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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | evennn02n 15701* | A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.) (Proof shortened by AV, 10-Jul-2022.) |
⊢ (𝑁 ∈ ℕ0 → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ0 (2 · 𝑛) = 𝑁)) | ||
Theorem | evennn2n 15702* | A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.) |
⊢ (𝑁 ∈ ℕ → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ (2 · 𝑛) = 𝑁)) | ||
Theorem | 2tp1odd 15703 | A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 = ((2 · 𝐴) + 1)) → ¬ 2 ∥ 𝐵) | ||
Theorem | mulsucdiv2z 15704 | An integer multiplied with its successor divided by 2 yields an integer, i.e. an integer multiplied with its successor is even. (Contributed by AV, 19-Jul-2021.) |
⊢ (𝑁 ∈ ℤ → ((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ) | ||
Theorem | sqoddm1div8z 15705 | A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (((𝑁↑2) − 1) / 8) ∈ ℤ) | ||
Theorem | 2teven 15706 | A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 = (2 · 𝐴)) → 2 ∥ 𝐵) | ||
Theorem | zeo5 15707 | An integer is either even or odd, version of zeo3 15688 avoiding the negation of the representation of an odd number. (Proposed by BJ, 21-Jun-2021.) (Contributed by AV, 26-Jun-2020.) |
⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ∨ 2 ∥ (𝑁 + 1))) | ||
Theorem | evend2 15708 | An integer is even iff its quotient with 2 is an integer. This is a representation of even numbers without using the divides relation, see zeo 12071 and zeo2 12072. (Contributed by AV, 22-Jun-2021.) |
⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ (𝑁 / 2) ∈ ℤ)) | ||
Theorem | oddp1d2 15709 | An integer is odd iff its successor divided by 2 is an integer. This is a representation of odd numbers without using the divides relation, see zeo 12071 and zeo2 12072. (Contributed by AV, 22-Jun-2021.) |
⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) | ||
Theorem | zob 15710 | Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.) |
⊢ (𝑁 ∈ ℤ → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 − 1) / 2) ∈ ℤ)) | ||
Theorem | oddm1d2 15711 | An integer is odd iff its predecessor divided by 2 is an integer. This is another representation of odd numbers without using the divides relation. (Contributed by AV, 18-Jun-2021.) (Proof shortened by AV, 22-Jun-2021.) |
⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 − 1) / 2) ∈ ℤ)) | ||
Theorem | ltoddhalfle 15712 | An integer is less than half of an odd number iff it is less than or equal to the half of the predecessor of the odd number (which is an even number). (Contributed by AV, 29-Jun-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ∧ 𝑀 ∈ ℤ) → (𝑀 < (𝑁 / 2) ↔ 𝑀 ≤ ((𝑁 − 1) / 2))) | ||
Theorem | halfleoddlt 15713 | An integer is greater than half of an odd number iff it is greater than or equal to the half of the odd number. (Contributed by AV, 1-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ∧ 𝑀 ∈ ℤ) → ((𝑁 / 2) ≤ 𝑀 ↔ (𝑁 / 2) < 𝑀)) | ||
Theorem | opoe 15714 | The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) ∧ (𝐵 ∈ ℤ ∧ ¬ 2 ∥ 𝐵)) → 2 ∥ (𝐴 + 𝐵)) | ||
Theorem | omoe 15715 | The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) ∧ (𝐵 ∈ ℤ ∧ ¬ 2 ∥ 𝐵)) → 2 ∥ (𝐴 − 𝐵)) | ||
Theorem | opeo 15716 | The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) ∧ (𝐵 ∈ ℤ ∧ 2 ∥ 𝐵)) → ¬ 2 ∥ (𝐴 + 𝐵)) | ||
Theorem | omeo 15717 | The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) ∧ (𝐵 ∈ ℤ ∧ 2 ∥ 𝐵)) → ¬ 2 ∥ (𝐴 − 𝐵)) | ||
