Theorem List for Intuitionistic Logic Explorer - 8101-8200 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | nnltlem1 8101 |
Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|
⊢ ((𝑀 ∈
ℕ ∧ 𝑁 ∈
ℕ) → (𝑀 <
𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
|
Theorem | nnm1ge0 8102 |
A positive integer decreased by 1 is greater than or equal to 0.
(Contributed by AV, 30-Oct-2018.)
|
⊢ (𝑁 ∈
ℕ → 0 ≤ (𝑁
− 1)) |
|
Theorem | nn0ge0div 8103 |
Division of a nonnegative integer by a positive number is not negative.
(Contributed by Alexander van der Vekens, 14-Apr-2018.)
|
⊢ ((𝐾 ∈
ℕ0 ∧ 𝐿 ∈
ℕ) → 0 ≤ (𝐾
/ 𝐿)) |
|
Theorem | zdiv 8104* |
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)) |
|
Theorem | zdivadd 8105 |
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) |
|
Theorem | zdivmul 8106 |
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) |
|
Theorem | zextle 8107* |
An extensionality-like property for integer ordering. (Contributed by
NM, 29-Oct-2005.)
|
⊢ ((𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ ∧ ∀𝑘 ∈ ℤ
(𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁)) → 𝑀 = 𝑁) |
|
Theorem | zextlt 8108* |
An extensionality-like property for integer ordering. (Contributed by
NM, 29-Oct-2005.)
|
⊢ ((𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ ∧ ∀𝑘 ∈ ℤ
(𝑘 < 𝑀 ↔ 𝑘 < 𝑁)) → 𝑀 = 𝑁) |
|
Theorem | recnz 8109 |
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) |
|
Theorem | btwnnz 8110 |
A number between an integer and its successor is not an integer.
(Contributed by NM, 3-May-2005.)
|
⊢ ((A ∈ ℤ ∧
A < B ∧ B < (A + 1))
→ ¬ B ∈ ℤ) |
|
Theorem | gtndiv 8111 |
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) |
|
Theorem | halfnz 8112 |
One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
|
⊢ ¬ (1 / 2) ∈ ℤ |
|
Theorem | prime 8113* |
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 < x ∧ x ≤ A ∧ (A / x) ∈ ℕ)
→ x = A))) |
|
Theorem | msqznn 8114 |
The square of a nonzero integer is a positive integer. (Contributed by
NM, 2-Aug-2004.)
|
⊢ ((A ∈ ℤ ∧
A ≠ 0) → (A · A)
∈ ℕ) |
|
Theorem | zneo 8115 |
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 | nneoor 8116 |
A positive integer is even or odd. (Contributed by Jim Kingdon,
15-Mar-2020.)
|
⊢ (𝑁 ∈
ℕ → ((𝑁 / 2)
∈ ℕ ∨
((𝑁 + 1) / 2) ∈ ℕ)) |
|
Theorem | nneo 8117 |
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 8118 |
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 8119 |
An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
|
⊢ (𝑁 ∈
ℤ → ((𝑁 / 2)
∈ ℤ ∨
((𝑁 + 1) / 2) ∈ ℤ)) |
|
Theorem | zeo2 8120 |
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 8121* |
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 | peano5uzti 8122* |
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 | peano5uzi 8123* |
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 | dfuzi 8124* |
An expression for the upper integers that start at 𝑁 that is
analogous to dfnn2 7697 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 8125* |
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 8126* |
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 8127* |
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 8128* |
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 | fzind 8129* |
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 8130* |
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 8131* |
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 8132* |
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 | btwnz 8133* |
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 | nn0zd 8134 |
A positive integer is an integer. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (φ
→ A ∈ ℕ0)
⇒ ⊢ (φ → A ∈
ℤ) |
|
Theorem | nnzd 8135 |
A nonnegative integer is an integer. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (φ
→ A ∈ ℕ) ⇒ ⊢ (φ → A ∈
ℤ) |
|
Theorem | zred 8136 |
An integer is a real number. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (φ
→ A ∈ ℤ) ⇒ ⊢ (φ → A ∈
ℝ) |
|
Theorem | zcnd 8137 |
An integer is a complex number. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (φ
→ A ∈ ℤ) ⇒ ⊢ (φ → A ∈
ℂ) |
|
Theorem | znegcld 8138 |
Closure law for negative integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (φ
→ A ∈ ℤ) ⇒ ⊢ (φ → -A ∈
ℤ) |
|
Theorem | peano2zd 8139 |
Deduction from second Peano postulate generalized to integers.
