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Theorem List for Metamath Proof Explorer - 40101-40200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem6gbe 40101 6 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
6 ∈ GoldbachEven

Theorem7gbo 40102 7 is an odd Goldbach number. (Contributed by AV, 20-Jul-2020.)
7 ∈ GoldbachOdd

Theorem8gbe 40103 8 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
8 ∈ GoldbachEven

Theorem9gboa 40104 9 is an odd Goldbach number. (Contributed by AV, 26-Jul-2020.)
9 ∈ GoldbachOddALTV

Theorem11gboa 40105 11 is an odd Goldbach number. (Contributed by AV, 29-Jul-2020.)
11 ∈ GoldbachOddALTV

Theoremstgoldbwt 40106 If the strong ternary Goldbach conjecture is valid, then the weak ternary Goldbach conjecture holds, too. (Contributed by AV, 27-Jul-2020.)
(∀𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOddALTV ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOdd ))

Theorembgoldbwt 40107* If the binary Goldbach conjecture is valid, then the (weak) ternary Goldbach conjecture holds, too. (Contributed by AV, 20-Jul-2020.)
(∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) → ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd ))

Theorembgoldbst 40108* If the binary Goldbach conjecture is valid, then the (strong) ternary Goldbach conjecture holds, too. (Contributed by AV, 26-Jul-2020.)
(∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) → ∀𝑚 ∈ Odd (7 < 𝑚𝑚 ∈ GoldbachOddALTV ))

Theoremsgoldbaltlem1 40109 Lemma 1 for sgoldbalt 40111: If an even number greater than 4 is the sum of two primes, one of the prime summands must be odd, i.e. not 2. (Contributed by AV, 22-Jul-2020.)
((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))

Theoremsgoldbaltlem2 40110 Lemma 2 for sgoldbalt 40111: If an even number greater than 4 is the sum of two primes, the primes must be odd, i.e. not 2. (Contributed by AV, 22-Jul-2020.)
((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → (𝑃 ∈ Odd ∧ 𝑄 ∈ Odd )))

Theoremsgoldbalt 40111* An alternate (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.)
(∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))

Theoremnnsum3primes4 40112* 4 is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)
𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 3 ∧ 4 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))

Theoremnnsum4primes4 40113* 4 is the sum of at most 4 (actually 2) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 4 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))

Theoremnnsum3primesprm 40114* Every prime is "the sum of at most 3" (actually one - the prime itself) primes. (Contributed by AV, 2-Aug-2020.) (Proof shortened by AV, 17-Apr-2021.)
(𝑃 ∈ ℙ → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))

Theoremnnsum4primesprm 40115* Every prime is "the sum of at most 4" (actually one - the prime itself) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
(𝑃 ∈ ℙ → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))

Theoremnnsum3primesgbe 40116* Any even Goldbach number is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)
(𝑁 ∈ GoldbachEven → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))

Theoremnnsum4primesgbe 40117* Any even Goldbach number is the sum of at most 4 (actually 2) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
(𝑁 ∈ GoldbachEven → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))

Theoremnnsum3primesle9 40118* Every integer greater than 1 and less than or equal to 8 is the sum of at most 3 primes. (Contributed by AV, 2-Aug-2020.)
((𝑁 ∈ (ℤ‘2) ∧ 𝑁 ≤ 8) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))

Theoremnnsum4primesle9 40119* Every integer greater than 1 and less than or equal to 8 is the sum of at most 4 primes. (Contributed by AV, 24-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
((𝑁 ∈ (ℤ‘2) ∧ 𝑁 ≤ 8) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))

Theoremnnsum4primesodd 40120* If the (weak) ternary Goldbach conjecture is valid, then every odd integer greater than 5 is the sum of 3 primes. (Contributed by AV, 2-Jul-2020.)
(∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd ) → ((𝑁 ∈ (ℤ‘6) ∧ 𝑁 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))𝑁 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))

