HomeHome Intuitionistic Logic Explorer
Theorem List (p. 85 of 106)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnneoi 8401 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 8402 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
(𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
 
Theoremzeo2 8403 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 8404* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ {𝑥 ∈ ℤ ∣ 𝐴𝑥}) → (𝐵 + 1) ∈ {𝑥 ∈ ℤ ∣ 𝐴𝑥})
 
Theorempeano5uzti 8405* 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 8406* 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 8407* An expression for the upper integers that start at 𝑁 that is analogous to dfnn2 7992 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 8408* 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 8409* 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 8410* 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 8411* 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 8412* 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 8413* 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 8414* 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 8415* 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 8416* 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 8417 A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ∈ ℤ)
 
Theoremnnzd 8418 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℤ)
 
Theoremzred 8419 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)       (𝜑𝐴 ∈ ℝ)
 
Theoremzcnd 8420 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)       (𝜑𝐴 ∈ ℂ)
 
Theoremznegcld 8421 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)       (𝜑 → -𝐴 ∈ ℤ)
 
Theorempeano2zd 8422 Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)       (𝜑 → (𝐴 + 1) ∈ ℤ)
 
Theoremzaddcld 8423 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)       (𝜑 → (𝐴 + 𝐵) ∈ ℤ)
 
Theoremzsubcld 8424 Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)       (𝜑 → (𝐴𝐵) ∈ ℤ)
 
Theoremzmulcld 8425 Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)       (𝜑 → (𝐴 · 𝐵) ∈ ℤ)
 
Theoremzadd2cl 8426 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 8427 Constant used for decimal constructor.
class 𝐴𝐵
 
Definitiondf-dec 8428 Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (1000 + 2000) = 3000 1kp2ke3k 10278. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
 
Theorem9p1e10 8429 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 8430 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)
 
Theoremdeceq1 8431 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)
 
Theoremdeceq2 8432 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)
 
Theoremdeceq1i 8433 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 = 𝐵       𝐴𝐶 = 𝐵𝐶
 
Theoremdeceq2i 8434 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 = 𝐵       𝐶𝐴 = 𝐶𝐵
 
Theoremdeceq12i 8435 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 = 𝐵    &   𝐶 = 𝐷       𝐴𝐶 = 𝐵𝐷
 
Theoremnumnncl 8436 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 8437 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0       (𝑇 · 𝐴) = ((𝑇 · 𝐴) + 0)
 
Theoremnum0h 8438 Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0       𝐴 = ((𝑇 · 0) + 𝐴)
 
Theoremnumcl 8439 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 8440 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 8441 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       𝐴𝐵 ∈ ℕ0
 
Theorem10nn 8442 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
10 ∈ ℕ
 
Theorem10pos 8443 The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
0 < 10
 
Theorem10nn0 8444 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
10 ∈ ℕ0
 
Theorem10re 8445 The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
10 ∈ ℝ
 
Theoremdecnncl 8446 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ       𝐴𝐵 ∈ ℕ
 
Theoremdec0u 8447 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
 
Theoremdec0h 8448 Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0       𝐴 = 0𝐴
 
Theoremnumnncl2 8449 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)
𝑇 ∈ ℕ    &   𝐴 ∈ ℕ       ((𝑇 · 𝐴) + 0) ∈ ℕ
 
Theoremdecnncl2 8450 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
 
Theoremnumlt 8451 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 8452 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)
 
Theoremle9lt10 8453 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
 
Theoremdeclt 8454 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
 
Theoremdecltc 8455 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
 
Theoremdeclth 8456 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
 
Theoremdecsuc 8457 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
 
Theorem3declth 8458 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
 
Theorem3decltc 8459 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
 
Theoremdecle 8460 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    &   𝐵𝐶       𝐴𝐵𝐴𝐶
 
Theoremdecleh 8461 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       𝐴𝐶𝐵𝐷
 
Theoremdeclei 8462 Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐶 ≤ 9       𝐶𝐴𝐵
 
Theoremnumlti 8463 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 8464 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
 
Theoremdecltdi 8465 Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐶 ≤ 9       𝐶 < 𝐴𝐵
 
Theoremnumsucc 8466 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 8467 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
 
Theorem1e0p1 8468 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
1 = (0 + 1)
 
Theoremdec10p 8469 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(10 + 𝐴) = 1𝐴
 
Theoremnumma 8470 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 8471 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 8472 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)
 
Theoremnumadd 8473 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)
 
Theoremnumaddc 8474 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)
 
Theoremnummul1c 8475 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)
 
Theoremnummul2c 8476 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)
 
Theoremdecma 8477 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
 
Theoremdecmac 8478 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
 
Theoremdecma2c 8479 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
 
Theoremdecadd 8480 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
 
Theoremdecaddc 8481 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
 
Theoremdecaddc2 8482 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
 
Theoremdecrmanc 8483 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
 
Theoremdecrmac 8484 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
 
Theoremdecaddm10 8485 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
 
Theoremdecaddi 8486 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
 
Theoremdecaddci 8487 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
 
Theoremdecaddci2 8488 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
 
Theoremdecsubi 8489 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    &   (𝐵𝑁) = 𝐶       (𝑀𝑁) = 𝐴𝐶
 
Theoremdecmul1 8490 The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
𝑃 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 = 𝐴𝐵    &   𝐷 ∈ ℕ0    &   (𝐴 · 𝑃) = 𝐶    &   (𝐵 · 𝑃) = 𝐷       (𝑁 · 𝑃) = 𝐶𝐷
 
Theoremdecmul1c 8491 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
 
Theoremdecmul2c 8492 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
 
Theoremdecmulnc 8493 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)
 
Theorem11multnc 8494 The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
𝑁 ∈ ℕ0       (𝑁 · 11) = 𝑁𝑁
 
Theoremdecmul10add 8495 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)
 
Theorem6p5lem 8496 Lemma for 6p5e11 8499 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   𝐵 = (𝐷 + 1)    &   𝐶 = (𝐸 + 1)    &   (𝐴 + 𝐷) = 1𝐸       (𝐴 + 𝐵) = 1𝐶
 
Theorem5p5e10 8497 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
(5 + 5) = 10
 
Theorem6p4e10 8498 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
(6 + 4) = 10
 
Theorem6p5e11 8499 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(6 + 5) = 11
 
Theorem6p6e12 8500 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 + 6) = 12
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10511
  Copyright terms: Public domain < Previous  Next >