Theorem | z0even 15718 | 2 divides 0. That means 0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.) |
⊢ 2 ∥ 0 | ||
Theorem | n2dvds1 15719 | 2 does not divide 1. That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.) (Proof shortened by Steven Nguyen, 3-May-2023.) |
⊢ ¬ 2 ∥ 1 | ||
Theorem | n2dvds1OLD 15720 | Obsolete version of n2dvds1 15719 as of 3-May-2023. 2 does not divide 1. That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ 2 ∥ 1 | ||
Theorem | n2dvdsm1 15721 | 2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.) |
⊢ ¬ 2 ∥ -1 | ||
Theorem | z2even 15722 | 2 divides 2. That means 2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.) |
⊢ 2 ∥ 2 | ||
Theorem | n2dvds3 15723 | 2 does not divide 3. That means 3 is odd. (Contributed by AV, 28-Feb-2021.) (Proof shortened by Steven Nguyen, 3-May-2023.) |
⊢ ¬ 2 ∥ 3 | ||
Theorem | n2dvds3OLD 15724 | Obsolete version of n2dvds3 15723 as of 3-May-2023. 2 does not divide 3. That means 3 is odd. (Contributed by AV, 28-Feb-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ 2 ∥ 3 | ||
Theorem | z4even 15725 | 2 divides 4. That means 4 is even. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.) |
⊢ 2 ∥ 4 | ||
Theorem | 4dvdseven 15726 | An integer which is divisible by 4 is divisible by 2, that is, is even. (Contributed by AV, 4-Jul-2021.) |
⊢ (4 ∥ 𝑁 → 2 ∥ 𝑁) | ||
Theorem | m1expe 15727 | Exponentiation of -1 by an even power. Variant of m1expeven 13479. (Contributed by AV, 25-Jun-2021.) |
⊢ (2 ∥ 𝑁 → (-1↑𝑁) = 1) | ||
Theorem | m1expo 15728 | Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (-1↑𝑁) = -1) | ||
Theorem | m1exp1 15729 | Exponentiation of negative one is one iff the exponent is even. (Contributed by AV, 20-Jun-2021.) |
⊢ (𝑁 ∈ ℤ → ((-1↑𝑁) = 1 ↔ 2 ∥ 𝑁)) | ||
Theorem | nn0enne 15730 | A positive integer is an even nonnegative integer iff it is an even positive integer. (Contributed by AV, 30-May-2020.) |
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ0 ↔ (𝑁 / 2) ∈ ℕ)) | ||
Theorem | nn0ehalf 15731 | The half of an even nonnegative integer is a nonnegative integer. (Contributed by AV, 22-Jun-2020.) (Revised by AV, 28-Jun-2021.) (Proof shortened by AV, 10-Jul-2022.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 2 ∥ 𝑁) → (𝑁 / 2) ∈ ℕ0) | ||
Theorem | nnehalf 15732 | The half of an even positive integer is a positive integer. (Contributed by AV, 28-Jun-2021.) |
⊢ ((𝑁 ∈ ℕ ∧ 2 ∥ 𝑁) → (𝑁 / 2) ∈ ℕ) | ||
Theorem | nn0onn 15733 | An odd nonnegative integer is positive. (Contributed by Steven Nguyen, 25-Mar-2023.) |
⊢ ((𝑁 ∈ ℕ0 ∧ ¬ 2 ∥ 𝑁) → 𝑁 ∈ ℕ) | ||
Theorem | nn0o1gt2 15734 | An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (𝑁 = 1 ∨ 2 < 𝑁)) | ||
Theorem | nno 15735 | An alternate characterization of an odd integer greater than 1. (Contributed by AV, 2-Jun-2020.) (Proof shortened by AV, 10-Jul-2022.) |
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ) | ||
Theorem | nn0o 15736 | An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0) | ||
Theorem | nn0ob 15737 | Alternate characterizations of an odd nonnegative integer. (Contributed by AV, 4-Jun-2020.) |
⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) / 2) ∈ ℕ0 ↔ ((𝑁 − 1) / 2) ∈ ℕ0)) | ||
Theorem | nn0oddm1d2 15738 | A positive integer is odd iff its predecessor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.) (Proof shortened by AV, 10-Jul-2022.) |
⊢ (𝑁 ∈ ℕ0 → (¬ 2 ∥ 𝑁 ↔ ((𝑁 − 1) / 2) ∈ ℕ0)) | ||
Theorem | nnoddm1d2 15739 | A positive integer is odd iff its successor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.) |
