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Theorem List for Metamath Proof Explorer - 15101-15200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem3dvdsdec 15101 A decimal number is divisible by three iff the sum of its two "digits" is divisible by three. The term "digits" in its narrow sense is only correct if 𝐴 and 𝐵 actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers 𝐴 and 𝐵, especially if 𝐴 is itself a decimal number, e.g. 𝐴 = 𝐶𝐷. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       (3 ∥ 𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵))

Theorem3dvdsdecOLD 15102 Obsolete proof of 3dvdsdec 15101 as of 8-Sep-2021. (Contributed by AV, 14-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       (3 ∥ 𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵))

Theorem3dvds2dec 15103 A decimal number is divisible by three iff the sum of its three "digits" is divisible by three. The term "digits" in its narrow sense is only correct if 𝐴, 𝐵 and 𝐶 actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers 𝐴, 𝐵 and 𝐶. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0       (3 ∥ 𝐴𝐵𝐶 ↔ 3 ∥ ((𝐴 + 𝐵) + 𝐶))

Theorem3dvds2decOLD 15104 Old version of 3dvds2dec 15103. Obsolete as of 1-Aug-2021. (Contributed by AV, 14-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0       (3 ∥ 𝐴𝐵𝐶 ↔ 3 ∥ ((𝐴 + 𝐵) + 𝐶))

Theoremfprodfvdvdsd 15105* A finite product of integers is divisible by any of its factors being function values. (Contributed by AV, 1-Aug-2021.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹:𝐵⟶ℤ)       (𝜑 → ∀𝑥𝐴 (𝐹𝑥) ∥ ∏𝑘𝐴 (𝐹𝑘))

Theoremfproddvdsd 15106* A finite product of integers is divisible by any of its factors. (Contributed by AV, 14-Aug-2020.) (Proof shortened by AV, 2-Aug-2021.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ ℤ)       (𝜑 → ∀𝑥𝐴 𝑥 ∥ ∏𝑘𝐴 𝑘)

6.1.4  Even and odd numbers

The set of integers can be partitioned into the set of even numbers and the set of odd numbers, see zeo4 15109. Instead of defining new class variables Even and Odd to represent these sets, we use the idiom 2 ∥ 𝑁 to say that "𝑁 is even" (which implies 𝑁 ∈ ℤ, see evenelz 15107) and ¬ 2 ∥ 𝑁 to say that "𝑁 is odd" (under the assumption that 𝑁 ∈ ℤ). The previously proven theorems about even and odd numbers, like zneo 11498, zeo 11501, zeo2 11502, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 15128 and oddp1d2 15129. The corresponding theorems are zeneo 15110, zeo3 15108 and zeo4 15109.

Theoremevenelz 15107 An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 15032. (Contributed by AV, 22-Jun-2021.)
(2 ∥ 𝑁𝑁 ∈ ℤ)

Theoremzeo3 15108 An integer is even or odd. With this representation of even and odd integers, this variant of zeo 11501 follows immediately from the law of excluded middle, see exmidd 431. (Contributed by AV, 17-Jun-2021.)
(𝑁 ∈ ℤ → (2 ∥ 𝑁 ∨ ¬ 2 ∥ 𝑁))

Theoremzeo4 15109 An integer is even or odd but not both. With this representation of even and odd integers, this variant of zeo2 11502 follows immediately from the principle of double negation, see notnotb 304. (Contributed by AV, 17-Jun-2021.)
(𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁))

Theoremzeneo 15110 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. This variant of zneo 11498 follows immediately from the fact that a contradiction implies anything, see pm2.21i 116. (Contributed by AV, 22-Jun-2021.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴𝐵))

Theoremodd2np1lem 15111* Lemma for odd2np1 15112. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝑁 ∈ ℕ0 → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁 ∨ ∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑁))

Theoremodd2np1 15112* An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁))

Theoremeven2n 15113* An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.)
(2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁)

Theoremoddm1even 15114 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
(𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1)))

Theoremoddp1even 15115 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
(𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 + 1)))

Theoremoexpneg 15116 The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → (-𝐴𝑁) = -(𝐴𝑁))

Theoremmod2eq0even 15117 An integer is 0 modulo 2 iff it is even (i.e. divisible by 2), see example 2 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.)
(𝑁 ∈ ℤ → ((𝑁 mod 2) = 0 ↔ 2 ∥ 𝑁))

Theoremmod2eq1n2dvds 15118 An integer is 1 modulo 2 iff it is odd (i.e. not divisible by 2), see example 3 in [ApostolNT] p. 107. (Contributed by AV, 24-May-2020.) (Proof shortened by AV, 5-Jul-2020.)
(𝑁 ∈ ℤ → ((𝑁 mod 2) = 1 ↔ ¬ 2 ∥ 𝑁))

