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Theorem List for Metamath Proof Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremznn0sub 11301 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 11220.) (Contributed by NM, 14-Jul-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ (𝑁𝑀) ∈ ℕ0))

Theoremnzadd 11302 The sum of a real number not being an integer and an integer is not an integer. (Contributed by AV, 19-Jul-2021.)
((𝐴 ∈ (ℝ ∖ ℤ) ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ (ℝ ∖ ℤ))

Theoremzmulcl 11303 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ)

Theoremzltp1le 11304 Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))

Theoremzleltp1 11305 Integer ordering relation. (Contributed by NM, 10-May-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 < (𝑁 + 1)))

Theoremzlem1lt 11306 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))

Theoremzltlem1 11307 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁𝑀 ≤ (𝑁 − 1)))

Theoremzgt0ge1 11308 An integer greater than 0 is greater than or equal to 1. (Contributed by AV, 14-Oct-2018.)
(𝑍 ∈ ℤ → (0 < 𝑍 ↔ 1 ≤ 𝑍))

Theoremnnleltp1 11309 Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴𝐵𝐴 < (𝐵 + 1)))

Theoremnnltp1le 11310 Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵))

Theoremnnaddm1cl 11311 Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) − 1) ∈ ℕ)

Theoremnn0ltp1le 11312 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))

Theoremnn0leltp1 11313 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁𝑀 < (𝑁 + 1)))

Theoremnn0ltlem1 11314 Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 < 𝑁𝑀 ≤ (𝑁 − 1)))

Theoremnn0sub2 11315 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝑁𝑀) ∈ ℕ0)

Theoremnn0lt10b 11316 A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by OpenAI, 25-Mar-2020.)
(𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0))

Theoremnn0lt2 11317 A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
((𝑁 ∈ ℕ0𝑁 < 2) → (𝑁 = 0 ∨ 𝑁 = 1))

Theoremnn0lem1lt 11318 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))

Theoremnnlem1lt 11319 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))

Theoremnnltlem1 11320 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁𝑀 ≤ (𝑁 − 1)))

Theoremnnm1ge0 11321 A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.)
(𝑁 ∈ ℕ → 0 ≤ (𝑁 − 1))

Theoremnn0ge0div 11322 Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
((𝐾 ∈ ℕ0𝐿 ∈ ℕ) → 0 ≤ (𝐾 / 𝐿))

Theoremzdiv 11323* Two ways to express "𝑀 divides 𝑁. (Contributed by NM, 3-Oct-2008.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ))

Theoremzdivadd 11324 Property of divisibility: if 𝐷 divides 𝐴 and 𝐵 then it divides 𝐴 + 𝐵. (Contributed by NM, 3-Oct-2008.)
(((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 / 𝐷) ∈ ℤ ∧ (𝐵 / 𝐷) ∈ ℤ)) → ((𝐴 + 𝐵) / 𝐷) ∈ ℤ)

Theoremzdivmul 11325 Property of divisibility: if 𝐷 divides 𝐴 then it divides 𝐵 · 𝐴. (Contributed by NM, 3-Oct-2008.)
(((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐴 / 𝐷) ∈ ℤ) → ((𝐵 · 𝐴) / 𝐷) ∈ ℤ)

Theoremzextle 11326* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘𝑀𝑘𝑁)) → 𝑀 = 𝑁)

Theoremzextlt 11327* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘 < 𝑀𝑘 < 𝑁)) → 𝑀 = 𝑁)

Theoremrecnz 11328 The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)
((𝐴 ∈ ℝ ∧ 1 < 𝐴) → ¬ (1 / 𝐴) ∈ ℤ)

Theorembtwnnz 11329 A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)
((𝐴 ∈ ℤ ∧ 𝐴 < 𝐵𝐵 < (𝐴 + 1)) → ¬ 𝐵 ∈ ℤ)

Theoremgtndiv 11330 A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ ∧ 𝐵 < 𝐴) → ¬ (𝐵 / 𝐴) ∈ ℤ)

Theoremhalfnz 11331 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
¬ (1 / 2) ∈ ℤ

Theorem3halfnz 11332 Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
¬ (3 / 2) ∈ ℤ