Theoremnnsum4primesoddALTV 40121* If the (strong) ternary Goldbach conjecture is valid, then every odd integer greater than 7 is the sum of 3 primes. (Contributed by AV, 26-Jul-2020.)
(∀𝑚 ∈ Odd (7 < 𝑚𝑚 ∈ GoldbachOddALTV ) → ((𝑁 ∈ (ℤ‘8) ∧ 𝑁 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))𝑁 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))

Theoremevengpop3 40122* If the (weak) ternary Goldbach conjecture is valid, then every even integer greater than 8 is the sum of an odd Goldbach number and 3. (Contributed by AV, 24-Jul-2020.)
(∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd ) → ((𝑁 ∈ (ℤ‘9) ∧ 𝑁 ∈ Even ) → ∃𝑜 ∈ GoldbachOdd 𝑁 = (𝑜 + 3)))

Theoremevengpoap3 40123* If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of an odd Goldbach number and 3. (Contributed by AV, 27-Jul-2020.) (Proof shortened by AV, 15-Sep-2021.)
(∀𝑚 ∈ Odd (7 < 𝑚𝑚 ∈ GoldbachOddALTV ) → ((𝑁 ∈ (ℤ12) ∧ 𝑁 ∈ Even ) → ∃𝑜 ∈ GoldbachOddALTV 𝑁 = (𝑜 + 3)))

Theoremnnsum4primeseven 40124* If the (weak) ternary Goldbach conjecture is valid, then every even integer greater than 8 is the sum of 4 primes. (Contributed by AV, 25-Jul-2020.)
(∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd ) → ((𝑁 ∈ (ℤ‘9) ∧ 𝑁 ∈ Even ) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓𝑘)))

Theoremnnsum4primesevenALTV 40125* If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of 4 primes. (Contributed by AV, 27-Jul-2020.)
(∀𝑚 ∈ Odd (7 < 𝑚𝑚 ∈ GoldbachOddALTV ) → ((𝑁 ∈ (ℤ12) ∧ 𝑁 ∈ Even ) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓𝑘)))

Theoremwtgoldbnnsum4prm 40126* If the (weak) ternary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 4 primes, showing that Schnirelmann's constant would be less than or equal to 4. See corollary 1.1 in [Helfgott] p. 4. (Contributed by AV, 25-Jul-2020.)
(∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd ) → ∀𝑛 ∈ (ℤ‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))

Theoremstgoldbnnsum4prm 40127* If the (strong) ternary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 4 primes. (Contributed by AV, 27-Jul-2020.)
(∀𝑚 ∈ Odd (7 < 𝑚𝑚 ∈ GoldbachOddALTV ) → ∀𝑛 ∈ (ℤ‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))

Theorembgoldbnnsum3prm 40128* If the binary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 3 primes, showing that Schnirelmann's constant would be equal to 3. (Contributed by AV, 2-Aug-2020.)
(∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))

Theorembgoldbtbndlem1 40129 Lemma 1 for bgoldbtbnd 40133: the odd numbers between 7 and 13 (exclusive) are (strong) odd Goldbach numbers. (Contributed by AV, 29-Jul-2020.)
((𝑁 ∈ Odd ∧ 7 < 𝑁𝑁 ∈ (7[,)13)) → 𝑁 ∈ GoldbachOddALTV )

Theorembgoldbtbndlem2 40130* Lemma 2 for bgoldbtbnd 40133. (Contributed by AV, 1-Aug-2020.)
(𝜑𝑀 ∈ (ℤ11))    &   (𝜑𝑁 ∈ (ℤ11))    &   (𝜑 → ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑁) → 𝑛 ∈ GoldbachEven ))    &   (𝜑𝐷 ∈ (ℤ‘3))    &   (𝜑𝐹 ∈ (RePart‘𝐷))    &   (𝜑 → ∀𝑖 ∈ (0..^𝐷)((𝐹𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝐹‘(𝑖 + 1)) − (𝐹𝑖)) < (𝑁 − 4) ∧ 4 < ((𝐹‘(𝑖 + 1)) − (𝐹𝑖))))    &   (𝜑 → (𝐹‘0) = 7)    &   (𝜑 → (𝐹‘1) = 13)    &   (𝜑𝑀 < (𝐹𝐷))    &   𝑆 = (𝑋 − (𝐹‘(𝐼 − 1)))       ((𝜑𝑋 ∈ Odd ∧ 𝐼 ∈ (1..^𝐷)) → ((𝑋 ∈ ((𝐹𝐼)[,)(𝐹‘(𝐼 + 1))) ∧ (𝑋 − (𝐹𝐼)) ≤ 4) → (𝑆 ∈ Even ∧ 𝑆 < 𝑁 ∧ 4 < 𝑆)))

