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Theorem sbgoldbalt 47768
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 12649 . . . . . 6 2 ∈ ℤ
2 evenz 47617 . . . . . 6 (𝑛 ∈ Even → 𝑛 ∈ ℤ)
3 zltp1le 12667 . . . . . 6 ((2 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
41, 2, 3sylancr 587 . . . . 5 (𝑛 ∈ Even → (2 < 𝑛 ↔ (2 + 1) ≤ 𝑛))
5 2p1e3 12408 . . . . . . 7 (2 + 1) = 3
65breq1i 5150 . . . . . 6 ((2 + 1) ≤ 𝑛 ↔ 3 ≤ 𝑛)
7 3re 12346 . . . . . . . . 9 3 ∈ ℝ
87a1i 11 . . . . . . . 8 (𝑛 ∈ Even → 3 ∈ ℝ)
92zred 12722 . . . . . . . 8 (𝑛 ∈ Even → 𝑛 ∈ ℝ)
108, 9leloed 11404 . . . . . . 7 (𝑛 ∈ Even → (3 ≤ 𝑛 ↔ (3 < 𝑛 ∨ 3 = 𝑛)))
11 3z 12650 . . . . . . . . . . . 12 3 ∈ ℤ
12 zltp1le 12667 . . . . . . . . . . . 12 ((3 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
1311, 2, 12sylancr 587 . . . . . . . . . . 11 (𝑛 ∈ Even → (3 < 𝑛 ↔ (3 + 1) ≤ 𝑛))
14 3p1e4 12411 . . . . . . . . . . . . 13 (3 + 1) = 4
1514breq1i 5150 . . . . . . . . . . . 12 ((3 + 1) ≤ 𝑛 ↔ 4 ≤ 𝑛)
16 4re 12350 . . . . . . . . . . . . . . 15 4 ∈ ℝ
1716a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ Even → 4 ∈ ℝ)
1817, 9leloed 11404 . . . . . . . . . . . . 13 (𝑛 ∈ Even → (4 ≤ 𝑛 ↔ (4 < 𝑛 ∨ 4 = 𝑛)))
19 pm3.35 803 . . . . . . . . . . . . . . . . . 18 ((4 < 𝑛 ∧ (4 < 𝑛𝑛 ∈ GoldbachEven )) → 𝑛 ∈ GoldbachEven )
20 isgbe 47738 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ GoldbachEven ↔ (𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
21 simp3 1139 . . . . . . . . . . . . . . . . . . . . . . . 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 3168 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ Even ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
2423reximdva 3168 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ Even → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
2524imp 406 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))
2620, 25sylbi 217 . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . 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 16729 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℙ
32 2p2e4 12401 . . . . . . . . . . . . . . . . . . . 20 (2 + 2) = 4
3332eqcomi 2746 . . . . . . . . . . . . . . . . . . 19 4 = (2 + 2)
34 rspceov 7480 . . . . . . . . . . . . . . . . . . 19 ((2 ∈ ℙ ∧ 2 ∈ ℙ ∧ 4 = (2 + 2)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞))
3531, 31, 33, 34mp3an 1463 . . . . . . . . . . . . . . . . . 18 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞)
36 eqeq1 2741 . . . . . . . . . . . . . . . . . . 19 (4 = 𝑛 → (4 = (𝑝 + 𝑞) ↔ 𝑛 = (𝑝 + 𝑞)))
37362rexbidv 3222 . . . . . . . . . . . . . . . . . 18 (4 = 𝑛 → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 4 = (𝑝 + 𝑞) ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
3835, 37mpbii 233 . . . . . . . . . . . . . . . . 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 858 . . . . . . . . . . . . . 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 240 . . . . . . . . . . . 12 (𝑛 ∈ Even → (4 ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
4415, 43biimtrid 242 . . . . . . . . . . 11 (𝑛 ∈ Even → ((3 + 1) ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
4513, 44sylbid 240 . . . . . . . . . 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 47695 . . . . . . . . . . . 12 3 ∈ Odd
48 eleq1 2829 . . . . . . . . . . . 12 (3 = 𝑛 → (3 ∈ Odd ↔ 𝑛 ∈ Odd ))
4947, 48mpbii 233 . . . . . . . . . . 11 (3 = 𝑛𝑛 ∈ Odd )
50 oddneven 47631 . . . . . . . . . . 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 858 . . . . . . . 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 240 . . . . . 6 (𝑛 ∈ Even → (3 ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
566, 55biimtrid 242 . . . . 5 (𝑛 ∈ Even → ((2 + 1) ≤ 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
574, 56sylbid 240 . . . 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 12441 . . . . . . . 8 2 < 4
60 2re 12340 . . . . . . . . . 10 2 ∈ ℝ
6160a1i 11 . . . . . . . . 9 (𝑛 ∈ Even → 2 ∈ ℝ)
62 lttr 11337 . . . . . . . . 9 ((2 ∈ ℝ ∧ 4 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
6361, 17, 9, 62syl3anc 1373 . . . . . . . 8 (𝑛 ∈ Even → ((2 < 4 ∧ 4 < 𝑛) → 2 < 𝑛))
6459, 63mpani 696 . . . . . . 7 (𝑛 ∈ Even → (4 < 𝑛 → 2 < 𝑛))
6564imp 406 . . . . . 6 ((𝑛 ∈ Even ∧ 4 < 𝑛) → 2 < 𝑛)
66 simpll 767 . . . . . . . . 9 (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)) → 𝑛 ∈ Even )
67 simpr 484 . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 15 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ))
70 simpll 767 . . . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . . 15 (((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) ∧ 𝑛 = (𝑝 + 𝑞)) → (𝑛 ∈ Even ∧ 4 < 𝑛𝑛 = (𝑝 + 𝑞)))
74 sbgoldbaltlem2 47767 . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 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 412 . . . . . . . . . . . 12 ((((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) ∧ 𝑞 ∈ ℙ) → (𝑛 = (𝑝 + 𝑞) → (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
8079reximdva 3168 . . . . . . . . . . 11 (((𝑛 ∈ Even ∧ 4 < 𝑛) ∧ 𝑝 ∈ ℙ) → (∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
8180reximdva 3168 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ 4 < 𝑛) → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑛 = (𝑝 + 𝑞))))
8281imp 406 . . . . . . . . 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 511 . . . . . . . 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 412 . . . . . . 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, 20imbitrrdi 252 . . . . . 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 412 . . . 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 212 . 2 (𝑛 ∈ Even → ((4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))))
9089ralbiia 3091 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 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070   class class class wbr 5143  (class class class)co 7431  cr 11154  1c1 11156   + caddc 11158   < clt 11295  cle 11296  2c2 12321  3c3 12322  4c4 12323  cz 12613  cprime 16708   Even ceven 47611   Odd codd 47612   GoldbachEven cgbe 47732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-dvds 16291  df-prm 16709  df-even 47613  df-odd 47614  df-gbe 47735
This theorem is referenced by:  sbgoldbb  47769  sbgoldbmb  47773
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