Theoremsuprzcl 11333* The supremum of a bounded-above set of integers is a member of the set. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥) → sup(𝐴, ℝ, < ) ∈ 𝐴)

Theoremprime 11334* Two ways to express "𝐴 is a prime number (or 1)." See also isprm 15225. (Contributed by NM, 4-May-2005.)
(𝐴 ∈ ℕ → (∀𝑥 ∈ ℕ ((𝐴 / 𝑥) ∈ ℕ → (𝑥 = 1 ∨ 𝑥 = 𝐴)) ↔ ∀𝑥 ∈ ℕ ((1 < 𝑥𝑥𝐴 ∧ (𝐴 / 𝑥) ∈ ℕ) → 𝑥 = 𝐴)))

Theoremmsqznn 11335 The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)
((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) → (𝐴 · 𝐴) ∈ ℕ)

Theoremzneo 11336 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐴) ≠ ((2 · 𝐵) + 1))

Theoremnneo 11337 A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)
(𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℕ))

Theoremnneoi 11338 A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.)
𝑁 ∈ ℕ       ((𝑁 / 2) ∈ ℕ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℕ)

Theoremzeo 11339 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
(𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))

Theoremzeo2 11340 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)
(𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℤ))

Theorempeano2uz2 11341* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ {𝑥 ∈ ℤ ∣ 𝐴𝑥}) → (𝐵 + 1) ∈ {𝑥 ∈ ℤ ∣ 𝐴𝑥})

Theorempeano5uzi 11342* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
𝑁 ∈ ℤ       ((𝑁𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → {𝑘 ∈ ℤ ∣ 𝑁𝑘} ⊆ 𝐴)

Theorempeano5uzti 11343* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
(𝑁 ∈ ℤ → ((𝑁𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → {𝑘 ∈ ℤ ∣ 𝑁𝑘} ⊆ 𝐴))

Theoremdfuzi 11344* An expression for the upper integers that start at 𝑁 that is analogous to dfnn2 10910 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
𝑁 ∈ ℤ       {𝑧 ∈ ℤ ∣ 𝑁𝑧} = {𝑥 ∣ (𝑁𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}

Theoremuzind 11345* Induction on the upper integers that start at 𝑀. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 5-Jul-2005.)
(𝑗 = 𝑀 → (𝜑𝜓))    &   (𝑗 = 𝑘 → (𝜑𝜒))    &   (𝑗 = (𝑘 + 1) → (𝜑𝜃))    &   (𝑗 = 𝑁 → (𝜑𝜏))    &   (𝑀 ∈ ℤ → 𝜓)    &   ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀𝑘) → (𝜒𝜃))       ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁) → 𝜏)

Theoremuzind2 11346* Induction on the upper integers that start after an integer 𝑀. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 25-Jul-2005.)
(𝑗 = (𝑀 + 1) → (𝜑𝜓))    &   (𝑗 = 𝑘 → (𝜑𝜒))    &   (𝑗 = (𝑘 + 1) → (𝜑𝜃))    &   (𝑗 = 𝑁 → (𝜑𝜏))    &   (𝑀 ∈ ℤ → 𝜓)    &   ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 < 𝑘) → (𝜒𝜃))       ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → 𝜏)

Theoremuzind3 11347* Induction on the upper integers that start at an integer 𝑀. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 26-Jul-2005.)
(𝑗 = 𝑀 → (𝜑𝜓))    &   (𝑗 = 𝑚 → (𝜑𝜒))    &   (𝑗 = (𝑚 + 1) → (𝜑𝜃))    &   (𝑗 = 𝑁 → (𝜑𝜏))    &   (𝑀 ∈ ℤ → 𝜓)    &   ((𝑀 ∈ ℤ ∧ 𝑚 ∈ {𝑘 ∈ ℤ ∣ 𝑀𝑘}) → (𝜒𝜃))       ((𝑀 ∈ ℤ ∧ 𝑁 ∈ {𝑘 ∈ ℤ ∣ 𝑀𝑘}) → 𝜏)

Theoremnn0ind 11348* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 13-May-2004.)
(𝑥 = 0 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ ℕ0 → (𝜒𝜃))       (𝐴 ∈ ℕ0𝜏)

