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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | zneo 12701 | 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)) | ||
| Theorem | nneo 12702 | 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)) | ||
| Theorem | nneoi 12703 | 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) | ||
| Theorem | zeo 12704 | An integer is even or odd. (Contributed by NM, 1-Jan-2006.) |
| ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) | ||
| Theorem | zeo2 12705 | 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)) | ||
| Theorem | peano2uz2 12706* | Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.) |
| ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ {𝑥 ∈ ℤ ∣ 𝐴 ≤ 𝑥}) → (𝐵 + 1) ∈ {𝑥 ∈ ℤ ∣ 𝐴 ≤ 𝑥}) | ||
| Theorem | peano5uzi 12707* | Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.) |
| ⊢ 𝑁 ∈ ℤ ⇒ ⊢ ((𝑁 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ⊆ 𝐴) | ||
| Theorem | peano5uzti 12708* | Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.) |
| ⊢ (𝑁 ∈ ℤ → ((𝑁 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ⊆ 𝐴)) | ||
| Theorem | dfuzi 12709* | An expression for the upper integers that start at 𝑁 that is analogous to dfnn2 12279 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.) |
| ⊢ 𝑁 ∈ ℤ ⇒ ⊢ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} = ∩ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} | ||
| Theorem | uzind 12710* | 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) & ⊢ (𝑀 ∈ ℤ → 𝜓) & ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) → (𝜒 → 𝜃)) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝜏) | ||
| Theorem | uzind2 12711* | 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) → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) & ⊢ (𝑀 ∈ ℤ → 𝜓) & ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 < 𝑘) → (𝜒 → 𝜃)) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → 𝜏) | ||
| Theorem | uzind3 12712* | 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑚 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑚 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) & ⊢ (𝑀 ∈ ℤ → 𝜓) & ⊢ ((𝑀 ∈ ℤ ∧ 𝑚 ∈ {𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘}) → (𝜒 → 𝜃)) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ {𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘}) → 𝜏) | ||
| Theorem | nn0ind 12713* | 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃)) ⇒ ⊢ (𝐴 ∈ ℕ0 → 𝜏) | ||
| Theorem | nn0indALT 12714* | 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 12713 or nn0indALT 12714 may be used; see comment for nnind 12284. (Contributed by NM, 28-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃)) & ⊢ 𝜓 & ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) ⇒ ⊢ (𝐴 ∈ ℕ0 → 𝜏) | ||
| Theorem | nn0indd 12715* | Principle of Mathematical Induction (inference schema) on nonnegative integers, a deduction version. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
| ⊢ (𝑥 = 0 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) & ⊢ (𝜑 → 𝜒) & ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ0) → 𝜂) | ||
| Theorem | fzind 12716* | 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏)) & ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝜓) & ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁)) → (𝜒 → 𝜃)) ⇒ ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝜏) | ||
| Theorem | fnn0ind 12717* | 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏)) & ⊢ (𝑁 ∈ ℕ0 → 𝜓) & ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ∧ 𝑦 < 𝑁) → (𝜒 → 𝜃)) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁) → 𝜏) | ||
| Theorem | nn0ind-raph 12718* | 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃)) ⇒ ⊢ (𝐴 ∈ ℕ0 → 𝜏) | ||
| Theorem | zindd 12719* | 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜏)) & ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) & ⊢ (𝜁 → 𝜓) & ⊢ (𝜁 → (𝑦 ∈ ℕ0 → (𝜒 → 𝜏))) & ⊢ (𝜁 → (𝑦 ∈ ℕ → (𝜒 → 𝜃))) ⇒ ⊢ (𝜁 → (𝐴 ∈ ℤ → 𝜂)) | ||
