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Theorem List for Metamath Proof Explorer - 15001-15100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmuldvds2 15001 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) ∥ 𝑁𝑀𝑁))

Theoremdvdscmul 15002 Multiplication by a constant maintains the divides relation. Theorem 1.1(d) in [ApostolNT] p. 14 (multiplication property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀𝑁 → (𝐾 · 𝑀) ∥ (𝐾 · 𝑁)))

Theoremdvdsmulc 15003 Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀𝑁 → (𝑀 · 𝐾) ∥ (𝑁 · 𝐾)))

Theoremdvdscmulr 15004 Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝐾 ∈ ℤ ∧ 𝐾 ≠ 0)) → ((𝐾 · 𝑀) ∥ (𝐾 · 𝑁) ↔ 𝑀𝑁))

Theoremdvdsmulcr 15005 Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝐾 ∈ ℤ ∧ 𝐾 ≠ 0)) → ((𝑀 · 𝐾) ∥ (𝑁 · 𝐾) ↔ 𝑀𝑁))

Theoremsummodnegmod 15006 The sum of two integers modulo a positive integer equals zero iff the first of the two integers equals the negative of the other integer modulo the positive integer. (Contributed by AV, 25-Jul-2021.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 𝐵) mod 𝑁) = 0 ↔ (𝐴 mod 𝑁) = (-𝐵 mod 𝑁)))

Theoremmodmulconst 15007 Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.)
(((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) ∧ 𝑀 ∈ ℕ) → ((𝐴 mod 𝑀) = (𝐵 mod 𝑀) ↔ ((𝐶 · 𝐴) mod (𝐶 · 𝑀)) = ((𝐶 · 𝐵) mod (𝐶 · 𝑀))))

Theoremdvds2ln 15008 If an integer divides each of two other integers, it divides any linear combination of them. Theorem 1.1(c) in [ApostolNT] p. 14 (linearity property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
(((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝐾𝑀𝐾𝑁) → 𝐾 ∥ ((𝐼 · 𝑀) + (𝐽 · 𝑁))))

Theoremdvds2add 15009 If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾𝑀𝐾𝑁) → 𝐾 ∥ (𝑀 + 𝑁)))

Theoremdvds2sub 15010 If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾𝑀𝐾𝑁) → 𝐾 ∥ (𝑀𝑁)))

Theoremdvds2subd 15011 Natural deduction form of dvds2sub 15010. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝐾𝑀)    &   (𝜑𝐾𝑁)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)       (𝜑𝐾 ∥ (𝑀𝑁))

Theoremdvdstr 15012 The divides relation is transitive. Theorem 1.1(b) in [ApostolNT] p. 14 (transitive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾𝑀𝑀𝑁) → 𝐾𝑁))

Theoremdvdsmultr1 15013 If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾𝑀𝐾 ∥ (𝑀 · 𝑁)))

Theoremdvdsmultr1d 15014 Natural deduction form of dvdsmultr1 15013. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐾𝑀)       (𝜑𝐾 ∥ (𝑀 · 𝑁))

Theoremdvdsmultr2 15015 If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾𝑁𝐾 ∥ (𝑀 · 𝑁)))

Theoremordvdsmul 15016 If an integer divides either of two others, it divides their product. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾𝑀𝐾𝑁) → 𝐾 ∥ (𝑀 · 𝑁)))

Theoremdvdssub2 15017 If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.)
(((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐾 ∥ (𝑀𝑁)) → (𝐾𝑀𝐾𝑁))

Theoremdvdsadd 15018 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 13-Jul-2014.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 ∥ (𝑀 + 𝑁)))

Theoremdvdsaddr 15019 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 ∥ (𝑁 + 𝑀)))

Theoremdvdssub 15020 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 ∥ (𝑀𝑁)))

Theoremdvdssubr 15021 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 ∥ (𝑁𝑀)))

Theoremdvdsadd2b 15022 Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝐶 ∈ ℤ ∧ 𝐴𝐶)) → (𝐴𝐵𝐴 ∥ (𝐶 + 𝐵)))

Theoremdvdsaddre2b 15023 Adding a multiple of the base does not affect divisibility. Variant of dvdsadd2b 15022 only requiring 𝐵 to be a real number (not necessarily an integer). (Contributed by AV, 19-Jul-2021.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℤ ∧ 𝐴𝐶)) → (𝐴𝐵𝐴 ∥ (𝐶 + 𝐵)))

Theoremfsumdvds 15024* If every term in a sum is divisible by 𝑁, then so is the sum. (Contributed by Mario Carneiro, 17-Jan-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℤ)    &   ((𝜑𝑘𝐴) → 𝑁𝐵)       (𝜑𝑁 ∥ Σ𝑘𝐴 𝐵)