Theorembgoldbtbndlem3 40131* Lemma 3 for bgoldbtbnd 40133. (Contributed by AV, 1-Aug-2020.)
(𝜑𝑀 ∈ (ℤ11))    &   (𝜑𝑁 ∈ (ℤ11))    &   (𝜑 → ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑁) → 𝑛 ∈ GoldbachEven ))    &   (𝜑𝐷 ∈ (ℤ‘3))    &   (𝜑𝐹 ∈ (RePart‘𝐷))    &   (𝜑 → ∀𝑖 ∈ (0..^𝐷)((𝐹𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝐹‘(𝑖 + 1)) − (𝐹𝑖)) < (𝑁 − 4) ∧ 4 < ((𝐹‘(𝑖 + 1)) − (𝐹𝑖))))    &   (𝜑 → (𝐹‘0) = 7)    &   (𝜑 → (𝐹‘1) = 13)    &   (𝜑𝑀 < (𝐹𝐷))    &   (𝜑 → (𝐹𝐷) ∈ ℝ)    &   𝑆 = (𝑋 − (𝐹𝐼))       ((𝜑𝑋 ∈ Odd ∧ 𝐼 ∈ (1..^𝐷)) → ((𝑋 ∈ ((𝐹𝐼)[,)(𝐹‘(𝐼 + 1))) ∧ 4 < 𝑆) → (𝑆 ∈ Even ∧ 𝑆 < 𝑁 ∧ 4 < 𝑆)))

Theorembgoldbtbndlem4 40132* Lemma 4 for bgoldbtbnd 40133. (Contributed by AV, 1-Aug-2020.)
(𝜑𝑀 ∈ (ℤ11))    &   (𝜑𝑁 ∈ (ℤ11))    &   (𝜑 → ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑁) → 𝑛 ∈ GoldbachEven ))    &   (𝜑𝐷 ∈ (ℤ‘3))    &   (𝜑𝐹 ∈ (RePart‘𝐷))    &   (𝜑 → ∀𝑖 ∈ (0..^𝐷)((𝐹𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝐹‘(𝑖 + 1)) − (𝐹𝑖)) < (𝑁 − 4) ∧ 4 < ((𝐹‘(𝑖 + 1)) − (𝐹𝑖))))    &   (𝜑 → (𝐹‘0) = 7)    &   (𝜑 → (𝐹‘1) = 13)    &   (𝜑𝑀 < (𝐹𝐷))    &   (𝜑 → (𝐹𝐷) ∈ ℝ)       (((𝜑𝐼 ∈ (1..^𝐷)) ∧ 𝑋 ∈ Odd ) → ((𝑋 ∈ ((𝐹𝐼)[,)(𝐹‘(𝐼 + 1))) ∧ (𝑋 − (𝐹𝐼)) ≤ 4) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑋 = ((𝑝 + 𝑞) + 𝑟))))

Theorembgoldbtbnd 40133* If the binary Goldbach conjecture is valid up to an integer 𝑁, and there is a series ("ladder") of primes with a difference of at most 𝑁 up to an integer 𝑀, then the strong ternary Goldbach conjecture is valid up to 𝑀, see section 1.2.2 in [Helfgott] p. 4 with N = 4 x 10^18, taken from [OeSilva], and M = 8.875 x 10^30. (Contributed by AV, 1-Aug-2020.)
(𝜑𝑀 ∈ (ℤ11))    &   (𝜑𝑁 ∈ (ℤ11))    &   (𝜑 → ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑁) → 𝑛 ∈ GoldbachEven ))    &   (𝜑𝐷 ∈ (ℤ‘3))    &   (𝜑𝐹 ∈ (RePart‘𝐷))    &   (𝜑 → ∀𝑖 ∈ (0..^𝐷)((𝐹𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝐹‘(𝑖 + 1)) − (𝐹𝑖)) < (𝑁 − 4) ∧ 4 < ((𝐹‘(𝑖 + 1)) − (𝐹𝑖))))    &   (𝜑 → (𝐹‘0) = 7)    &   (𝜑 → (𝐹‘1) = 13)    &   (𝜑𝑀 < (𝐹𝐷))    &   (𝜑 → (𝐹𝐷) ∈ ℝ)       (𝜑 → ∀𝑛 ∈ Odd ((7 < 𝑛𝑛 < 𝑀) → 𝑛 ∈ GoldbachOddALTV ))