⊢ (𝑁 ∈ ℕ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℕ)) | ||
Theorem | sumeven 15740* | If every term in a sum is even, then so is the sum. (Contributed by AV, 14-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 2 ∥ 𝐵) ⇒ ⊢ (𝜑 → 2 ∥ Σ𝑘 ∈ 𝐴 𝐵) | ||
Theorem | sumodd 15741* | If every term in a sum is odd, then the sum is even iff the number of terms in the sum is even. (Contributed by AV, 14-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 2 ∥ 𝐵) ⇒ ⊢ (𝜑 → (2 ∥ (♯‘𝐴) ↔ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵)) | ||
Theorem | evensumodd 15742* | If every term in a sum with an even number of terms is odd, then the sum is even. (Contributed by AV, 14-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 2 ∥ 𝐵) & ⊢ (𝜑 → 2 ∥ (♯‘𝐴)) ⇒ ⊢ (𝜑 → 2 ∥ Σ𝑘 ∈ 𝐴 𝐵) | ||
Theorem | oddsumodd 15743* | If every term in a sum with an odd number of terms is odd, then the sum is odd. (Contributed by AV, 14-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 2 ∥ 𝐵) & ⊢ (𝜑 → ¬ 2 ∥ (♯‘𝐴)) ⇒ ⊢ (𝜑 → ¬ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵) | ||
Theorem | pwp1fsum 15744* | The n-th power of a number increased by 1 expressed by a product with a finite sum. (Contributed by AV, 15-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (((-1↑(𝑁 − 1)) · (𝐴↑𝑁)) + 1) = ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑁 − 1))((-1↑𝑘) · (𝐴↑𝑘)))) | ||
Theorem | oddpwp1fsum 15745* | An odd power of a number increased by 1 expressed by a product with a finite sum. (Contributed by AV, 15-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → ¬ 2 ∥ 𝑁) ⇒ ⊢ (𝜑 → ((𝐴↑𝑁) + 1) = ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑁 − 1))((-1↑𝑘) · (𝐴↑𝑘)))) | ||
Theorem | divalglem0 15746 | Lemma for divalg 15756. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ ⇒ ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐷 ∥ (𝑁 − 𝑅) → 𝐷 ∥ (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))))) | ||
Theorem | divalglem1 15747 | Lemma for divalg 15756. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 ⇒ ⊢ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷))) | ||
Theorem | divalglem2 15748* | Lemma for divalg 15756. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by AV, 2-Oct-2020.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 & ⊢ 𝑆 = {𝑟 ∈ ℕ0 ∣ 𝐷 ∥ (𝑁 − 𝑟)} ⇒ ⊢ inf(𝑆, ℝ, < ) ∈ 𝑆 | ||
Theorem | divalglem4 15749* | Lemma for divalg 15756. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 & ⊢ 𝑆 = {𝑟 ∈ ℕ0 ∣ 𝐷 ∥ (𝑁 − 𝑟)} ⇒ ⊢ 𝑆 = {𝑟 ∈ ℕ0 ∣ ∃𝑞 ∈ ℤ 𝑁 = ((𝑞 · 𝐷) + 𝑟)} | ||
Theorem | divalglem5 15750* | Lemma for divalg 15756. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by AV, 2-Oct-2020.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 & ⊢ 𝑆 = {𝑟 ∈ ℕ0 ∣ 𝐷 ∥ (𝑁 − 𝑟)} & ⊢ 𝑅 = inf(𝑆, ℝ, < ) ⇒ ⊢ (0 ≤ 𝑅 ∧ 𝑅 < (abs‘𝐷)) | ||
Theorem | divalglem6 15751 | Lemma for divalg 15756. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝑋 ∈ (0...(𝐴 − 1)) & ⊢ 𝐾 ∈ ℤ ⇒ ⊢ (𝐾 ≠ 0 → ¬ (𝑋 + (𝐾 · 𝐴)) ∈ (0...(𝐴 − 1))) | ||
Theorem | divalglem7 15752 | Lemma for divalg 15756. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 ⇒ ⊢ ((𝑋 ∈ (0...((abs‘𝐷) − 1)) ∧ 𝐾 ∈ ℤ) → (𝐾 ≠ 0 → ¬ (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1)))) | ||
Theorem | divalglem8 15753* | Lemma for divalg 15756. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 & ⊢ 𝑆 = {𝑟 ∈ ℕ0 ∣ 𝐷 ∥ (𝑁 − 𝑟)} ⇒ ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑋 < (abs‘𝐷) ∧ 𝑌 < (abs‘𝐷))) → (𝐾 ∈ ℤ → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) → 𝑋 = 𝑌))) | ||