Theoremdvdslelem 15025 Lemma for dvdsle 15026. (Contributed by Paul Chapman, 21-Mar-2011.)
𝑀 ∈ ℤ    &   𝑁 ∈ ℕ    &   𝐾 ∈ ℤ       (𝑁 < 𝑀 → (𝐾 · 𝑀) ≠ 𝑁)

Theoremdvdsle 15026 The divisors of a positive integer are bounded by it. The proof does not use /. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀𝑁𝑀𝑁))

Theoremdvdsleabs 15027 The divisors of a nonzero integer are bounded by its absolute value. Theorem 1.1(i) in [ApostolNT] p. 14 (comparison property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑀𝑁𝑀 ≤ (abs‘𝑁)))

Theoremdvdsleabs2 15028 Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑀𝑁 → (abs‘𝑀) ≤ (abs‘𝑁)))

Theoremdvdsabseq 15029 If two integers divide each other, they must be equal, up to a difference in sign. Theorem 1.1(j) in [ApostolNT] p. 14. (Contributed by Mario Carneiro, 30-May-2014.) (Revised by AV, 7-Aug-2021.)
((𝑀𝑁𝑁𝑀) → (abs‘𝑀) = (abs‘𝑁))

Theoremdvdseq 15030 If two nonnegative integers divide each other, they must be equal. (Contributed by Mario Carneiro, 30-May-2014.) (Proof shortened by AV, 7-Aug-2021.)
(((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀𝑁𝑁𝑀)) → 𝑀 = 𝑁)

Theoremdivconjdvds 15031 If a nonzero integer 𝑀 divides another integer 𝑁, the other integer 𝑁 divided by the nonzero integer 𝑀 (i.e. the divisor conjugate of 𝑁 to 𝑀) divides the other integer 𝑁. Theorem 1.1(k) in [ApostolNT] p. 14. (Contributed by AV, 7-Aug-2021.)
((𝑀𝑁𝑀 ≠ 0) → (𝑁 / 𝑀) ∥ 𝑁)

Theoremdvdsdivcl 15032* The complement of a divisor of 𝑁 is also a divisor of 𝑁. (Contributed by Mario Carneiro, 2-Jul-2015.) (Proof shortened by AV, 9-Aug-2021.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → (𝑁 / 𝐴) ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁})

Theoremdvdsflip 15033* An involution of the divisors of a number. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 13-May-2016.)
𝐴 = {𝑥 ∈ ℕ ∣ 𝑥𝑁}    &   𝐹 = (𝑦𝐴 ↦ (𝑁 / 𝑦))       (𝑁 ∈ ℕ → 𝐹:𝐴1-1-onto𝐴)

Theoremdvdsssfz1 15034* The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝𝐴} ⊆ (1...𝐴))

Theoremdvds1 15035 The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝑀 ∈ ℕ0 → (𝑀 ∥ 1 ↔ 𝑀 = 1))

Theoremalzdvds 15036* Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.)
(𝑁 ∈ ℤ → (∀𝑥 ∈ ℤ 𝑥𝑁𝑁 = 0))

Theoremdvdsext 15037* Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℕ0 (𝐴𝑥𝐵𝑥)))

Theoremfzm1ndvds 15038 No number between 1 and 𝑀 − 1 divides 𝑀. (Contributed by Mario Carneiro, 24-Jan-2015.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → ¬ 𝑀𝑁)

Theoremfzo0dvdseq 15039 Zero is the only one of the first 𝐴 nonnegative integers that is divisible by 𝐴. (Contributed by Stefan O'Rear, 6-Sep-2015.)
(𝐵 ∈ (0..^𝐴) → (𝐴𝐵𝐵 = 0))

Theoremfzocongeq 15040 Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.)
((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → ((𝐷𝐶) ∥ (𝐴𝐵) ↔ 𝐴 = 𝐵))

TheoremaddmodlteqALT 15041 Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. Shorter proof of addmodlteq 12740 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝐼 = 𝐽))

Theoremdvdsfac 15042 A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)
((𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝐾)) → 𝐾 ∥ (!‘𝑁))

Theoremdvdsexp 15043 A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.)
((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) → (𝐴𝑀) ∥ (𝐴𝑁))

Theoremdvdsmod 15044 Any number 𝐾 whose mod base 𝑁 is divisible by a divisor 𝑃 of the base is also divisible by 𝑃. This means that primes will also be relatively prime to the base when reduced mod 𝑁 for any base. (Contributed by Mario Carneiro, 13-Mar-2014.)
(((𝑃 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) ∧ 𝑃𝑁) → (𝑃 ∥ (𝐾 mod 𝑁) ↔ 𝑃𝐾))