Axiomax-bgbltosilva 40134 The binary Goldbach conjecture is valid for all even numbers less than or equal to 4x10^18, see result of [OeSilva] p. ?. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.)
((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 ≤ (4 · (10↑18))) → 𝑁 ∈ GoldbachEven )

Theorembgoldbachlt 40135* The binary Goldbach conjecture is valid for small even numbers (i.e. for all even numbers less than or equal to a fixed big 𝑚). This is verified for m = 4 x 10^18 by Oliveira e Silva, see ax-bgbltosilva 40134. (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.)
𝑚 ∈ ℕ ((4 · (10↑18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ))

Axiomax-hgprmladder 40136 There is a partition ("ladder") of primes from 7 to 8.8 x 10^30 with parts ("rungs") having lengths of at least 4 and at most N - 4, see section 1.2.2 in [Helfgott] p. 4. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.)
𝑑 ∈ (ℤ‘3)∃𝑓 ∈ (RePart‘𝑑)(((𝑓‘0) = 7 ∧ (𝑓‘1) = 13 ∧ (𝑓𝑑) = (89 · (10↑29))) ∧ ∀𝑖 ∈ (0..^𝑑)((𝑓𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓𝑖)) < ((4 · (10↑18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓𝑖))))

Theoremtgblthelfgott 40137 The ternary Goldbach conjecture is valid for all odd numbers less than 8.8 x 10^30 (actually 8.875694 x 10^30, see section 1.2.2 in [Helfgott] p. 4, using bgoldbachlt 40135, ax-hgprmladder 40136 and bgoldbtbnd 40133. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)
((𝑁 ∈ Odd ∧ 7 < 𝑁𝑁 < (88 · (10↑29))) → 𝑁 ∈ GoldbachOddALTV )

Theoremtgoldbachlt 40138* The ternary Goldbach conjecture is valid for small odd numbers (i.e. for all odd numbers less than a fixed big 𝑚 greater than 8 x 10^30). This is verified for m = 8.875694 x 10^30 by Helfgott, see tgblthelfgott 40137. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)
𝑚 ∈ ℕ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑛 ∈ Odd ((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ))

Axiomax-tgoldbachgt 40139* The ternary Goldbach conjecture is valid for big odd numbers (i.e. for all odd numbers greater than a fixed 𝑚). This is proven by Helfgott (see section 7.4 in [Helfgott] p. 70) for m = 10^27. Temporarily provided as "axiom". (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.)
𝑚 ∈ ℕ (𝑚 ≤ (10↑27) ∧ ∀𝑛 ∈ Odd (𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ))

Theoremtgoldbach 40140 The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 40138 and ax-tgoldbachgt 40139. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.)
𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOddALTV )

Axiomax-bgbltosilvaOLD 40141 Obsolete version of ax-bgbltosilva 40134 as of 9-Sep-2021. (Contributed by AV, 3-Aug-2020.) (New usage is discouraged.)
((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 ≤ (4 · (10↑18))) → 𝑁 ∈ GoldbachEven )

TheorembgoldbachltOLD 40142* Obsolete version of bgoldbachlt 40135 as of 9-Sep-2021. (Contributed by AV, 3-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑚 ∈ ℕ ((4 · (10↑18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ))

