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Theorem List for Intuitionistic Logic Explorer - 8301-8400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremzltlem1 8301 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁𝑀 ≤ (𝑁 − 1)))

Theoremzgt0ge1 8302 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 8303 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 8304 Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵))

Theoremnnaddm1cl 8305 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 8306 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 8307 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁𝑀 < (𝑁 + 1)))

Theoremnn0ltlem1 8308 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)))

Theoremznn0sub 8309 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 8310.) (Contributed by NM, 14-Jul-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ (𝑁𝑀) ∈ ℕ0))

Theoremnn0sub 8310 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁 ↔ (𝑁𝑀) ∈ ℕ0))

Theoremnn0n0n1ge2 8311 A nonnegative integer which is neither 0 nor 1 is greater than or equal to 2. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
((𝑁 ∈ ℕ0𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → 2 ≤ 𝑁)

Theoremelz2 8312* Membership in the set of integers. Commonly used in constructions of the integers as equivalence classes under subtraction of the positive integers. (Contributed by Mario Carneiro, 16-May-2014.)
(𝑁 ∈ ℤ ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑁 = (𝑥𝑦))

Theoremdfz2 8313 Alternative definition of the integers, based on elz2 8312. (Contributed by Mario Carneiro, 16-May-2014.)
ℤ = ( − “ (ℕ × ℕ))

Theoremnn0sub2 8314 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝑁𝑀) ∈ ℕ0)

Theoremzapne 8315 Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁𝑀𝑁))

Theoremzdceq 8316 Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 = 𝐵)

Theoremzdcle 8317 Integer is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴𝐵)

Theoremzdclt 8318 Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 < 𝐵)

Theoremzltlen 8319 Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7621 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (𝐴𝐵𝐵𝐴)))

Theoremnn0n0n1ge2b 8320 A nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
(𝑁 ∈ ℕ0 → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁))

Theoremnn0lt10b 8321 A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0))

Theoremnn0lt2 8322 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 8323 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))

Theoremnnlem1lt 8324 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))

Theoremnnltlem1 8325 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁𝑀 ≤ (𝑁 − 1)))

Theoremnnm1ge0 8326 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 8327 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 8328* 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 8329 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 8330 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 8331* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘𝑀𝑘𝑁)) → 𝑀 = 𝑁)

Theoremzextlt 8332* 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 8333 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 8334 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 8335 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 8336 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
¬ (1 / 2) ∈ ℤ

Theoremprime 8337* Two ways to express "𝐴 is a prime number (or 1)." (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 8338 The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)
((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) → (𝐴 · 𝐴) ∈ ℕ)

Theoremzneo 8339 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))

Theoremnneoor 8340 A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
(𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ))

Theoremnneo 8341 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 8342 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 8343 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
(𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))

Theoremzeo2 8344 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 8345* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ {𝑥 ∈ ℤ ∣ 𝐴𝑥}) → (𝐵 + 1) ∈ {𝑥 ∈ ℤ ∣ 𝐴𝑥})

Theorempeano5uzti 8346* 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))

Theorempeano5uzi 8347* 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)

Theoremdfuzi 8348* An expression for the upper integers that start at 𝑁 that is analogous to dfnn2 7916 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 8349* 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 8350* 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 8351* 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 8352* 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)

Theoremfzind 8353* 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 8354* 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 8355* 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 8356* 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 8357* 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 8358 A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ∈ ℤ)

Theoremnnzd 8359 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℤ)

Theoremzred 8360 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)       (𝜑𝐴 ∈ ℝ)

Theoremzcnd 8361 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)       (𝜑𝐴 ∈ ℂ)

Theoremznegcld 8362 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)       (𝜑 → -𝐴 ∈ ℤ)

Theorempeano2zd 8363 Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)       (𝜑 → (𝐴 + 1) ∈ ℤ)

Theoremzaddcld 8364 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)       (𝜑 → (𝐴 + 𝐵) ∈ ℤ)

Theoremzsubcld 8365 Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)       (𝜑 → (𝐴𝐵) ∈ ℤ)

Theoremzmulcld 8366 Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)       (𝜑 → (𝐴 · 𝐵) ∈ ℤ)

Theoremzadd2cl 8367 Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
(𝑁 ∈ ℤ → (𝑁 + 2) ∈ ℤ)

3.4.9  Decimal arithmetic

Syntaxcdc 8368 Constant used for decimal constructor.
class 𝐴𝐵

Definitiondf-dec 8369 Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (1000 + 2000) = 3000. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴𝐵 = ((10 · 𝐴) + 𝐵)

Theoremdeceq1 8370 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
(𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)

Theoremdeceq2 8371 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
(𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)

Theoremdeceq1i 8372 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 = 𝐵       𝐴𝐶 = 𝐵𝐶