Theoremmulmoddvds 15045 If an integer is divisible by a positive integer, the product of this integer with another integer modulo the positive integer is 0. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑁𝐴 → ((𝐴 · 𝐵) mod 𝑁) = 0))

Theorem3dvds 15046* A rule for divisibility by 3 of a number written in base 10. This is Metamath 100 proof #85. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.) (Revised by AV, 8-Sep-2021.)
((𝑁 ∈ ℕ0𝐹:(0...𝑁)⟶ℤ) → (3 ∥ Σ𝑘 ∈ (0...𝑁)((𝐹𝑘) · (10↑𝑘)) ↔ 3 ∥ Σ𝑘 ∈ (0...𝑁)(𝐹𝑘)))

Theorem3dvdsOLD 15047* Obsolete version of 3dvds 15046 as of 8-Sep-2021. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑁 ∈ ℕ0𝐹:(0...𝑁)⟶ℤ) → (3 ∥ Σ𝑘 ∈ (0...𝑁)((𝐹𝑘) · (10↑𝑘)) ↔ 3 ∥ Σ𝑘 ∈ (0...𝑁)(𝐹𝑘)))

Theorem3dvdsdec 15048 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 15049 Obsolete proof of 3dvdsdec 15048 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 15050 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 15051 Old version of 3dvds2dec 15050. 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 15052* 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 15053* 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 15056. 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 15054) 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 11457, zeo 11460, zeo2 11461, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 15075 and oddp1d2 15076. The corresponding theorems are zeneo 15057, zeo3 15055 and zeo4 15056.

Theoremevenelz 15054 An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 14982. (Contributed by AV, 22-Jun-2021.)
(2 ∥ 𝑁𝑁 ∈ ℤ)

Theoremzeo3 15055 An integer is even or odd. With this representation of even and odd integers, this variant of zeo 11460 follows immediatly from the law of excluded middle, see exmidd 432. (Contributed by AV, 17-Jun-2021.)
(𝑁 ∈ ℤ → (2 ∥ 𝑁 ∨ ¬ 2 ∥ 𝑁))

Theoremzeo4 15056 An integer is even or odd but not both. With this representation of even and odd integers, this variant of zeo2 11461 follows immediatly from the principle of double negation, see notnotb 304. (Contributed by AV, 17-Jun-2021.)
(𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁))

Theoremzeneo 15057 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 11457 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 15058* Lemma for odd2np1 15059. (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 15059* 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 15060* An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.)
(2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁)

Theoremoddm1even 15061 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
(𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1)))

Theoremoddp1even 15062 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
(𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 + 1)))

Theoremoexpneg 15063 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 15064 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 15065 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 15066* 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 15067* 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 15068* 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 15069* 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 15070 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 15071 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 15072 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 15073 A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.)
((𝐴 ∈ ℤ ∧ 𝐵 = (2 · 𝐴)) → 2 ∥ 𝐵)

Theoremzeo5 15074 An integer is either even or odd, version of zeo3 15055 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 15075 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 11460 and zeo2 11461. (Contributed by AV, 22-Jun-2021.)
(𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ (𝑁 / 2) ∈ ℤ))

Theoremoddp1d2 15076 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 11460 and zeo2 11461. (Contributed by AV, 22-Jun-2021.)
(𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ))

Theoremzob 15077 Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.)
(𝑁 ∈ ℤ → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 − 1) / 2) ∈ ℤ))

Theoremoddm1d2 15078 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 15079 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 15080 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 15081 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 15082 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 15083 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 15084 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 15085 Exponentiation of -1 by an even power. Variant of m1expeven 12902. (Contributed by AV, 25-Jun-2021.)
(2 ∥ 𝑁 → (-1↑𝑁) = 1)

Theoremm1expo 15086 Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.)
((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (-1↑𝑁) = -1)

Theoremm1exp1 15087 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 15088 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 15089 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 15090 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 15091 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 15092 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 15093 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 15094 Alternate characterizations of an odd nonnegative integer. (Contributed by AV, 4-Jun-2020.)
(𝑁 ∈ ℕ0 → (((𝑁 + 1) / 2) ∈ ℕ0 ↔ ((𝑁 − 1) / 2) ∈ ℕ0))

Theoremnn0oddm1d2 15095 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 15096 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 15097 0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.)
2 ∥ 0

Theoremn2dvds1 15098 2 does not divide 1 (common case). That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.)
¬ 2 ∥ 1

Theoremn2dvdsm1 15099 2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.)
¬ 2 ∥ -1

Theoremz2even 15100 2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.)
2 ∥ 2

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