(Contributed by Mario Carneiro, 28-May-2016.)
|
⊢ (φ
→ A ∈ ℤ) ⇒ ⊢ (φ → (A + 1) ∈
ℤ) |
|
Theorem | zaddcld 8140 |
Closure of addition of integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (φ
→ A ∈ ℤ) & ⊢ (φ → B ∈
ℤ) ⇒ ⊢ (φ → (A + B) ∈ ℤ) |
|
Theorem | zsubcld 8141 |
Closure of subtraction of integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (φ
→ A ∈ ℤ) & ⊢ (φ → B ∈
ℤ) ⇒ ⊢ (φ → (A − B)
∈ ℤ) |
|
Theorem | zmulcld 8142 |
Closure of multiplication of integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (φ
→ A ∈ ℤ) & ⊢ (φ → B ∈
ℤ) ⇒ ⊢ (φ → (A · B)
∈ ℤ) |
|
Theorem | zadd2cl 8143 |
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
|
|
Syntax | cdc 8144 |
Constant used for decimal constructor.
|
class ;AB |
|
Definition | df-dec 8145 |
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) |
|
Theorem | deceq1 8146 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
⊢ (A =
B → ;A𝐶 = ;B𝐶) |
|
Theorem | deceq2 8147 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
⊢ (A =
B → ;𝐶A =
;𝐶B) |
|
Theorem | deceq1i 8148 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
⊢ A =
B ⇒ ⊢ ;A𝐶 = ;B𝐶 |
|
Theorem | deceq2i 8149 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
⊢ A =
B ⇒ ⊢ ;𝐶A =
;𝐶B |
|
Theorem | deceq12i 8150 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
⊢ A =
B & ⊢ 𝐶 = 𝐷 ⇒ ⊢ ;A𝐶 = ;B𝐷 |
|
Theorem | numnncl 8151 |
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 8152 |
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 8153 |
Add a zero in the higher places. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
⊢ 𝑇 ∈
ℕ0
& ⊢ A ∈ ℕ0
⇒ ⊢ A = ((𝑇 · 0) + A) |
|
Theorem | numcl 8154 |
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 8155 |
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 | decnncl 8156 |
Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
|
⊢ A ∈ ℕ0 & ⊢ B ∈
ℕ ⇒ ⊢ ;AB ∈
ℕ |
|
Theorem | deccl 8157 |
Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
|
⊢ A ∈ ℕ0 & ⊢ B ∈
ℕ0 ⇒ ⊢ ;AB ∈
ℕ0 |
|
Theorem | dec0u 8158 |
Add a zero in the units place. (Contributed by Mario Carneiro,
17-Apr-2015.)
|
⊢ A ∈ ℕ0
⇒ ⊢ (10 · A) = ;A0 |
|
Theorem | dec0h 8159 |
Add a zero in the higher places. (Contributed by Mario Carneiro,
17-Apr-2015.)
|
⊢ A ∈ ℕ0
⇒ ⊢ A = ;0A |
|
Theorem | numnncl2 8160 |
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 8161 |
Closure for a decimal integer (zero units place). (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
⊢ A ∈ ℕ ⇒ ⊢ ;A0 ∈ ℕ |
|
Theorem | numlt 8162 |
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 8163 |
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 | declt 8164 |
Comparing two decimal integers (equal higher places). (Contributed by
Mario Carneiro, 17-Apr-2015.)
|
⊢ A ∈ ℕ0 & ⊢ B ∈
ℕ0
& ⊢ 𝐶 ∈
ℕ
& ⊢ B <
𝐶
⇒ ⊢ ;AB < ;A𝐶 |
|
Theorem | decltc 8165 |
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 |
|
Theorem | decsuc 8166 |
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 |
|
Theorem | numlti 8167 |
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 8168 |
Comparing a digit to a decimal integer. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
⊢ A ∈ ℕ & ⊢ B ∈
ℕ0
& ⊢ 𝐶 ∈
ℕ0
& ⊢ 𝐶 < 10 ⇒ ⊢ 𝐶 < ;AB |
|
Theorem | numsucc 8169 |
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 8170 |
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 |
|
Theorem | 1e0p1 8171 |
The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
|
⊢ 1 = (0 + 1) |
|
Theorem | dec10p 8172 |
Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (10 + A) =
;1A |
|
Theorem | dec10 8173 |
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 |
|
Theorem | numma 8174 |
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 8175 |
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 8176 |
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 8177 |
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 8178 |
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 8179 |
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 8180 |
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 8181 |
Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed
multiplicand 𝑃 (no carry). (Contributed by Mario
Carneiro,
18-Feb-2014.)
