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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nneoi 12701 | 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 12702 | An integer is even or odd. (Contributed by NM, 1-Jan-2006.) |
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) | ||
Theorem | zeo2 12703 | 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 12704* | 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 12705* | 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 12706* | 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 12707* | 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 12708* | 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 12709* | 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 12710* | 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 12711* | 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 12712* | 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 12711 or nn0indALT 12712 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 12713* | 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 12714* | 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 12715* | 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 12716* | 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 12717* | 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 12718* | 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 12719* | 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 12720 | An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | zcnd 12721 | An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) | ||
Theorem | znegcld 12722 | Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → -𝐴 ∈ ℤ) | ||
Theorem | peano2zd 12723 | Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ) | ||
Theorem | zaddcld 12724 | Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ) | ||
Theorem | zsubcld 12725 | Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) | ||
Theorem | zmulcld 12726 | Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ) | ||
Theorem | znnn0nn 12727 | 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 12728 | 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 12729* | 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 12730 | 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 12731 | Constant used for decimal constructor. |
class ;𝐴𝐵 | ||
Definition | df-dec 12732 | Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 30382. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.) |
⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵) | ||
Theorem | 9p1e10 12733 | 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 12734 | 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 12735 | A decimal number is a set. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ ;𝐴𝐵 ∈ V | ||
Theorem | deceq1 12736 | 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 12737 | 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 12738 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ;𝐴𝐶 = ;𝐵𝐶 | ||
Theorem | deceq2i 12739 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ;𝐶𝐴 = ;𝐶𝐵 | ||
Theorem | deceq12i 12740 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ ;𝐴𝐶 = ;𝐵𝐷 | ||
Theorem | numnncl 12741 | 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 12742 | 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 12743 | Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑇 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ 𝐴 = ((𝑇 · 0) + 𝐴) | ||
Theorem | numcl 12744 | 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 12745 | 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 12746 | 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 12747 | 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.) |
⊢ ;10 ∈ ℕ | ||
Theorem | 10pos 12748 | The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.) |
⊢ 0 < ;10 | ||
Theorem | 10nn0 12749 | 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ ;10 ∈ ℕ0 | ||
Theorem | 10re 12750 | 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 12751 | 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 12752 | 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 12753 | 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 12754 | 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 12755 | 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 12756 | 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 12757 | 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 12758 | 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 12759 | 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 12760 | 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 12761 | 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 12762 | 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 12763 | 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 12764 | 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 12765 | 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 12766 | 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 12767 | 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 12768 | 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 12769 | 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 12770 | 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 12771 | 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 12772 | 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 12773 | The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 1 = (0 + 1) | ||
Theorem | dec10p 12774 | Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (;10 + 𝐴) = ;1𝐴 | ||
Theorem | numma 12775 | 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 12776 | 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 12777 | 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 12778 | 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 12779 | 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 12780 | 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 12781 | 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 12782 | 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 12783 | 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 12784 | 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 12785 | 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 12786 | 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 12787 | 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 12788 | 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 12789 | 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 12790 | 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 12791 | 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 12792 | 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 12793 | 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 12794 | 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 12795 | 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 12796 | 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 12797 | 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 12798 | 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) | ||
Theorem | 11multnc 12799 | The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.) |
⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝑁 · ;11) = ;𝑁𝑁 | ||
Theorem | decmul10add 12800 | A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑀 ∈ ℕ0 & ⊢ 𝐸 = (𝑀 · 𝐴) & ⊢ 𝐹 = (𝑀 · 𝐵) ⇒ ⊢ (𝑀 · ;𝐴𝐵) = (;𝐸0 + 𝐹) |
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