Theorem List for Intuitionistic Logic Explorer - 8901-9000 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | iseqfn 8901* |
The sequence builder function is a function. (Contributed by Jim
Kingdon, 30-May-2020.)
|
⊢ (φ
→ 𝑀 ∈ ℤ) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ ((φ ∧
x ∈
(ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧
(x ∈
𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)
⇒ ⊢ (φ → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ≥‘𝑀)) |
|
Theorem | iseq1 8902* |
Value of the sequence builder function at its initial value.
(Contributed by Jim Kingdon, 31-May-2020.)
|
⊢ (φ
→ 𝑀 ∈ ℤ) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ ((φ ∧
x ∈
(ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧
(x ∈
𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)
⇒ ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹‘𝑀)) |
|
Theorem | iseqcl 8903* |
Closure properties of the recursive sequence builder. (Contributed by
Jim Kingdon, 1-Jun-2020.)
|
⊢ (φ
→ 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ ((φ ∧
x ∈
(ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧
(x ∈
𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)
⇒ ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) ∈ 𝑆) |
|
Theorem | iseqp1 8904* |
Value of the sequence builder function at a successor. (Contributed by
Jim Kingdon, 31-May-2020.)
|
⊢ (φ
→ 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ ((φ ∧
x ∈
(ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧
(x ∈
𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)
⇒ ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1)))) |
|
Theorem | iseqfveq2 8905* |
Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
|
⊢ (φ
→ 𝐾 ∈ (ℤ≥‘𝑀)) & ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺‘𝐾)) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ ((φ ∧
x ∈
(ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧
x ∈
(ℤ≥‘𝐾)) → (𝐺‘x) ∈ 𝑆) & ⊢ ((φ ∧
(x ∈
𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆) & ⊢ (φ → 𝑁 ∈
(ℤ≥‘𝐾)) & ⊢ ((φ ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘)) ⇒ ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁)) |
|
Theorem | iseqfeq2 8906* |
Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
|
⊢ (φ
→ 𝐾 ∈ (ℤ≥‘𝑀)) & ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺‘𝐾)) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ ((φ ∧
x ∈
(ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧
x ∈
(ℤ≥‘𝐾)) → (𝐺‘x) ∈ 𝑆) & ⊢ ((φ ∧
(x ∈
𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆) & ⊢ ((φ ∧ 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) → (𝐹‘𝑘) = (𝐺‘𝑘)) ⇒ ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆) ↾
(ℤ≥‘𝐾)) = seq𝐾( + , 𝐺, 𝑆)) |
|
Theorem | iseqfveq 8907* |
Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
|
⊢ (φ
→ 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((φ ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))
& ⊢ (φ
→ 𝑆 ∈ 𝑉)
& ⊢ ((φ
∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧
x ∈
(ℤ≥‘𝑀)) → (𝐺‘x) ∈ 𝑆) & ⊢ ((φ ∧
(x ∈
𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)
⇒ ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝑀( + , 𝐺, 𝑆)‘𝑁)) |
|
3.5.9 Integer powers
|
|
Syntax | cexp 8908 |
Extend class notation to include exponentiation of a complex number to an
integer power.
|
class ↑ |
|
Definition | df-iexp 8909* |
Define exponentiation to nonnegative integer powers. This definition is
not meant to be used directly; instead, exp0 8913
and expp1 8916 provide the
standard recursive definition. The up-arrow notation is used by Donald
Knuth for iterated exponentiation (Science 194, 1235-1242, 1976)
and
is convenient for us since we don't have superscripts. 10-Jun-2005: The
definition was extended to include zero exponents, so that 0↑0 = 1
per the convention of Definition 10-4.1 of [Gleason] p. 134.