Theorem | divalglem9 15754* | Lemma for divalg 15756. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by AV, 2-Oct-2020.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 & ⊢ 𝑆 = {𝑟 ∈ ℕ0 ∣ 𝐷 ∥ (𝑁 − 𝑟)} & ⊢ 𝑅 = inf(𝑆, ℝ, < ) ⇒ ⊢ ∃!𝑥 ∈ 𝑆 𝑥 < (abs‘𝐷) | ||
Theorem | divalglem10 15755* | Lemma for divalg 15756. (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by AV, 2-Oct-2020.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 & ⊢ 𝑆 = {𝑟 ∈ ℕ0 ∣ 𝐷 ∥ (𝑁 − 𝑟)} ⇒ ⊢ ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) | ||
Theorem | divalg 15756* | The division algorithm (theorem). Dividing an integer 𝑁 by a nonzero integer 𝐷 produces a (unique) quotient 𝑞 and a unique remainder 0 ≤ 𝑟 < (abs‘𝐷). Theorem 1.14 in [ApostolNT] p. 19. The proof does not use / or ⌊ or mod. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) | ||
Theorem | divalgb 15757* | Express the division algorithm as stated in divalg 15756 in terms of ∥. (Contributed by Paul Chapman, 31-Mar-2011.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ ∃!𝑟 ∈ ℕ0 (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟)))) | ||
Theorem | divalg2 15758* | The division algorithm (theorem) for a positive divisor. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → ∃!𝑟 ∈ ℕ0 (𝑟 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑟))) | ||
Theorem | divalgmod 15759 | The result of the mod operator satisfies the requirements for the remainder 𝑅 in the division algorithm for a positive divisor (compare divalg2 15758 and divalgb 15757). This demonstration theorem justifies the use of mod to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by AV, 21-Aug-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → (𝑅 = (𝑁 mod 𝐷) ↔ (𝑅 ∈ ℕ0 ∧ (𝑅 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑅))))) | ||
Theorem | divalgmodcl 15760 | The result of the mod operator satisfies the requirements for the remainder 𝑅 in the division algorithm for a positive divisor. Variant of divalgmod 15759. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by AV, 21-Aug-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ 𝑅 ∈ ℕ0) → (𝑅 = (𝑁 mod 𝐷) ↔ (𝑅 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑅)))) | ||
Theorem | modremain 15761* | The result of the modulo operation is the remainder of the division algorithm. (Contributed by AV, 19-Aug-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → ((𝑁 mod 𝐷) = 𝑅 ↔ ∃𝑧 ∈ ℤ ((𝑧 · 𝐷) + 𝑅) = 𝑁)) | ||
Theorem | ndvdssub 15762 | Corollary of the division algorithm. If an integer 𝐷 greater than 1 divides 𝑁, then it does not divide any of 𝑁 − 1, 𝑁 − 2... 𝑁 − (𝐷 − 1). (Contributed by Paul Chapman, 31-Mar-2011.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐾 < 𝐷)) → (𝐷 ∥ 𝑁 → ¬ 𝐷 ∥ (𝑁 − 𝐾))) | ||
Theorem | ndvdsadd 15763 | Corollary of the division algorithm. If an integer 𝐷 greater than 1 divides 𝑁, then it does not divide any of 𝑁 + 1, 𝑁 + 2... 𝑁 + (𝐷 − 1). (Contributed by Paul Chapman, 31-Mar-2011.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐾 < 𝐷)) → (𝐷 ∥ 𝑁 → ¬ 𝐷 ∥ (𝑁 + 𝐾))) | ||
Theorem | ndvdsp1 15764 | Special case of ndvdsadd 15763. If an integer 𝐷 greater than 1 divides 𝑁, it does not divide 𝑁 + 1. (Contributed by Paul Chapman, 31-Mar-2011.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ 1 < 𝐷) → (𝐷 ∥ 𝑁 → ¬ 𝐷 ∥ (𝑁 + 1))) | ||
Theorem | ndvdsi 15765 | A quick test for non-divisibility. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝑄 ∈ ℕ0 & ⊢ 𝑅 ∈ ℕ & ⊢ ((𝐴 · 𝑄) + 𝑅) = 𝐵 & ⊢ 𝑅 < 𝐴 ⇒ ⊢ ¬ 𝐴 ∥ 𝐵 | ||