Theoremnn0indALT 11349* Principle of Mathematical Induction (inference schema) on nonnegative integers. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either nn0ind 11348 or nn0indALT 11349 may be used; see comment for nnind 10915. (Contributed by NM, 28-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑦 ∈ ℕ0 → (𝜒𝜃))    &   𝜓    &   (𝑥 = 0 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))       (𝐴 ∈ ℕ0𝜏)

Theoremnn0indd 11350* Principle of Mathematical Induction (inference schema) on nonnegative integers, a deduction version. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝑥 = 0 → (𝜓𝜒))    &   (𝑥 = 𝑦 → (𝜓𝜃))    &   (𝑥 = (𝑦 + 1) → (𝜓𝜏))    &   (𝑥 = 𝐴 → (𝜓𝜂))    &   (𝜑𝜒)    &   (((𝜑𝑦 ∈ ℕ0) ∧ 𝜃) → 𝜏)       ((𝜑𝐴 ∈ ℕ0) → 𝜂)

Theoremfzind 11351* Induction on the integers from 𝑀 to 𝑁 inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.)
(𝑥 = 𝑀 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐾 → (𝜑𝜏))    &   ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁) → 𝜓)    &   (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 𝑀𝑦𝑦 < 𝑁)) → (𝜒𝜃))       (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀𝐾𝐾𝑁)) → 𝜏)

Theoremfnn0ind 11352* Induction on the integers from 0 to 𝑁 inclusive. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.)
(𝑥 = 0 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐾 → (𝜑𝜏))    &   (𝑁 ∈ ℕ0𝜓)    &   ((𝑁 ∈ ℕ0𝑦 ∈ ℕ0𝑦 < 𝑁) → (𝜒𝜃))       ((𝑁 ∈ ℕ0𝐾 ∈ ℕ0𝐾𝑁) → 𝜏)

Theoremnn0ind-raph 11353* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.)
(𝑥 = 0 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ ℕ0 → (𝜒𝜃))       (𝐴 ∈ ℕ0𝜏)

Theoremzindd 11354* Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.)
(𝑥 = 0 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜏))    &   (𝑥 = -𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜂))    &   (𝜁𝜓)    &   (𝜁 → (𝑦 ∈ ℕ0 → (𝜒𝜏)))    &   (𝜁 → (𝑦 ∈ ℕ → (𝜒𝜃)))       (𝜁 → (𝐴 ∈ ℤ → 𝜂))

Theorembtwnz 11355* Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)
(𝐴 ∈ ℝ → (∃𝑥 ∈ ℤ 𝑥 < 𝐴 ∧ ∃𝑦 ∈ ℤ 𝐴 < 𝑦))

Theoremnn0zd 11356 A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ∈ ℤ)

Theoremnnzd 11357 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℤ)

Theoremzred 11358 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)       (𝜑𝐴 ∈ ℝ)

Theoremzcnd 11359 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)       (𝜑𝐴 ∈ ℂ)

Theoremznegcld 11360 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)       (𝜑 → -𝐴 ∈ ℤ)

Theorempeano2zd 11361 Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)       (𝜑 → (𝐴 + 1) ∈ ℤ)

Theoremzaddcld 11362 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)       (𝜑 → (𝐴 + 𝐵) ∈ ℤ)

Theoremzsubcld 11363 Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)       (𝜑 → (𝐴𝐵) ∈ ℤ)

Theoremzmulcld 11364 Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)       (𝜑 → (𝐴 · 𝐵) ∈ ℤ)

Theoremznnn0nn 11365 The negative of a negative integer, is a natural number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝑁 ∈ ℤ ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℕ)

Theoremzadd2cl 11366 Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
(𝑁 ∈ ℤ → (𝑁 + 2) ∈ ℤ)

Theoremzriotaneg 11367* The negative of the unique integer such that 𝜑. (Contributed by AV, 1-Dec-2018.)
(𝑥 = -𝑦 → (𝜑𝜓))       (∃!𝑥 ∈ ℤ 𝜑 → (𝑥 ∈ ℤ 𝜑) = -(𝑦 ∈ ℤ 𝜓))