4-Jun-2014: The definition was extended to include negative integer
exponents. The case x = 0, y <
0 gives the value (1 / 0),
so we will avoid this case in our theorems. (Contributed by Jim
Kingdon, 7-Jun-2020.)
|
⊢ ↑ = (x
∈ ℂ, y ∈ ℤ
↦ if(y = 0, 1, if(0 < y, (seq1( · , (ℕ × {x}), ℂ)‘y), (1 / (seq1( · , (ℕ ×
{x}), ℂ)‘-y))))) |
|
Theorem | expivallem 8910 |
Lemma for expival 8911. If we take a complex number apart from zero
and
raise it to a positive integer power, the result is apart from zero.
(Contributed by Jim Kingdon, 7-Jun-2020.)
|
⊢ ((A ∈ ℂ ∧
A # 0 ∧
𝑁 ∈ ℕ) → (seq1( · , (ℕ
× {A}), ℂ)‘𝑁) # 0) |
|
Theorem | expival 8911 |
Value of exponentiation to integer powers. (Contributed by Jim Kingdon,
7-Jun-2020.)
|
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℤ ∧
(A # 0 ∨
0 ≤ 𝑁)) →
(A↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ ×
{A}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ ×
{A}), ℂ)‘-𝑁))))) |
|
Theorem | expinnval 8912 |
Value of exponentiation to positive integer powers. (Contributed by Jim
Kingdon, 8-Jun-2020.)
|
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ) → (A↑𝑁) = (seq1( · , (ℕ ×
{A}), ℂ)‘𝑁)) |
|
Theorem | exp0 8913 |
Value of a complex number raised to the 0th power. Note that under our
definition, 0↑0 = 1, following the convention
used by Gleason.
Part of Definition 10-4.1 of [Gleason] p.
134. (Contributed by NM,
20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)
|
⊢ (A ∈ ℂ → (A↑0) = 1) |
|
Theorem | 0exp0e1 8914 |
0↑0 = 1 (common case). This is our convention.
It follows the
convention used by Gleason; see Part of Definition 10-4.1 of [Gleason]
p. 134. (Contributed by David A. Wheeler, 8-Dec-2018.)
|
⊢ (0↑0) = 1 |
|
Theorem | exp1 8915 |
Value of a complex number raised to the first power. (Contributed by
NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.)
|
⊢ (A ∈ ℂ → (A↑1) = A) |
|
Theorem | expp1 8916 |
Value of a complex number raised to a nonnegative integer power plus
one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by
NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.)
|
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (A↑(𝑁 + 1)) = ((A↑𝑁) · A)) |
|
Theorem | expnegap0 8917 |
Value of a complex number raised to a negative integer power.
(Contributed by Jim Kingdon, 8-Jun-2020.)
|
⊢ ((A ∈ ℂ ∧
A # 0 ∧
𝑁 ∈ ℕ0) → (A↑-𝑁) = (1 / (A↑𝑁))) |
|
Theorem | expineg2 8918 |
Value of a complex number raised to a negative integer power.
(Contributed by Jim Kingdon, 8-Jun-2020.)
|
⊢ (((A ∈ ℂ ∧
A # 0) ∧
(𝑁 ∈ ℂ ∧
-𝑁 ∈ ℕ0)) → (A↑𝑁) = (1 / (A↑-𝑁))) |
|
Theorem | expn1ap0 8919 |
A number to the negative one power is the reciprocal. (Contributed by Jim
Kingdon, 8-Jun-2020.)
|
⊢ ((A ∈ ℂ ∧
A # 0) → (A↑-1) = (1 / A)) |
|
Theorem | expcllem 8920* |
Lemma for proving nonnegative integer exponentiation closure laws.