Axiomax-hgprmladderOLD 40143 Obsolete version of ax-hgprmladder 40136 as of 9-Sep-2021. (Contributed by AV, 3-Aug-2020.) (New usage is discouraged.)
𝑑 ∈ (ℤ‘3)∃𝑓 ∈ (RePart‘𝑑)(((𝑓‘0) = 7 ∧ (𝑓‘1) = 13 ∧ (𝑓𝑑) = (89 · (10↑29))) ∧ ∀𝑖 ∈ (0..^𝑑)((𝑓𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓𝑖)) < ((4 · (10↑18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓𝑖))))

TheoremtgblthelfgottOLD 40144 Obsolete version of tgblthelfgott 40137 as of 9-Sep-2021. (Contributed by AV, 4-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑁 ∈ Odd ∧ 7 < 𝑁𝑁 < (88 · (10↑29))) → 𝑁 ∈ GoldbachOddALTV )

TheoremtgoldbachltOLD 40145* Obsolete version of tgoldbachlt 40138 as of 9-Sep-2021. (Contributed by AV, 4-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑚 ∈ ℕ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑛 ∈ Odd ((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV ))

Axiomax-tgoldbachgtOLD 40146* Obsolete version of ax-tgoldbachgt 40139 as of 9-Sep-2021. (Contributed by AV, 2-Aug-2020.) (New usage is discouraged.)
𝑚 ∈ ℕ (𝑚 ≤ (10↑27) ∧ ∀𝑛 ∈ Odd (𝑚 < 𝑛𝑛 ∈ GoldbachOddALTV ))

TheoremtgoldbachOLD 40147 Obsolete version of tgoldbach 40140 as of 9-Sep-2021. (Contributed by AV, 2-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOddALTV )

20.34.6  Words over a set (extension)

20.34.6.1  Truncated words

Theoremwrdred1 40148 A word truncated by a symbol is a word. (Contributed by AV, 29-Jan-2021.)
(𝐹 ∈ Word 𝑆 → (𝐹 ↾ (0..^((#‘𝐹) − 1))) ∈ Word 𝑆)

Theoremwrdred1hash 40149 The length of a word truncated by a symbol. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.)
((𝐹 ∈ Word 𝑆 ∧ 1 ≤ (#‘𝐹)) → (#‘(𝐹 ↾ (0..^((#‘𝐹) − 1)))) = ((#‘𝐹) − 1))

20.34.6.2  Last symbol of a word (extension)

Theoremlswn0 40150 The last symbol of a not empty word exists. The empty set must be excluded as symbol, because otherwise, it cannot be distinguished between valid cases ( is the last symbol) and invalid cases ( means that no last symbol exists. This is because of the special definition of a function in set.mm. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
((𝑊 ∈ Word 𝑉 ∧ ∅ ∉ 𝑉 ∧ (#‘𝑊) ≠ 0) → ( lastS ‘𝑊) ≠ ∅)

20.34.6.3  Concatenations with singleton words (extension)

Theoremccatw2s1cl 40151 The concatenation of a word with two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉𝑌𝑉) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) ∈ Word 𝑉)

20.34.6.4  Prefixes of a word

In https://www.allacronyms.com/prefix/abbreviated, "pfx" is proposed as abbreviation for "prefix". Regarding the meaning of "prefix", it is different in computer science (automata theory/formal languages) compared with linguistics: in linguistics, a prefix has a meaning (see Wikipedia "Prefix" https://en.wikipedia.org/wiki/Prefix), whereas in computer science, a prefix is an arbitrary substring/subword starting at the beginning of a string/word (see Wikipedia "Substring" https://en.wikipedia.org/wiki/Substring#Prefix or https://math.stackexchange.com/questions/2190559/ is-there-standard-terminology-notation-for-the-prefix-of-a-word ).