Theoremoddnn02np1 15119* A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.)
(𝑁 ∈ ℕ0 → (¬ 2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ0 ((2 · 𝑛) + 1) = 𝑁))

Theoremoddge22np1 15120* An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.)
(𝑁 ∈ (ℤ‘2) → (¬ 2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ ((2 · 𝑛) + 1) = 𝑁))

Theoremevennn02n 15121* A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.)
(𝑁 ∈ ℕ0 → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ0 (2 · 𝑛) = 𝑁))

Theoremevennn2n 15122* 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))

Theorem2tp1odd 15123 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)

Theoremmulsucdiv2z 15124 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)

Theoremsqoddm1div8z 15125 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)

Theorem2teven 15126 A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.)
((𝐴 ∈ ℤ ∧ 𝐵 = (2 · 𝐴)) → 2 ∥ 𝐵)

Theoremzeo5 15127 An integer is either even or odd, version of zeo3 15108 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)))

Theoremevend2 15128 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 11501 and zeo2 11502. (Contributed by AV, 22-Jun-2021.)
(𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ (𝑁 / 2) ∈ ℤ))

Theoremoddp1d2 15129 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 11501 and zeo2 11502. (Contributed by AV, 22-Jun-2021.)
(𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ))

Theoremzob 15130 Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.)
(𝑁 ∈ ℤ → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 − 1) / 2) ∈ ℤ))

Theoremoddm1d2 15131 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))

Theoremltoddhalfle 15132 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 ∥ 𝑁𝑀 ∈ ℤ) → (𝑀 < (𝑁 / 2) ↔ 𝑀 ≤ ((𝑁 − 1) / 2)))

Theoremhalfleoddlt 15133 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 ∥ 𝑁𝑀 ∈ ℤ) → ((𝑁 / 2) ≤ 𝑀 ↔ (𝑁 / 2) < 𝑀))

Theoremopoe 15134 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))

Theoremomoe 15135 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 ∥ (𝐴𝐵))

Theoremopeo 15136 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))

Theoremomeo 15137 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 ∥ (𝐴𝐵))

Theoremm1expe 15138 Exponentiation of -1 by an even power. Variant of m1expeven 12947. (Contributed by AV, 25-Jun-2021.)
(2 ∥ 𝑁 → (-1↑𝑁) = 1)

Theoremm1expo 15139 Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.)
((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (-1↑𝑁) = -1)

Theoremm1exp1 15140 Exponentiation of negative one is one iff the exponent is even. (Contributed by AV, 20-Jun-2021.)
(𝑁 ∈ ℤ → ((-1↑𝑁) = 1 ↔ 2 ∥ 𝑁))

Theoremnn0enne 15141 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))

Theoremnn0ehalf 15142 The half of an even nonnegative integer is a nonnegative integer. (Contributed by AV, 22-Jun-2020.) (Revised by AV, 28-Jun-2021.)
((𝑁 ∈ ℕ0 ∧ 2 ∥ 𝑁) → (𝑁 / 2) ∈ ℕ0)

Theoremnnehalf 15143 The half of an even positive integer is a positive integer. (Contributed by AV, 28-Jun-2021.)
((𝑁 ∈ ℕ ∧ 2 ∥ 𝑁) → (𝑁 / 2) ∈ ℕ)

Theoremnn0o1gt2 15144 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))

Theoremnno 15145 An alternate characterization of an odd integer greater than 1. (Contributed by AV, 2-Jun-2020.)
((𝑁 ∈ (ℤ‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ)

Theoremnn0o 15146 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)

Theoremnn0ob 15147 Alternate characterizations of an odd nonnegative integer. (Contributed by AV, 4-Jun-2020.)
(𝑁 ∈ ℕ0 → (((𝑁 + 1) / 2) ∈ ℕ0 ↔ ((𝑁 − 1) / 2) ∈ ℕ0))

Theoremnn0oddm1d2 15148 A positive integer is odd iff its predecessor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.)
(𝑁 ∈ ℕ0 → (¬ 2 ∥ 𝑁 ↔ ((𝑁 − 1) / 2) ∈ ℕ0))

Theoremnnoddm1d2 15149 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))

Theoremz0even 15150 0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.)
2 ∥ 0

Theoremn2dvds1 15151 2 does not divide 1. That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.)
¬ 2 ∥ 1

Theoremn2dvdsm1 15152 2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.)
¬ 2 ∥ -1

Theoremz2even 15153 2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.)
2 ∥ 2

Theoremn2dvds3 15154 2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.)
¬ 2 ∥ 3

Theoremz4even 15155 4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.)
2 ∥ 4

Theorem4dvdseven 15156 An integer which is divisible by 4 is an even integer. (Contributed by AV, 4-Jul-2021.)
(4 ∥ 𝑁 → 2 ∥ 𝑁)