(Contributed by NM, 14-Dec-2005.)
|
⊢ 𝐹 ⊆ ℂ & ⊢ ((x ∈ 𝐹 ∧ y ∈ 𝐹) → (x · y)
∈ 𝐹)
& ⊢ 1 ∈ 𝐹 ⇒ ⊢ ((A ∈ 𝐹 ∧ B ∈ ℕ0) → (A↑B) ∈ 𝐹) |
|
Theorem | expcl2lemap 8921* |
Lemma for proving integer exponentiation closure laws. (Contributed by
Jim Kingdon, 8-Jun-2020.)
|
⊢ 𝐹 ⊆ ℂ & ⊢ ((x ∈ 𝐹 ∧ y ∈ 𝐹) → (x · y)
∈ 𝐹)
& ⊢ 1 ∈ 𝐹 & ⊢ ((x ∈ 𝐹 ∧ x # 0) →
(1 / x) ∈ 𝐹) ⇒ ⊢ ((A ∈ 𝐹 ∧ A # 0 ∧ B ∈ ℤ) → (A↑B) ∈ 𝐹) |
|
Theorem | nnexpcl 8922 |
Closure of exponentiation of nonnegative integers. (Contributed by NM,
16-Dec-2005.)
|
⊢ ((A ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈
ℕ) |
|
Theorem | nn0expcl 8923 |
Closure of exponentiation of nonnegative integers. (Contributed by NM,
14-Dec-2005.)
|
⊢ ((A ∈ ℕ0 ∧ 𝑁 ∈
ℕ0) → (A↑𝑁) ∈ ℕ0) |
|
Theorem | zexpcl 8924 |
Closure of exponentiation of integers. (Contributed by NM,
16-Dec-2005.)
|
⊢ ((A ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈
ℤ) |
|
Theorem | qexpcl 8925 |
Closure of exponentiation of rationals. (Contributed by NM,
16-Dec-2005.)
|
⊢ ((A ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈
ℚ) |
|
Theorem | reexpcl 8926 |
Closure of exponentiation of reals. (Contributed by NM,
14-Dec-2005.)
|
⊢ ((A ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈
ℝ) |
|
Theorem | expcl 8927 |
Closure law for nonnegative integer exponentiation. (Contributed by NM,
26-May-2005.)
|
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈
ℂ) |
|
Theorem | rpexpcl 8928 |
Closure law for exponentiation of positive reals. (Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
|
⊢ ((A ∈ ℝ+ ∧ 𝑁 ∈
ℤ) → (A↑𝑁) ∈
ℝ+) |
|
Theorem | reexpclzap 8929 |
Closure of exponentiation of reals. (Contributed by Jim Kingdon,
9-Jun-2020.)
|
⊢ ((A ∈ ℝ ∧
A # 0 ∧
𝑁 ∈ ℤ) → (A↑𝑁) ∈
ℝ) |
|
Theorem | qexpclz 8930 |
Closure of exponentiation of rational numbers. (Contributed by Mario
Carneiro, 9-Sep-2014.)
|
⊢ ((A ∈ ℚ ∧
A ≠ 0 ∧ 𝑁 ∈
ℤ) → (A↑𝑁) ∈
ℚ) |
|
Theorem | m1expcl2 8931 |
Closure of exponentiation of negative one. (Contributed by Mario
Carneiro, 18-Jun-2015.)
|
⊢ (𝑁 ∈
ℤ → (-1↑𝑁)
∈ {-1, 1}) |
|
Theorem | m1expcl 8932 |
Closure of exponentiation of negative one. (Contributed by Mario
Carneiro, 18-Jun-2015.)
|
⊢ (𝑁 ∈
ℤ → (-1↑𝑁)
∈ ℤ) |
|
Theorem | expclzaplem 8933* |
Closure law for integer exponentiation. Lemma for expclzap 8934 and
expap0i 8941. (Contributed by Jim Kingdon, 9-Jun-2020.)
|
⊢ ((A ∈ ℂ ∧
A # 0 ∧
𝑁 ∈ ℤ) → (A↑𝑁) ∈
{z ∈
ℂ ∣ z # 0}) |
|
Theorem | expclzap 8934 |
Closure law for integer exponentiation. (Contributed by Jim Kingdon,
9-Jun-2020.)
|
⊢ ((A ∈ ℂ ∧
A # 0 ∧
𝑁 ∈ ℤ) → (A↑𝑁) ∈
ℂ) |
|
Theorem | nn0expcli 8935 |
Closure of exponentiation of nonnegative integers. (Contributed by
Mario Carneiro, 17-Apr-2015.)
|
⊢ A ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0
⇒ ⊢ (A↑𝑁) ∈
ℕ0 |
|
Theorem | nn0sqcl 8936 |
The square of a nonnegative integer is a nonnegative integer.