Syntaxcpfx 40152 Syntax for the prefix operator.
class prefix

Definitiondf-pfx 40153* Define an operation which extracts prefixes of words, i.e. subwords starting at the beginning of a word. Definition in section 9.1 of [AhoHopUll] p. 318. "pfx" is used as label fragment. (Contributed by AV, 2-May-2020.)
prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))

Theorempfxval 40154 Value of a prefix. (Contributed by AV, 2-May-2020.)
((𝑆𝑉𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))

Theorempfx00 40155 A zero length prefix. (Contributed by AV, 2-May-2020.)
(𝑆 prefix 0) = ∅

Theorempfx0 40156 A prefix of an empty set is always the empty set. (Contributed by AV, 3-May-2020.)
(∅ prefix 𝐿) = ∅

Theorempfxcl 40157 Closure of the prefix extractor. (Contributed by AV, 2-May-2020.)
(𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝐿) ∈ Word 𝐴)

Theorempfxmpt 40158* Value of the prefix extractor as mapping. (Contributed by AV, 2-May-2020.)
((𝑆 ∈ Word 𝐴𝐿 ∈ (0...(#‘𝑆))) → (𝑆 prefix 𝐿) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆𝑥)))

Theorempfxres 40159 Value of the prefix extractor as restriction. Could replace swrd0val 13132. (Contributed by AV, 2-May-2020.)
((𝑆 ∈ Word 𝐴𝐿 ∈ (0...(#‘𝑆))) → (𝑆 prefix 𝐿) = (𝑆 ↾ (0..^𝐿)))

Theorempfxf 40160 A prefix of a word is a function from a half-open range of nonnegative integers of the same length as the prefix to the set of symbols for the original word. Could replace swrd0f 13138. (Contributed by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝑊))) → (𝑊 prefix 𝐿):(0..^𝐿)⟶𝑉)

Theorempfxfn 40161 Value of the prefix extractor as function with domain. (Contributed by AV, 2-May-2020.)
((𝑆 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝑆))) → (𝑆 prefix 𝐿) Fn (0..^𝐿))

Theorempfxlen 40162 Length of a prefix. Could replace swrd0len 13133. (Contributed by AV, 2-May-2020.)
((𝑆 ∈ Word 𝐴𝐿 ∈ (0...(#‘𝑆))) → (#‘(𝑆 prefix 𝐿)) = 𝐿)

Theorempfxid 40163 A word is a prefix of itself. (Contributed by AV, 2-May-2020.)
(𝑆 ∈ Word 𝐴 → (𝑆 prefix (#‘𝑆)) = 𝑆)

Theorempfxrn 40164 The range of a prefix of a word is a subset of the set of symbols for the word. (Contributed by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝑊))) → ran (𝑊 prefix 𝐿) ⊆ 𝑉)

Theorempfxn0 40165 A prefix consisting of at least one symbol is not empty. Could replace swrdn0 13141. (Contributed by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ ℕ ∧ 𝐿 ≤ (#‘𝑊)) → (𝑊 prefix 𝐿) ≠ ∅)

Theorempfxnd 40166 The value of the prefix extractor is the empty set (undefined) if the argument is not within the range of the word. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ ℕ0 ∧ (#‘𝑊) < 𝐿) → (𝑊 prefix 𝐿) = ∅)

Theorempfxlen0 40167 Length of a prefix of a word reduced by a single symbol. Could replace swrd0len0 13147. TODO-AV: Really useful? swrd0len0 13147 is only used in wwlknred 25989. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ ℕ0 ∧ (#‘𝑊) = (𝐿 + 1)) → (#‘(𝑊 prefix 𝐿)) = 𝐿)

Theoremaddlenrevpfx 40168 The sum of the lengths of two reversed parts of a word is the length of the word. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → ((#‘(𝑊 substr ⟨𝑀, (#‘𝑊)⟩)) + (#‘(𝑊 prefix 𝑀))) = (#‘𝑊))

Theoremaddlenpfx 40169 The sum of the lengths of two parts of a word is the length of the word. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → ((#‘(𝑊 prefix 𝑀)) + (#‘(𝑊 substr ⟨𝑀, (#‘𝑊)⟩))) = (#‘𝑊))

Theorempfxfv 40170 A symbol in a prefix of a word, indexed using the prefix' indices. Could replace swrd0fv 13150. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → ((𝑊 prefix 𝐿)‘𝐼) = (𝑊𝐼))