Theoremsumeven 15157* If every term in a sum is even, then so is the sum. (Contributed by AV, 14-Aug-2021.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℤ)    &   ((𝜑𝑘𝐴) → 2 ∥ 𝐵)       (𝜑 → 2 ∥ Σ𝑘𝐴 𝐵)

Theoremsumodd 15158* 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.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℤ)    &   ((𝜑𝑘𝐴) → ¬ 2 ∥ 𝐵)       (𝜑 → (2 ∥ (#‘𝐴) ↔ 2 ∥ Σ𝑘𝐴 𝐵))

Theoremevensumodd 15159* 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.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℤ)    &   ((𝜑𝑘𝐴) → ¬ 2 ∥ 𝐵)    &   (𝜑 → 2 ∥ (#‘𝐴))       (𝜑 → 2 ∥ Σ𝑘𝐴 𝐵)

Theoremoddsumodd 15160* 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.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℤ)    &   ((𝜑𝑘𝐴) → ¬ 2 ∥ 𝐵)    &   (𝜑 → ¬ 2 ∥ (#‘𝐴))       (𝜑 → ¬ 2 ∥ Σ𝑘𝐴 𝐵)

Theorempwp1fsum 15161* The n-th power of a number increased by 1 expressed by a product with a finite sum. (Contributed by AV, 15-Aug-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (((-1↑(𝑁 − 1)) · (𝐴𝑁)) + 1) = ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑁 − 1))((-1↑𝑘) · (𝐴𝑘))))

Theoremoddpwp1fsum 15162* An odd power of a number increased by 1 expressed by a product with a finite sum. (Contributed by AV, 15-Aug-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝑁)       (𝜑 → ((𝐴𝑁) + 1) = ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑁 − 1))((-1↑𝑘) · (𝐴𝑘))))

6.1.5  The division algorithm

Theoremdivalglem0 15163 Lemma for divalg 15173. (Contributed by Paul Chapman, 21-Mar-2011.)
𝑁 ∈ ℤ    &   𝐷 ∈ ℤ       ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐷 ∥ (𝑁𝑅) → 𝐷 ∥ (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷))))))

Theoremdivalglem1 15164 Lemma for divalg 15173. (Contributed by Paul Chapman, 21-Mar-2011.)
𝑁 ∈ ℤ    &   𝐷 ∈ ℤ    &   𝐷 ≠ 0       0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷)))

Theoremdivalglem2 15165* Lemma for divalg 15173. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by AV, 2-Oct-2020.)
𝑁 ∈ ℤ    &   𝐷 ∈ ℤ    &   𝐷 ≠ 0    &   𝑆 = {𝑟 ∈ ℕ0𝐷 ∥ (𝑁𝑟)}       inf(𝑆, ℝ, < ) ∈ 𝑆

Theoremdivalglem4 15166* Lemma for divalg 15173. (Contributed by Paul Chapman, 21-Mar-2011.)
𝑁 ∈ ℤ    &   𝐷 ∈ ℤ    &   𝐷 ≠ 0    &   𝑆 = {𝑟 ∈ ℕ0𝐷 ∥ (𝑁𝑟)}       𝑆 = {𝑟 ∈ ℕ0 ∣ ∃𝑞 ∈ ℤ 𝑁 = ((𝑞 · 𝐷) + 𝑟)}

Theoremdivalglem5 15167* Lemma for divalg 15173. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by AV, 2-Oct-2020.)
𝑁 ∈ ℤ    &   𝐷 ∈ ℤ    &   𝐷 ≠ 0    &   𝑆 = {𝑟 ∈ ℕ0𝐷 ∥ (𝑁𝑟)}    &   𝑅 = inf(𝑆, ℝ, < )       (0 ≤ 𝑅𝑅 < (abs‘𝐷))

Theoremdivalglem6 15168 Lemma for divalg 15173. (Contributed by Paul Chapman, 21-Mar-2011.)
𝐴 ∈ ℕ    &   𝑋 ∈ (0...(𝐴 − 1))    &   𝐾 ∈ ℤ       (𝐾 ≠ 0 → ¬ (𝑋 + (𝐾 · 𝐴)) ∈ (0...(𝐴 − 1)))

Theoremdivalglem7 15169 Lemma for divalg 15173. (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))))

Theoremdivalglem8 15170* Lemma for divalg 15173. (Contributed by Paul Chapman, 21-Mar-2011.)
𝑁 ∈ ℤ    &   𝐷 ∈ ℤ    &   𝐷 ≠ 0    &   𝑆 = {𝑟 ∈ ℕ0𝐷 ∥ (𝑁𝑟)}       (((𝑋𝑆𝑌𝑆) ∧ (𝑋 < (abs‘𝐷) ∧ 𝑌 < (abs‘𝐷))) → (𝐾 ∈ ℤ → ((𝐾 · (abs‘𝐷)) = (𝑌𝑋) → 𝑋 = 𝑌)))