(Contributed by Stefan O'Rear, 16-Oct-2014.)
|
⊢ (A ∈ ℕ0 → (A↑2) ∈
ℕ0) |
|
Theorem | expm1t 8937 |
Exponentiation in terms of predecessor exponent. (Contributed by NM,
19-Dec-2005.)
|
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ) → (A↑𝑁) = ((A↑(𝑁 − 1)) · A)) |
|
Theorem | 1exp 8938 |
Value of one raised to a nonnegative integer power. (Contributed by NM,
15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
|
⊢ (𝑁 ∈
ℤ → (1↑𝑁)
= 1) |
|
Theorem | expap0 8939 |
Positive integer exponentiation is apart from zero iff its mantissa is
apart from zero. That it is easier to prove this first, and then prove
expeq0 8940 in terms of it, rather than the other way
around, is perhaps an
illustration of the maxim "In constructive analysis, the apartness
is
more basic [ than ] equality." ([Geuvers], p. 1). (Contributed by Jim
Kingdon, 10-Jun-2020.)
|
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((A↑𝑁) # 0 ↔ A # 0)) |
|
Theorem | expeq0 8940 |
Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed
by NM, 23-Feb-2005.)
|
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((A↑𝑁) = 0 ↔ A = 0)) |
|
Theorem | expap0i 8941 |
Integer exponentiation is apart from zero if its mantissa is apart from
zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
|
⊢ ((A ∈ ℂ ∧
A # 0 ∧
𝑁 ∈ ℤ) → (A↑𝑁) # 0) |
|
Theorem | expgt0 8942 |
Nonnegative integer exponentiation with a positive mantissa is positive.
(Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro,
4-Jun-2014.)
|
⊢ ((A ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 0
< A) → 0 < (A↑𝑁)) |
|
Theorem | expnegzap 8943 |
Value of a complex number raised to a negative power. (Contributed by
Mario Carneiro, 4-Jun-2014.)
|
⊢ ((A ∈ ℂ ∧
A # 0 ∧
𝑁 ∈ ℤ) → (A↑-𝑁) = (1 / (A↑𝑁))) |
|
Theorem | 0exp 8944 |
Value of zero raised to a positive integer power. (Contributed by NM,
19-Aug-2004.)
|
⊢ (𝑁 ∈
ℕ → (0↑𝑁)
= 0) |
|
Theorem | expge0 8945 |
Nonnegative integer exponentiation with a nonnegative mantissa is
nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario
Carneiro, 4-Jun-2014.)
|
⊢ ((A ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ A)
→ 0 ≤ (A↑𝑁)) |
|
Theorem | expge1 8946 |
Nonnegative integer exponentiation with a mantissa greater than or equal
to 1 is greater than or equal to 1. (Contributed by NM, 21-Feb-2005.)
(Revised by Mario Carneiro, 4-Jun-2014.)
|
⊢ ((A ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 1 ≤ A)
→ 1 ≤ (A↑𝑁)) |
|
Theorem | expgt1 8947 |
Positive integer exponentiation with a mantissa greater than 1 is greater
than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro,
4-Jun-2014.)
|
⊢ ((A ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1
< A) → 1 < (A↑𝑁)) |
|
Theorem | mulexp 8948 |
Positive integer exponentiation of a product. Proposition 10-4.2(c) of
[Gleason] p. 135, restricted to
nonnegative integer exponents.