Theorem | flodddiv4 15766 | The floor of an odd integer divided by 4. (Contributed by AV, 17-Jun-2021.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 = ((2 · 𝑀) + 1)) → (⌊‘(𝑁 / 4)) = if(2 ∥ 𝑀, (𝑀 / 2), ((𝑀 − 1) / 2))) | ||
Theorem | fldivndvdslt 15767 | The floor of an integer divided by a nonzero integer not dividing the first integer is less than the integer divided by the positive integer. (Contributed by AV, 4-Jul-2021.) |
⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℤ ∧ 𝐿 ≠ 0) ∧ ¬ 𝐿 ∥ 𝐾) → (⌊‘(𝐾 / 𝐿)) < (𝐾 / 𝐿)) | ||
Theorem | flodddiv4lt 15768 | The floor of an odd number divided by 4 is less than the odd number divided by 4. (Contributed by AV, 4-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (⌊‘(𝑁 / 4)) < (𝑁 / 4)) | ||
Theorem | flodddiv4t2lthalf 15769 | The floor of an odd number divided by 4, multiplied by 2 is less than the half of the odd number. (Contributed by AV, 4-Jul-2021.) (Proof shortened by AV, 10-Jul-2022.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < (𝑁 / 2)) | ||
Syntax | cbits 15770 | Define the binary bits of an integer. |
class bits | ||
Syntax | csad 15771 | Define the sequence addition on bit sequences. |
class sadd | ||
Syntax | csmu 15772 | Define the sequence multiplication on bit sequences. |
class smul | ||
Definition | df-bits 15773* | Define the binary bits of an integer. The expression 𝑀 ∈ (bits‘𝑁) means that the 𝑀-th bit of 𝑁 is 1 (and its negation means the bit is 0). (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ bits = (𝑛 ∈ ℤ ↦ {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑚)))}) | ||
Theorem | bitsfval 15774* | Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ (𝑁 ∈ ℤ → (bits‘𝑁) = {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))}) | ||
Theorem | bitsval 15775 | Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ (𝑀 ∈ (bits‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑀))))) | ||
Theorem | bitsval2 15776 | Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (𝑀 ∈ (bits‘𝑁) ↔ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑀))))) | ||
Theorem | bitsss 15777 | The set of bits of an integer is a subset of ℕ0. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ (bits‘𝑁) ⊆ ℕ0 | ||
Theorem | bitsf 15778 | The bits function is a function from integers to subsets of nonnegative integers. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ bits:ℤ⟶𝒫 ℕ0 | ||
Theorem | bits0 15779 | Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ (𝑁 ∈ ℤ → (0 ∈ (bits‘𝑁) ↔ ¬ 2 ∥ 𝑁)) | ||
Theorem | bits0e 15780 | The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ (𝑁 ∈ ℤ → ¬ 0 ∈ (bits‘(2 · 𝑁))) | ||
Theorem | bits0o 15781 | The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ (𝑁 ∈ ℤ → 0 ∈ (bits‘((2 · 𝑁) + 1))) | ||
Theorem | bitsp1 15782 | The 𝑀 + 1-th bit of 𝑁 is the 𝑀-th bit of ⌊(𝑁 / 2). (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘𝑁) ↔ 𝑀 ∈ (bits‘(⌊‘(𝑁 / 2))))) | ||
Theorem | bitsp1e 15783 | The 𝑀 + 1-th bit of 2𝑁 is the 𝑀-th bit of 𝑁. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘(2 · 𝑁)) ↔ 𝑀 ∈ (bits‘𝑁))) | ||
Theorem | bitsp1o 15784 | The 𝑀 + 1-th bit of 2𝑁 + 1 is the 𝑀-th bit of 𝑁. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘((2 · 𝑁) + 1)) ↔ 𝑀 ∈ (bits‘𝑁))) | ||
Theorem | bitsfzolem 15785* | Lemma for bitsfzo 15786. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 1-Oct-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → (bits‘𝑁) ⊆ (0..^𝑀)) & ⊢ 𝑆 = inf({𝑛 ∈ ℕ0 ∣ 𝑁 < (2↑𝑛)}, ℝ, < ) ⇒ ⊢ (𝜑 → 𝑁 ∈ (0..^(2↑𝑀))) | ||