Theoremdivalglem9 15171* Lemma for divalg 15173. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by AV, 2-Oct-2020.)
𝑁 ∈ ℤ    &   𝐷 ∈ ℤ    &   𝐷 ≠ 0    &   𝑆 = {𝑟 ∈ ℕ0𝐷 ∥ (𝑁𝑟)}    &   𝑅 = inf(𝑆, ℝ, < )       ∃!𝑥𝑆 𝑥 < (abs‘𝐷)

Theoremdivalglem10 15172* Lemma for divalg 15173. (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by AV, 2-Oct-2020.)
𝑁 ∈ ℤ    &   𝐷 ∈ ℤ    &   𝐷 ≠ 0    &   𝑆 = {𝑟 ∈ ℕ0𝐷 ∥ (𝑁𝑟)}       ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))

Theoremdivalg 15173* 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 mod. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))

Theoremdivalgb 15174* Express the division algorithm as stated in divalg 15173 in terms of . (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ ∃!𝑟 ∈ ℕ0 (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁𝑟))))

Theoremdivalg2 15175* The division algorithm (theorem) for a positive divisor. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → ∃!𝑟 ∈ ℕ0 (𝑟 < 𝐷𝐷 ∥ (𝑁𝑟)))

Theoremdivalgmod 15176 The result of the mod operator satisfies the requirements for the remainder 𝑅 in the division algorithm for a positive divisor (compare divalg2 15175 and divalgb 15174). 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 < 𝐷𝐷 ∥ (𝑁𝑅)))))

TheoremdivalgmodOLD 15177* Obsolete proof of divalgmod 15176 as of 21-Aug-2021. (Contributed by Paul Chapman, 31-Mar-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → (𝑟 = (𝑁 mod 𝐷) ↔ (𝑟 ∈ ℕ0 ∧ (𝑟 < 𝐷𝐷 ∥ (𝑁𝑟)))))

Theoremdivalgmodcl 15178 The result of the mod operator satisfies the requirements for the remainder 𝑅 in the division algorithm for a positive divisor. Variant of divalgmod 15176. (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 < 𝐷𝐷 ∥ (𝑁𝑅))))

Theoremmodremain 15179* The result of the modulo operation is the remainder of the division algorithm. (Contributed by AV, 19-Aug-2021.)
((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0𝑅 < 𝐷)) → ((𝑁 mod 𝐷) = 𝑅 ↔ ∃𝑧 ∈ ℤ ((𝑧 · 𝐷) + 𝑅) = 𝑁))

Theoremndvdssub 15180 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)) → (𝐷𝑁 → ¬ 𝐷 ∥ (𝑁𝐾)))

Theoremndvdsadd 15181 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)) → (𝐷𝑁 → ¬ 𝐷 ∥ (𝑁 + 𝐾)))

Theoremndvdsp1 15182 Special case of ndvdsadd 15181. 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) → (𝐷𝑁 → ¬ 𝐷 ∥ (𝑁 + 1)))

Theoremndvdsi 15183 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        ¬ 𝐴𝐵

Theoremflodddiv4 15184 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)))

Theoremfldivndvdslt 15185 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) ∧ ¬ 𝐿𝐾) → (⌊‘(𝐾 / 𝐿)) < (𝐾 / 𝐿))

Theoremflodddiv4lt 15186 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))

Theoremflodddiv4t2lthalf 15187 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.)
((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < (𝑁 / 2))

6.1.6  Bit sequences

Syntaxcbits 15188 Define the binary bits of an integer.
class bits

Syntaxcsmu 15190 Define the sequence multiplication on bit sequences.
class smul

Definitiondf-bits 15191* 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)))})

Theorembitsfval 15192* 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)))})

Theorembitsval 15193 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)))))

Theorembitsval2 15194 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)))))

Theorembitsss 15195 The set of bits of an integer is a subset of 0. (Contributed by Mario Carneiro, 5-Sep-2016.)
(bits‘𝑁) ⊆ ℕ0

Theorembitsf 15196 The bits function is a function from integers to subsets of nonnegative integers. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits:ℤ⟶𝒫 ℕ0

Theorembits0 15197 Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.)
(𝑁 ∈ ℤ → (0 ∈ (bits‘𝑁) ↔ ¬ 2 ∥ 𝑁))

Theorembits0e 15198 The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.)
(𝑁 ∈ ℤ → ¬ 0 ∈ (bits‘(2 · 𝑁)))

Theorembits0o 15199 The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.)
(𝑁 ∈ ℤ → 0 ∈ (bits‘((2 · 𝑁) + 1)))

Theorembitsp1 15200 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)))))

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