(Contributed by NM, 13-Feb-2005.)
|
⊢ ((A ∈ ℂ ∧
B ∈
ℂ ∧ 𝑁 ∈
ℕ0) → ((A ·
B)↑𝑁) = ((A↑𝑁) · (B↑𝑁))) |
|
Theorem | mulexpzap 8949 |
Integer exponentiation of a product. (Contributed by Jim Kingdon,
10-Jun-2020.)
|
⊢ (((A ∈ ℂ ∧
A # 0) ∧
(B ∈
ℂ ∧ B # 0) ∧ 𝑁 ∈ ℤ) → ((A · B)↑𝑁) = ((A↑𝑁) · (B↑𝑁))) |
|
Theorem | exprecap 8950 |
Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim
Kingdon, 10-Jun-2020.)
|
⊢ ((A ∈ ℂ ∧
A # 0 ∧
𝑁 ∈ ℤ) → ((1 / A)↑𝑁) = (1 / (A↑𝑁))) |
|
Theorem | expadd 8951 |
Sum of exponents law for nonnegative integer exponentiation.
Proposition 10-4.2(a) of [Gleason] p.
135. (Contributed by NM,
30-Nov-2004.)
|
⊢ ((A ∈ ℂ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈
ℕ0) → (A↑(𝑀 + 𝑁)) = ((A↑𝑀) · (A↑𝑁))) |
|
Theorem | expaddzaplem 8952 |
Lemma for expaddzap 8953. (Contributed by Jim Kingdon, 10-Jun-2020.)
|
⊢ (((A ∈ ℂ ∧
A # 0) ∧
(𝑀 ∈ ℝ ∧
-𝑀 ∈ ℕ) ∧
𝑁 ∈ ℕ0) → (A↑(𝑀 + 𝑁)) = ((A↑𝑀) · (A↑𝑁))) |
|
Theorem | expaddzap 8953 |
Sum of exponents law for integer exponentiation. (Contributed by Jim
Kingdon, 10-Jun-2020.)
|
⊢ (((A ∈ ℂ ∧
A # 0) ∧
(𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (A↑(𝑀 + 𝑁)) = ((A↑𝑀) · (A↑𝑁))) |
|
Theorem | expmul 8954 |
Product of exponents law for positive integer exponentiation.
Proposition 10-4.2(b) of [Gleason] p.
135, restricted to nonnegative
integer exponents. (Contributed by NM, 4-Jan-2006.)
|
⊢ ((A ∈ ℂ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈
ℕ0) → (A↑(𝑀 · 𝑁)) = ((A↑𝑀)↑𝑁)) |
|
Theorem | expmulzap 8955 |
Product of exponents law for integer exponentiation. (Contributed by
Jim Kingdon, 11-Jun-2020.)
|
⊢ (((A ∈ ℂ ∧
A # 0) ∧
(𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (A↑(𝑀 · 𝑁)) = ((A↑𝑀)↑𝑁)) |
|
Theorem | expsubap 8956 |
Exponent subtraction law for nonnegative integer exponentiation.
(Contributed by Jim Kingdon, 11-Jun-2020.)
|
⊢ (((A ∈ ℂ ∧
A # 0) ∧
(𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (A↑(𝑀 − 𝑁)) = ((A↑𝑀) / (A↑𝑁))) |
|
Theorem | expp1zap 8957 |
Value of a nonzero complex number raised to an integer power plus one.
(Contributed by Jim Kingdon, 11-Jun-2020.)
|
⊢ ((A ∈ ℂ ∧
A # 0 ∧
𝑁 ∈ ℤ) → (A↑(𝑁 + 1)) = ((A↑𝑁) · A)) |
|
Theorem | expm1ap 8958 |
Value of a complex number raised to an integer power minus one.