| Theorem | fzindd 12720* | Induction on the integers from M to N inclusive, a deduction version. (Contributed by metakunt, 12-May-2024.) |
| ⊢ (𝑥 = 𝑀 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁) ∧ 𝜃) → 𝜏) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ ℤ ∧ 𝑀 ≤ 𝐴 ∧ 𝐴 ≤ 𝑁)) → 𝜂) | ||
| Theorem | btwnz 12721* | 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)) | ||
| Theorem | zred 12722 | An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
| Theorem | zcnd 12723 | An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) | ||
| Theorem | znegcld 12724 | Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → -𝐴 ∈ ℤ) | ||
| Theorem | peano2zd 12725 | Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ) | ||
| Theorem | zaddcld 12726 | Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ) | ||
| Theorem | zsubcld 12727 | Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) | ||
| Theorem | zmulcld 12728 | Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ) | ||
| Theorem | znnn0nn 12729 | The negative of a negative integer, is a natural number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((𝑁 ∈ ℤ ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℕ) | ||
| Theorem | zadd2cl 12730 | Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.) |
| ⊢ (𝑁 ∈ ℤ → (𝑁 + 2) ∈ ℤ) | ||
| Theorem | zriotaneg 12731* | The negative of the unique integer such that 𝜑. (Contributed by AV, 1-Dec-2018.) |
| ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ ℤ 𝜑 → (℩𝑥 ∈ ℤ 𝜑) = -(℩𝑦 ∈ ℤ 𝜓)) | ||
| Theorem | suprfinzcl 12732 | 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) | ||
| Syntax | cdc 12733 | Constant used for decimal constructor. |
| class ;𝐴𝐵 | ||
| Definition | df-dec 12734 | Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 30465. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.) |
| ⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵) | ||
| Theorem | 9p1e10 12735 | 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 | ||
| Theorem | dfdec10 12736 | 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) | ||
| Theorem | decex 12737 | A decimal number is a set. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
| ⊢ ;𝐴𝐵 ∈ V | ||
| Theorem | deceq1 12738 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
| ⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶) | ||
| Theorem | deceq2 12739 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
| ⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵) | ||
| Theorem | deceq1i 12740 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) |
| ⊢ 𝐴 = 𝐵 ⇒ ⊢ ;𝐴𝐶 = ;𝐵𝐶 | ||
| Theorem | deceq2i 12741 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) |
| ⊢ 𝐴 = 𝐵 ⇒ ⊢ ;𝐶𝐴 = ;𝐶𝐵 | ||
| Theorem | deceq12i 12742 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) |
| ⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ ;𝐴𝐶 = ;𝐵𝐷 | ||
| Theorem | numnncl 12743 | Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.) |
| ⊢ 𝑇 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ ⇒ ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ | ||
| Theorem | num0u 12744 | Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.) |
| ⊢ 𝑇 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ (𝑇 · 𝐴) = ((𝑇 · 𝐴) + 0) | ||
| Theorem | num0h 12745 | Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.) |
| ⊢ 𝑇 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ 𝐴 = ((𝑇 · 0) + 𝐴) | ||
| Theorem | numcl 12746 | 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 | ||
| Theorem | numsuc 12747 | 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) | ||
| Theorem | deccl 12748 | Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 ⇒ ⊢ ;𝐴𝐵 ∈ ℕ0 | ||
| Theorem | 10nn 12749 | 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.) |
| ⊢ ;10 ∈ ℕ | ||
| Theorem | 10pos 12750 | The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.) |
| ⊢ 0 < ;10 | ||
| Theorem | 10nn0 12751 | 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
| ⊢ ;10 ∈ ℕ0 | ||
| Theorem | 10re 12752 | The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.) |