Theoremdeceq2i 8373 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 = 𝐵       𝐶𝐴 = 𝐶𝐵

Theoremdeceq12i 8374 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 = 𝐵    &   𝐶 = 𝐷       𝐴𝐶 = 𝐵𝐷

Theoremnumnncl 8375 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 8376 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0       (𝑇 · 𝐴) = ((𝑇 · 𝐴) + 0)

Theoremnum0h 8377 Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0       𝐴 = ((𝑇 · 0) + 𝐴)

Theoremnumcl 8378 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 8379 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)

Theoremdecnncl 8380 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ       𝐴𝐵 ∈ ℕ

Theoremdeccl 8381 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       𝐴𝐵 ∈ ℕ0

Theoremdec0u 8382 Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0       (10 · 𝐴) = 𝐴0

Theoremdec0h 8383 Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0       𝐴 = 0𝐴

Theoremnumnncl2 8384 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)
𝑇 ∈ ℕ    &   𝐴 ∈ ℕ       ((𝑇 · 𝐴) + 0) ∈ ℕ

Theoremdecnncl2 8385 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ       𝐴0 ∈ ℕ

Theoremnumlt 8386 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ    &   𝐵 < 𝐶       ((𝑇 · 𝐴) + 𝐵) < ((𝑇 · 𝐴) + 𝐶)

Theoremnumltc 8387 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐶 < 𝑇    &   𝐴 < 𝐵       ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷)

Theoremdeclt 8388 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ    &   𝐵 < 𝐶       𝐴𝐵 < 𝐴𝐶

Theoremdecltc 8389 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐶 < 10    &   𝐴 < 𝐵       𝐴𝐶 < 𝐵𝐷

Theoremdecsuc 8390 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   (𝐵 + 1) = 𝐶    &   𝑁 = 𝐴𝐵       (𝑁 + 1) = 𝐴𝐶

Theoremnumlti 8391 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ    &   𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐶 < 𝑇       𝐶 < ((𝑇 · 𝐴) + 𝐵)

Theoremdeclti 8392 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐶 < 10       𝐶 < 𝐴𝐵

Theoremnumsucc 8393 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑌 ∈ ℕ0    &   𝑇 = (𝑌 + 1)    &   𝐴 ∈ ℕ0    &   (𝐴 + 1) = 𝐵    &   𝑁 = ((𝑇 · 𝐴) + 𝑌)       (𝑁 + 1) = ((𝑇 · 𝐵) + 0)

Theoremdecsucc 8394 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0    &   (𝐴 + 1) = 𝐵    &   𝑁 = 𝐴9       (𝑁 + 1) = 𝐵0

Theorem1e0p1 8395 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
1 = (0 + 1)

Theoremdec10p 8396 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
(10 + 𝐴) = 1𝐴

Theoremdec10 8397 The decimal form of 10. NB: In our presentations of large numbers later on, we will use our symbol for 10 at the highest digits when advantageous, because we can use this theorem to convert back to "long form" (where each digit is in the range 0-9) with no extra effort. However, we cannot do this for lower digits while maintaining the ease of use of the decimal system, since it requires nontrivial number knowledge (more than just equality theorems) to convert back. (Contributed by Mario Carneiro, 18-Feb-2014.)
10 = 10

Theoremnumma 8398 Perform a multiply-add of two decimal integers 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = ((𝑇 · 𝐴) + 𝐵)    &   𝑁 = ((𝑇 · 𝐶) + 𝐷)    &   𝑃 ∈ ℕ0    &   ((𝐴 · 𝑃) + 𝐶) = 𝐸    &   ((𝐵 · 𝑃) + 𝐷) = 𝐹       ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹)

Theoremnummac 8399 Perform a multiply-add of two decimal integers 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = ((𝑇 · 𝐴) + 𝐵)    &   𝑁 = ((𝑇 · 𝐶) + 𝐷)    &   𝑃 ∈ ℕ0    &   𝐹 ∈ ℕ0    &   𝐺 ∈ ℕ0    &   ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸    &   ((𝐵 · 𝑃) + 𝐷) = ((𝑇 · 𝐺) + 𝐹)       ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹)

Theoremnumma2c 8400 Perform a multiply-add of two decimal integers 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = ((𝑇 · 𝐴) + 𝐵)    &   𝑁 = ((𝑇 · 𝐶) + 𝐷)    &   𝑃 ∈ ℕ0    &   𝐹 ∈ ℕ0    &   𝐺 ∈ ℕ0    &   ((𝑃 · 𝐴) + (𝐶 + 𝐺)) = 𝐸    &   ((𝑃 · 𝐵) + 𝐷) = ((𝑇 · 𝐺) + 𝐹)       ((𝑃 · 𝑀) + 𝑁) = ((𝑇 · 𝐸) + 𝐹)

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