|
⊢ 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 8182 |
Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed
multiplicand 𝑃 (with carry). (Contributed by Mario
Carneiro,
18-Feb-2014.)
|
⊢ 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 8183 |
Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed
multiplicand 𝑃 (with carry). (Contributed by Mario
Carneiro,
18-Feb-2014.)
|
⊢ 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 8184 |
Add two numerals 𝑀 and 𝑁 (no carry).
(Contributed by Mario
Carneiro, 18-Feb-2014.)
|
⊢ A ∈ ℕ0 & ⊢ B ∈
ℕ0
& ⊢ 𝐶 ∈
ℕ0
& ⊢ 𝐷 ∈
ℕ0
& ⊢ 𝑀 = ;AB
& ⊢ 𝑁 = ;𝐶𝐷
& ⊢ (A + 𝐶) = 𝐸
& ⊢ (B + 𝐷) = 𝐹 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐸𝐹 |
|
Theorem | decaddc 8185 |
Add two numerals 𝑀 and 𝑁 (with carry).
(Contributed by Mario
Carneiro, 18-Feb-2014.)
|
⊢ 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 8186 |
Add two numerals 𝑀 and 𝑁 (with carry).
(Contributed by Mario
Carneiro, 18-Feb-2014.)
|
⊢ A ∈ ℕ0 & ⊢ B ∈
ℕ0
& ⊢ 𝐶 ∈
ℕ0
& ⊢ 𝐷 ∈
ℕ0
& ⊢ 𝑀 = ;AB
& ⊢ 𝑁 = ;𝐶𝐷
& ⊢ ((A + 𝐶) + 1) = 𝐸
& ⊢ (B + 𝐷) = 10
⇒ ⊢ (𝑀 + 𝑁) = ;𝐸0 |
|
Theorem | decaddi 8187 |
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 8188 |
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 8189 |
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 & ⊢ (B + 𝑁) = 10 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐷0 |
|
Theorem | decmul1c 8190 |
The product of a numeral with a number. (Contributed by Mario
Carneiro, 18-Feb-2014.)
|
⊢ 𝑃 ∈
ℕ0
& ⊢ A ∈ ℕ0 & ⊢ B ∈
ℕ0
& ⊢ 𝑁 = ;AB
& ⊢ 𝐷 ∈
ℕ0
& ⊢ 𝐸 ∈
ℕ0
& ⊢ ((A
· 𝑃) + 𝐸) = 𝐶
& ⊢ (B ·
𝑃) = ;𝐸𝐷 ⇒ ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
|
Theorem | decmul2c 8191 |
The product of a numeral with a number (with carry). (Contributed by
Mario Carneiro, 18-Feb-2014.)
|
⊢ 𝑃 ∈
ℕ0
& ⊢ A ∈ ℕ0 & ⊢ B ∈
ℕ0
& ⊢ 𝑁 = ;AB
& ⊢ 𝐷 ∈
ℕ0
& ⊢ 𝐸 ∈
ℕ0
& ⊢ ((𝑃 · A) + 𝐸) = 𝐶
& ⊢ (𝑃 · B) = ;𝐸𝐷 ⇒ ⊢ (𝑃 · 𝑁) = ;𝐶𝐷 |
|
Theorem | 6p5lem 8192 |
Lemma for 6p5e11 8193 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 |
|
Theorem | 6p5e11 8193 |
6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (6 + 5) = ;11 |
|
Theorem | 6p6e12 8194 |
6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (6 + 6) = ;12 |
|
Theorem | 7p4e11 8195 |
7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (7 + 4) = ;11 |
|
Theorem | 7p5e12 8196 |
7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (7 + 5) = ;12 |
|
Theorem | 7p6e13 8197 |
7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (7 + 6) = ;13 |
|
Theorem | 7p7e14 8198 |
7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (7 + 7) = ;14 |
|
Theorem | 8p3e11 8199 |
8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (8 + 3) = ;11 |
|
Theorem | 8p4e12 8200 |
8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (8 + 4) = ;12 |