| ⊢ ;10 ∈ ℝ | ||
| Theorem | decnncl 12753 | Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ ⇒ ⊢ ;𝐴𝐵 ∈ ℕ | ||
| Theorem | dec0u 12754 | 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 | ||
| Theorem | dec0h 12755 | Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ 𝐴 = ;0𝐴 | ||
| Theorem | numnncl2 12756 | Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.) |
| ⊢ 𝑇 ∈ ℕ & ⊢ 𝐴 ∈ ℕ ⇒ ⊢ ((𝑇 · 𝐴) + 0) ∈ ℕ | ||
| Theorem | decnncl2 12757 | Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ ⇒ ⊢ ;𝐴0 ∈ ℕ | ||
| Theorem | numlt 12758 | 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) | ||
| Theorem | numltc 12759 | 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) | ||
| Theorem | le9lt10 12760 | A "decimal digit" (i.e. a nonnegative integer less than or equal to 9) is less then 10. (Contributed by AV, 8-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐴 ≤ 9 ⇒ ⊢ 𝐴 < ;10 | ||
| Theorem | declt 12761 | Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ & ⊢ 𝐵 < 𝐶 ⇒ ⊢ ;𝐴𝐵 < ;𝐴𝐶 | ||
| Theorem | decltc 12762 | Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐶 < ;10 & ⊢ 𝐴 < 𝐵 ⇒ ⊢ ;𝐴𝐶 < ;𝐵𝐷 | ||
| Theorem | declth 12763 | Comparing two decimal integers (unequal higher places). (Contributed by AV, 8-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐶 ≤ 9 & ⊢ 𝐴 < 𝐵 ⇒ ⊢ ;𝐴𝐶 < ;𝐵𝐷 | ||
| Theorem | decsuc 12764 | The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ (𝐵 + 1) = 𝐶 & ⊢ 𝑁 = ;𝐴𝐵 ⇒ ⊢ (𝑁 + 1) = ;𝐴𝐶 | ||
| Theorem | 3declth 12765 | Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 8-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐴 < 𝐵 & ⊢ 𝐶 ≤ 9 & ⊢ 𝐸 ≤ 9 ⇒ ⊢ ;;𝐴𝐶𝐸 < ;;𝐵𝐷𝐹 | ||
| Theorem | 3decltc 12766 | Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 15-Jun-2021.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐴 < 𝐵 & ⊢ 𝐶 < ;10 & ⊢ 𝐸 < ;10 ⇒ ⊢ ;;𝐴𝐶𝐸 < ;;𝐵𝐷𝐹 | ||
| Theorem | decle 12767 | Comparing two decimal integers (equal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐵 ≤ 𝐶 ⇒ ⊢ ;𝐴𝐵 ≤ ;𝐴𝐶 | ||
| Theorem | decleh 12768 | Comparing two decimal integers (unequal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐶 ≤ 9 & ⊢ 𝐴 < 𝐵 ⇒ ⊢ ;𝐴𝐶 ≤ ;𝐵𝐷 | ||
| Theorem | declei 12769 | Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.) |
| ⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐶 ≤ 9 ⇒ ⊢ 𝐶 ≤ ;𝐴𝐵 | ||
| Theorem | numlti 12770 | 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) | ||
| Theorem | declti 12771 | Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐶 < ;10 ⇒ ⊢ 𝐶 < ;𝐴𝐵 | ||
| Theorem | decltdi 12772 | Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐶 ≤ 9 ⇒ ⊢ 𝐶 < ;𝐴𝐵 | ||
| Theorem | numsucc 12773 | 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) | ||
| Theorem | decsucc 12774 | The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ (𝐴 + 1) = 𝐵 & ⊢ 𝑁 = ;𝐴9 ⇒ ⊢ (𝑁 + 1) = ;𝐵0 | ||
| Theorem | 1e0p1 12775 | The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.) |
| ⊢ 1 = (0 + 1) | ||
| Theorem | dec10p 12776 | Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
| ⊢ (;10 + 𝐴) = ;1𝐴 | ||
| Theorem | numma 12777 | 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) | ||
| Theorem | nummac 12778 | 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) | ||
| Theorem | numma2c 12779 | 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) | ||
| Theorem | numadd 12780 | Add two decimal integers 𝑀 and 𝑁 (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) & ⊢ (𝐴 + 𝐶) = 𝐸 & ⊢ (𝐵 + 𝐷) = 𝐹 ⇒ ⊢ (𝑀 + 𝑁) = ((𝑇 · 𝐸) + 𝐹) | ||
| Theorem | numaddc 12781 | Add two decimal integers 𝑀 and 𝑁 (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) & ⊢ 𝐹 ∈ ℕ0 & ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 & ⊢ (𝐵 + 𝐷) = ((𝑇 · 1) + 𝐹) ⇒ ⊢ (𝑀 + 𝑁) = ((𝑇 · 𝐸) + 𝐹) | ||
| Theorem | nummul1c 12782 | The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.) |
| ⊢ 𝑇 ∈ ℕ0 & ⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝐵) & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 & ⊢ (𝐵 · 𝑃) = ((𝑇 · 𝐸) + 𝐷) ⇒ ⊢ (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷) | ||
| Theorem | nummul2c 12783 | The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
| ⊢ 𝑇 ∈ ℕ0 & ⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝐵) & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 & ⊢ (𝑃 · 𝐵) = ((𝑇 · 𝐸) + 𝐷) ⇒ ⊢ (𝑃 · 𝑁) = ((𝑇 · 𝐶) + 𝐷) | ||
| Theorem | decma 12784 | Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ 𝑃 ∈ ℕ0 & ⊢ ((𝐴 · 𝑃) + 𝐶) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝐷) = 𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 | ||
| Theorem | decmac 12785 | Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐺 ∈ ℕ0 & ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝐷) = ;𝐺𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 | ||
| Theorem | decma2c 12786 | Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplier 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐺 ∈ ℕ0 & ⊢ ((𝑃 · 𝐴) + (𝐶 + 𝐺)) = 𝐸 & ⊢ ((𝑃 · 𝐵) + 𝐷) = ;𝐺𝐹 ⇒ ⊢ ((𝑃 · 𝑀) + 𝑁) = ;𝐸𝐹 | ||
| Theorem | decadd 12787 | Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ (𝐴 + 𝐶) = 𝐸 & ⊢ (𝐵 + 𝐷) = 𝐹 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐸𝐹 | ||
| Theorem | decaddc 12788 | Add two numerals 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 & ⊢ 𝐹 ∈ ℕ0 & ⊢ (𝐵 + 𝐷) = ;1𝐹 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐸𝐹 | ||
| Theorem | decaddc2 12789 | Add two numerals 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 & ⊢ (𝐵 + 𝐷) = ;10 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐸0 | ||
| Theorem | decrmanc 12790 | Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no carry). (Contributed by AV, 16-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑃 ∈ ℕ0 & ⊢ (𝐴 · 𝑃) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝑁) = 𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 | ||
| Theorem | decrmac 12791 | Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by AV, 16-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐺 ∈ ℕ0 & ⊢ ((𝐴 · 𝑃) + 𝐺) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝑁) = ;𝐺𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 | ||
| Theorem | decaddm10 12792 | The sum of two multiples of 10 is a multiple of 10. (Contributed by AV, 30-Jul-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 ⇒ ⊢ (;𝐴0 + ;𝐵0) = ;(𝐴 + 𝐵)0 | ||
| Theorem | decaddi 12793 | Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ (𝐵 + 𝑁) = 𝐶 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐴𝐶 | ||
| Theorem | decaddci 12794 | Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ (𝐴 + 1) = 𝐷 & ⊢ 𝐶 ∈ ℕ0 & ⊢ (𝐵 + 𝑁) = ;1𝐶 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐷𝐶 | ||
| Theorem | decaddci2 12795 | Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ (𝐴 + 1) = 𝐷 & ⊢ (𝐵 + 𝑁) = ;10 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐷0 | ||
| Theorem | decsubi 12796 | Difference between a numeral 𝑀 and a nonnegative integer 𝑁 (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ (𝐴 + 1) = 𝐷 & ⊢ (𝐵 − 𝑁) = 𝐶 ⇒ ⊢ (𝑀 − 𝑁) = ;𝐴𝐶 | ||
| Theorem | decmul1 12797 | The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) Remove hypothesis 𝐷 ∈ ℕ0. (Revised by Steven Nguyen, 7-Dec-2022.) |
| ⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 = ;𝐴𝐵 & ⊢ (𝐴 · 𝑃) = 𝐶 & ⊢ (𝐵 · 𝑃) = 𝐷 ⇒ ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 | ||
| Theorem | decmul1c 12798 | The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 = ;𝐴𝐵 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 & ⊢ (𝐵 · 𝑃) = ;𝐸𝐷 ⇒ ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 | ||
| Theorem | decmul2c 12799 | The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
| ⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 = ;𝐴𝐵 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 & ⊢ (𝑃 · 𝐵) = ;𝐸𝐷 ⇒ ⊢ (𝑃 · 𝑁) = ;𝐶𝐷 | ||
| Theorem | decmulnc 12800 | The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021.) |
| ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 ⇒ ⊢ (𝑁 · ;𝐴𝐵) = ;(𝑁 · 𝐴)(𝑁 · 𝐵) | ||
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