even → 𝑛 ∈ ℤ)
3 zltp1le 12553 . . . . . 6 ((2 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
41, 2, 3sylancr 587 . . . . 5 (𝑛 ∈ Even → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
5 2p1e3 12295 . . . . . . 7 (2 + 1) = 3
65breq1i 5112 . . . . . 6 ((2 + 1) ≤ 𝑛 ↔ 3 ≤ 𝑛)
7 3re 12233 . . . . . . . . 9 3 ∈ ℝ
87a1i 11 . . . . . . . 8 (𝑛 ∈ Even → 3 ∈ ℝ)
92zred 12607 . . . . . . . 8 (𝑛 ∈ Even → 𝑛 ∈ ℝ)
108, 9leloed 11298 . . . . . . 7 (𝑛 ∈ Even → (3 ≤ 𝑛 ↔ (3 < 𝑛 ∨ 3 = 𝑛)))
11 3z 12536 . . . . . . . . . . . 12 3 ∈ ℤ
12 zltp1le 12553 . . . . . . . . . . . 12 ((3 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
1311, 2, 12sylancr 587 . . . . . . . . . . 11 (𝑛 ∈ Even → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
14 3p1e4 12298 . . . . . . . . . . . . 13 (3 + 1) = 4
1514breq1i 5112 . . . . . . . . . . . 12 ((3 + 1) ≤ 𝑛 ↔ 4 ≤ 𝑛)
16 4re 12237 . . . . . . . . . . . . . . 15 4 ∈ ℝ
1716a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ Even → 4 ∈ ℝ)
1817, 9leloed 11298 . . . . . . . . . . . . 13 (𝑛 ∈ Even → (4 ≤ 𝑛 ↔ (4 < 𝑛 ∨ 4 = 𝑛)))
19 pm3.35 801 . . . . . . . . . . . . . . . . . 18 ((4 < 𝑛 ∧ (4 < 𝑛𝑛 ∈ GoldbachEven )) → 𝑛 ∈ GoldbachEven )
20 isgbe 45933 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ GoldbachEven ↔ (𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
21 simp3 1138 . . . . . . . . . . . . . . . . . . . . . . . 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 3165 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Even ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
2423reximdva 3165 . . . . . . . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . . . . . . . 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 16568 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℙ
32 2p2e4 12288 . . . . . . . . . . . . . . . . . . . 20 (2 + 2) = 4
3332eqcomi 2745 . . . . . . . . . . . . . . . . . . 19 4 = (2 + 2)
34 rspceov 7404 . . . . . . . . . . . . . . . . . . 19 ((2 ∈ ℙ ∧ 2 ∈ ℙ ∧ 4 = (2 + 2)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞))
3531, 31, 33, 34mp3an 1461 . . . . . . . . . . . . . . . . . 18 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞)
36 eqeq1 2740 . . . . . . . . . . . . . . . . . . 19 (4 = 𝑛 → (4 = (𝑝 + 𝑞) ↔ 𝑛 = (𝑝 + 𝑞)))
37362rexbidv 3213 . . . . . . . . . . . . . . . . . 18 (4 = 𝑛 → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞) ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
3835, 37mpbii 232 . . . . . . . . . . . . . . . . 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 855 . . . . . . . . . . . . . 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 239 . . . . . . . . . . . 12 (𝑛 ∈ Even → (4 ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
4415, 43biimtrid 241 . . . . . . . . . . 11 (𝑛 ∈ Even → ((3 + 1) ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
4513, 44sylbid 239 . . . . . . . . . 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 45890 . . . . . . . . . . . 12 3 ∈ Odd
48 eleq1 2825 . . . . . . . . . . . 12 (3 = 𝑛 → (3 ∈ Odd ↔ 𝑛 ∈ Odd ))
4947, 48mpbii 232 . . . . . . . . . . 11 (3 = 𝑛𝑛 ∈ Odd )
50 oddneven 45826 . . . . . . . . . . 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 855 . . . . . . . 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 239 . . . . . 6 (𝑛 ∈ Even → (3 ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
566, 55biimtrid 241 . . . . 5 (𝑛 ∈ Even → ((2 + 1) ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
574, 56sylbid 239 . . . 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 12328 . . . . . . . 8 2 < 4
60 2re 12227 . . . . . . . . . 10 2 ∈ ℝ
6160a1i 11 . . . . . . . . 9 (𝑛 ∈ Even → 2 ∈ ℝ)
62 lttr 11231 . . . . . . . . 9 ((2 ∈ ℝ ∧ 4 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
6361, 17, 9, 62syl3anc 1371 . . . . . . . 8 (𝑛 ∈ Even → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
6459, 63mpani 694 . . . . . . 7 (𝑛 ∈ Even → (4 < 𝑛 → 2 < 𝑛))
6564imp 407 . . . . . 6 ((𝑛 ∈ Even ∧ 4 < 𝑛) → 2 < 𝑛)
66 simpll 765 . . . . . . . . 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 615 . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 ∈ Even ∧ 4 < 𝑛))
7170anim1i 615 . . . . . . . . . . . . . . . 16 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
72 df-3an 1089 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Even ∧ 4 < 𝑛𝑛 = (𝑝 + 𝑞)) ↔ ((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑛 = (𝑝 + 𝑞)))
7371, 72sylibr 233 . . . . . . . . . . . . . . 15 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑛 ∈ Even ∧ 4 < 𝑛𝑛 = (𝑝 + 𝑞)))
74 sbgoldbaltlem2 45962 . . . . . . . . . . . . . . 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 1089 . . . . . . . . . . . . . 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 3165 . . . . . . . . . . 11 (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
8180reximdva 3165 . . . . . . . . . 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 251 . . . . . 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 211 . 2 (𝑛 ∈ Even → ((4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
9089ralbiia 3094 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 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073   class class class wbr 5105  (class class class)co 7357  cr 11050  1c1 11052   + caddc 11054   < clt 11189  cle 11190  2c2 12208  3c3 12209  4c4 12210  cz 12499  cprime 16547   Even ceven 45806   Odd codd 45807   GoldbachEven cgbe 45927
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-dvds 16137  df-prm 16548  df-even 45808  df-odd 45809  df-gbe 45930
This theorem is referenced by:  sbgoldbb  45964  sbgoldbmb  45968
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