Theoremsuprfinzcl 11368 The supremum of a nonempty finite set of integers is a member of the set. (Contributed by AV, 1-Oct-2019.)
((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → sup(𝐴, ℝ, < ) ∈ 𝐴)

5.4.10  Decimal arithmetic

Syntaxcdc 11369 Constant used for decimal constructor.
class 𝐴𝐵

Definitiondf-dec 11370 Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (1000 + 2000) = 3000 1kp2ke3k 26695. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)

TheoremdfdecOLD 11371 Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. Obsolete version of df-dec 11370 as of 1-Aug-2021. (Contributed by Mario Carneiro, 17-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴𝐵 = ((10 · 𝐴) + 𝐵)

Theorem9p1e10 11372 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.)
(9 + 1) = 10

Theoremdfdec10 11373 Version of the definition of the "decimal constructor" using 10 instead of the symbol 10. Of course, this statement cannot be used as definition, because it uses the "decimal constructor". (Contributed by AV, 1-Aug-2021.)
𝐴𝐵 = ((10 · 𝐴) + 𝐵)

Theoremdecex 11374 A decimal number is a set. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴𝐵 ∈ V

TheoremdecexOLD 11375 Obsolete proof of decex 11374 as of 6-Sep-2021. (Contributed by Mario Carneiro, 17-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴𝐵 ∈ V

Theoremdeceq1 11376 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)

Theoremdeceq1OLD 11377 Obsolete proof of deceq1 11376 as of 6-Sep-2021. (Contributed by Mario Carneiro, 17-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)

Theoremdeceq2 11378 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)

Theoremdeceq2OLD 11379 Obsolete proof of deceq1 11376 as of 6-Sep-2021. (Contributed by Mario Carneiro, 17-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)

Theoremdeceq1i 11380 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 = 𝐵       𝐴𝐶 = 𝐵𝐶

Theoremdeceq2i 11381 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 = 𝐵       𝐶𝐴 = 𝐶𝐵

Theoremdeceq12i 11382 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 = 𝐵    &   𝐶 = 𝐷       𝐴𝐶 = 𝐵𝐷

Theoremnumnncl 11383 Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ       ((𝑇 · 𝐴) + 𝐵) ∈ ℕ

Theoremnum0u 11384 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0       (𝑇 · 𝐴) = ((𝑇 · 𝐴) + 0)

Theoremnum0h 11385 Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0       𝐴 = ((𝑇 · 0) + 𝐴)

Theoremnumcl 11386 Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0

Theoremnumsuc 11387 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   (𝐵 + 1) = 𝐶    &   𝑁 = ((𝑇 · 𝐴) + 𝐵)       (𝑁 + 1) = ((𝑇 · 𝐴) + 𝐶)

Theoremdeccl 11388 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       𝐴𝐵 ∈ ℕ0

TheoremdecclOLD 11389 Obsolete proof of deccl 11388 as of 6-Sep-2021. (Contributed by Mario Carneiro, 17-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       𝐴𝐵 ∈ ℕ0

Theorem10nn 11390 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
10 ∈ ℕ

Theorem10pos 11391 The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
0 < 10

Theorem10nn0 11392 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
10 ∈ ℕ0

Theorem10re 11393 The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
10 ∈ ℝ

Theoremdecnncl 11394 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ       𝐴𝐵 ∈ ℕ

TheoremdecnnclOLD 11395 Obsolete proof of decnncl 11394 as of 6-Sep-2021. (Contributed by Mario Carneiro, 17-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ       𝐴𝐵 ∈ ℕ

Theoremdec0u 11396 Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0       (10 · 𝐴) = 𝐴0

Theoremdec0uOLD 11397 Obsolete version of dec0u 11396 as of 6-Sep-2021. (Contributed by Mario Carneiro, 17-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0       (10 · 𝐴) = 𝐴0

Theoremdec0h 11398 Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0       𝐴 = 0𝐴

Theoremdec0hOLD 11399 Obsolete proof of dec0h 11398 as of 6-Sep-2021. (Contributed by Mario Carneiro, 17-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0       𝐴 = 0𝐴

Theoremnumnncl2 11400 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)
𝑇 ∈ ℕ    &   𝐴 ∈ ℕ       ((𝑇 · 𝐴) + 0) ∈ ℕ

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