Theorempfxfv0 40171 The first symbol in a prefix of a word. Could replace swrd0fv0 13151. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (1...(#‘𝑊))) → ((𝑊 prefix 𝐿)‘0) = (𝑊‘0))

Theorempfxtrcfv 40172 A symbol in a word truncated by one symbol. Could replace swrdtrcfv 13152. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅ ∧ 𝐼 ∈ (0..^((#‘𝑊) − 1))) → ((𝑊 prefix ((#‘𝑊) − 1))‘𝐼) = (𝑊𝐼))

Theorempfxtrcfv0 40173 The first symbol in a word truncated by one symbol. Could replace swrdtrcfv0 13153. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → ((𝑊 prefix ((#‘𝑊) − 1))‘0) = (𝑊‘0))

Theorempfxfvlsw 40174 The last symbol in a (not empty) prefix of a word. Could replace swrd0fvlsw 13154. (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (1...(#‘𝑊))) → ( lastS ‘(𝑊 prefix 𝐿)) = (𝑊‘(𝐿 − 1)))

Theorempfxeq 40175* The prefixes of two words are equal iff they have the same length and the same symbols at each position. Could replace swrdeq 13155. (Contributed by AV, 4-May-2020.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 prefix 𝑀) = (𝑈 prefix 𝑁) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊𝑖) = (𝑈𝑖))))

Theorempfxtrcfvl 40176 The last symbol in a word truncated by one symbol. Could replace swrdtrcfvl 13161. (Contributed by AV, 5-May-2020.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → ( lastS ‘(𝑊 prefix ((#‘𝑊) − 1))) = (𝑊‘((#‘𝑊) − 2)))

Theorempfxsuffeqwrdeq 40177 Two words are equal if and only if they have the same prefix and the same suffix. Could replace 2swrdeqwrdeq 13164. (Contributed by AV, 5-May-2020.)
((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 = 𝑆 ↔ ((#‘𝑊) = (#‘𝑆) ∧ ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))))

Theorempfxsuff1eqwrdeq 40178 Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. Could replace 2swrd1eqwrdeq 13165. (Contributed by AV, 6-May-2020.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 prefix ((#‘𝑊) − 1)) = (𝑈 prefix ((#‘𝑊) − 1)) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))

Theoremdisjwrdpfx 40179* Sets of words are disjoint if each set contains extensions of distinct words of a fixed length. Could replace disjxwrd 13166. (Contributed by AV, 6-May-2020.)
Disj 𝑦𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}

Theoremccatpfx 40180 Joining a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.)
((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 prefix 𝑍))

Theorempfxccat1 40181 Recover the left half of a concatenated word. Could replace swrdccat1 13168. (Contributed by AV, 6-May-2020.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) prefix (#‘𝑆)) = 𝑆)

Theorempfx1 40182 A prefix of length 1. (Contributed by AV, 15-May-2020.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (𝑊 prefix 1) = ⟨“(𝑊‘0)”⟩)

Theorempfx2 40183 A prefix of length 2. (Contributed by AV, 15-May-2020.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)

Theorempfxswrd 40184 A prefix of a subword. Could replace swrd0swrd 13172. (Contributed by AV, 8-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → (𝐿 ∈ (0...(𝑁𝑀)) → ((𝑊 substr ⟨𝑀, 𝑁⟩) prefix 𝐿) = (𝑊 substr ⟨𝑀, (𝑀 + 𝐿)⟩)))

Theoremswrdpfx 40185 A subword of a prefix. Could replace swrdswrd0 13173. (Contributed by AV, 8-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊))) → ((𝐾 ∈ (0...𝑁) ∧ 𝐿 ∈ (𝐾...𝑁)) → ((𝑊 prefix 𝑁) substr ⟨𝐾, 𝐿⟩) = (𝑊 substr ⟨𝐾, 𝐿⟩)))

Theorempfxpfx 40186 A prefix of a prefix. Could replace swrd0swrd0 13174. (Contributed by AV, 8-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊)) ∧ 𝐿 ∈ (0...𝑁)) → ((𝑊 prefix 𝑁) prefix 𝐿) = (𝑊 prefix 𝐿))