Theorem | bitsfzo 15786 | The bits of a number are all less than 𝑀 iff the number is nonnegative and less than 2↑𝑀. (Contributed by Mario Carneiro, 5-Sep-2016.) (Proof shortened by AV, 1-Oct-2020.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (𝑁 ∈ (0..^(2↑𝑀)) ↔ (bits‘𝑁) ⊆ (0..^𝑀))) | ||
Theorem | bitsmod 15787 | Truncating the bit sequence after some 𝑀 is equivalent to reducing the argument mod 2↑𝑀. (Contributed by Mario Carneiro, 6-Sep-2016.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (bits‘(𝑁 mod (2↑𝑀))) = ((bits‘𝑁) ∩ (0..^𝑀))) | ||
Theorem | bitsfi 15788 | Every number is associated with a finite set of bits. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ (𝑁 ∈ ℕ0 → (bits‘𝑁) ∈ Fin) | ||
Theorem | bitscmp 15789 | The bit complement of 𝑁 is -𝑁 − 1. (Thus, by bitsfi 15788, all negative numbers have cofinite bits representations.) (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ (𝑁 ∈ ℤ → (ℕ0 ∖ (bits‘𝑁)) = (bits‘(-𝑁 − 1))) | ||
Theorem | 0bits 15790 | The bits of zero. (Contributed by Mario Carneiro, 6-Sep-2016.) |
⊢ (bits‘0) = ∅ | ||
Theorem | m1bits 15791 | The bits of negative one. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ (bits‘-1) = ℕ0 | ||
Theorem | bitsinv1lem 15792 | Lemma for bitsinv1 15793. (Contributed by Mario Carneiro, 22-Sep-2016.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (𝑁 mod (2↑(𝑀 + 1))) = ((𝑁 mod (2↑𝑀)) + if(𝑀 ∈ (bits‘𝑁), (2↑𝑀), 0))) | ||
Theorem | bitsinv1 15793* | There is an explicit inverse to the bits function for nonnegative integers (which can be extended to negative integers using bitscmp 15789), part 1. (Contributed by Mario Carneiro, 7-Sep-2016.) |
⊢ (𝑁 ∈ ℕ0 → Σ𝑛 ∈ (bits‘𝑁)(2↑𝑛) = 𝑁) | ||
Theorem | bitsinv2 15794* | There is an explicit inverse to the bits function for nonnegative integers, part 2. (Contributed by Mario Carneiro, 8-Sep-2016.) |
⊢ (𝐴 ∈ (𝒫 ℕ0 ∩ Fin) → (bits‘Σ𝑛 ∈ 𝐴 (2↑𝑛)) = 𝐴) | ||
Theorem | bitsf1ocnv 15795* | The bits function restricted to nonnegative integers is a bijection from the integers to the finite sets of integers. It is in fact the inverse of the Ackermann bijection ackbijnn 15185. (Contributed by Mario Carneiro, 8-Sep-2016.) |
⊢ ((bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin) ∧ ◡(bits ↾ ℕ0) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 (2↑𝑛))) | ||
Theorem | bitsf1o 15796 | The bits function restricted to nonnegative integers is a bijection from the integers to the finite sets of integers. It is in fact the inverse of the Ackermann bijection ackbijnn 15185. (Contributed by Mario Carneiro, 8-Sep-2016.) |
⊢ (bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin) | ||
Theorem | bitsf1 15797 | The bits function is an injection from ℤ to 𝒫 ℕ0. It is obviously not a bijection (by Cantor's theorem canth2 8672), and in fact its range is the set of finite and cofinite subsets of ℕ0. (Contributed by Mario Carneiro, 22-Sep-2016.) |
⊢ bits:ℤ–1-1→𝒫 ℕ0 | ||
Theorem | 2ebits 15798 | The bits of a power of two. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ (𝑁 ∈ ℕ0 → (bits‘(2↑𝑁)) = {𝑁}) | ||
Theorem | bitsinv 15799* | The inverse of the bits function. (Contributed by Mario Carneiro, 8-Sep-2016.) |
⊢ 𝐾 = ◡(bits ↾ ℕ0) ⇒ ⊢ (𝐴 ∈ (𝒫 ℕ0 ∩ Fin) → (𝐾‘𝐴) = Σ𝑘 ∈ 𝐴 (2↑𝑘)) | ||
Theorem | bitsinvp1 15800 | Recursive definition of the inverse of the bits function. (Contributed by Mario Carneiro, 8-Sep-2016.) |
⊢ 𝐾 = ◡(bits ↾ ℕ0) ⇒ ⊢ ((𝐴 ⊆ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐴, (2↑𝑁), 0))) |
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