Theorempfxpfxid 40187 A prefix of a prefix with the same length is the prefix. Could replace swrd0swrdid 13175. (Contributed by AV, 8-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊))) → ((𝑊 prefix 𝑁) prefix 𝑁) = (𝑊 prefix 𝑁))

Theorempfxcctswrd 40188 The concatenation of the prefix of a word and the rest of the word yields the word itself. Could replace wrdcctswrd 13176. (Contributed by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → ((𝑊 prefix 𝑀) ++ (𝑊 substr ⟨𝑀, (#‘𝑊)⟩)) = 𝑊)

Theoremlenpfxcctswrd 40189 The length of the concatenation of the prefix of a word and the rest of the word is the length of the word. Could replace lencctswrd 13177. (Contributed by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → (#‘((𝑊 prefix 𝑀) ++ (𝑊 substr ⟨𝑀, (#‘𝑊)⟩))) = (#‘𝑊))

Theoremlenrevpfxcctswrd 40190 The length of the concatenation of the rest of a word and the prefix of the word is the length of the word. Could replace lenrevcctswrd 13178. (Contributed by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → (#‘((𝑊 substr ⟨𝑀, (#‘𝑊)⟩) ++ (𝑊 prefix 𝑀))) = (#‘𝑊))

Theorempfxlswccat 40191 Reconstruct a nonempty word from its prefix and last symbol. Could replace swrdccatwrd 13179. (Contributed by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → ((𝑊 prefix ((#‘𝑊) − 1)) ++ ⟨“( lastS ‘𝑊)”⟩) = 𝑊)

Theoremccats1pfxeq 40192 The last symbol of a word concatenated with the word with the last symbol removed having results in the word itself. Could replace ccats1swrdeq 13180. (Contributed by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 prefix (#‘𝑊)) → 𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)))

Theoremccats1pfxeqrex 40193* There exists a symbol such that its concatenation with the prefix obtained by deleting the last symbol of a nonempty word results in the word itself. Could replace ccats1swrdeqrex 13189. (Contributed by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 prefix (#‘𝑊)) → ∃𝑠𝑉 𝑈 = (𝑊 ++ ⟨“𝑠”⟩)))

Theorempfxccatin12lem1 40194 Lemma 1 for pfxccatin12 40196. Could replace swrdccatin12lem2b 13196. (Contributed by AV, 9-May-2020.)
((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...𝑋)) → ((𝐾 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿𝑀))) → (𝐾 − (𝐿𝑀)) ∈ (0..^(𝑁𝐿))))

Theorempfxccatin12lem2 40195 Lemma 2 for pfxccatin12 40196. Could replace swrdccatin12lem2 13199. (Contributed by AV, 9-May-2020.)
𝐿 = (#‘𝐴)       (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐾 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝐾) = ((𝐵 prefix (𝑁𝐿))‘(𝐾 − (#‘(𝐴 substr ⟨𝑀, 𝐿⟩))))))

Theorempfxccatin12 40196 The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12 13201. (Contributed by AV, 9-May-2020.)
𝐿 = (#‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))))

Theorempfxccat3 40197 The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. Could replace swrdccat3 13202. (Contributed by AV, 10-May-2020.)
𝐿 = (#‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = if(𝑁𝐿, (𝐴 substr ⟨𝑀, 𝑁⟩), if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))))))

Theorempfxccatpfx1 40198 A prefix of a concatenation being a prefix of the first concatenated word. (Contributed by AV, 10-May-2020.)
𝐿 = (#‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ (0...𝐿)) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 prefix 𝑁))

Theorempfxccatpfx2 40199 A prefix of a concatenation of two words being the first word concatenated with a prefix of the second word. (Contributed by AV, 10-May-2020.)
𝐿 = (#‘𝐴)    &   𝑀 = (#‘𝐵)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 ++ (𝐵 prefix (𝑁𝐿))))

Theorempfxccat3a 40200 A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. Could replace swrdccat3a 13204. (Contributed by AV, 10-May-2020.)
𝐿 = (#‘𝐴)    &   𝑀 = (#‘𝐵)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + 𝑀)) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿))))))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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