(Contributed by Jim Kingdon, 11-Jun-2020.)
|
⊢ ((A ∈ ℂ ∧
A # 0 ∧
𝑁 ∈ ℤ) → (A↑(𝑁 − 1)) = ((A↑𝑁) / A)) |
|
Theorem | expdivap 8959 |
Nonnegative integer exponentiation of a quotient. (Contributed by Jim
Kingdon, 11-Jun-2020.)
|
⊢ ((A ∈ ℂ ∧
(B ∈
ℂ ∧ B # 0) ∧ 𝑁 ∈ ℕ0) → ((A / B)↑𝑁) = ((A↑𝑁) / (B↑𝑁))) |
|
Theorem | ltexp2a 8960 |
Ordering relationship for exponentiation. (Contributed by NM,
2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
|
⊢ (((A ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1
< A ∧
𝑀 < 𝑁)) → (A↑𝑀) < (A↑𝑁)) |
|
Theorem | leexp2a 8961 |
Weak ordering relationship for exponentiation. (Contributed by NM,
14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)
|
⊢ ((A ∈ ℝ ∧ 1
≤ A ∧
𝑁 ∈ (ℤ≥‘𝑀)) → (A↑𝑀) ≤ (A↑𝑁)) |
|
Theorem | leexp2r 8962 |
Weak ordering relationship for exponentiation. (Contributed by Paul
Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
|
⊢ (((A ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈
(ℤ≥‘𝑀)) ∧ (0
≤ A ∧
A ≤ 1)) → (A↑𝑁) ≤ (A↑𝑀)) |
|
Theorem | leexp1a 8963 |
Weak mantissa ordering relationship for exponentiation. (Contributed by
NM, 18-Dec-2005.)
|
⊢ (((A ∈ ℝ ∧
B ∈
ℝ ∧ 𝑁 ∈
ℕ0) ∧ (0 ≤ A ∧ A ≤ B))
→ (A↑𝑁) ≤ (B↑𝑁)) |
|
Theorem | exple1 8964 |
Nonnegative integer exponentiation with a mantissa between 0 and 1
inclusive is less than or equal to 1. (Contributed by Paul Chapman,
29-Dec-2007.) (Revised by Mario Carneiro, 5-Jun-2014.)
|
⊢ (((A ∈ ℝ ∧ 0
≤ A ∧
A ≤ 1) ∧ 𝑁 ∈
ℕ0) → (A↑𝑁) ≤ 1) |
|
Theorem | expubnd 8965 |
An upper bound on A↑𝑁 when 2 ≤
A. (Contributed by NM,
19-Dec-2005.)
|
⊢ ((A ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 2 ≤ A)
→ (A↑𝑁) ≤ ((2↑𝑁) · ((A − 1)↑𝑁))) |
|
Theorem | sqval 8966 |
Value of the square of a complex number. (Contributed by Raph Levien,
10-Apr-2004.)
|
⊢ (A ∈ ℂ → (A↑2) = (A
· A)) |
|
Theorem | sqneg 8967 |
The square of the negative of a number.) (Contributed by NM,
15-Jan-2006.)
|
⊢ (A ∈ ℂ → (-A↑2) = (A↑2)) |
|
Theorem | sqsubswap 8968 |
Swap the order of subtraction in a square. (Contributed by Scott Fenton,
10-Jun-2013.)
|
⊢ ((A ∈ ℂ ∧
B ∈
ℂ) → ((A − B)↑2) = ((B − A)↑2)) |
|
Theorem | sqcl 8969 |
Closure of square. (Contributed by NM, 10-Aug-1999.)
|
⊢ (A ∈ ℂ → (A↑2) ∈
ℂ) |
|
Theorem | sqmul 8970 |
Distribution of square over multiplication. (Contributed by NM,
21-Mar-2008.)
|
⊢ ((A ∈ ℂ ∧
B ∈
ℂ) → ((A · B)↑2) = ((A↑2) · (B↑2))) |
|
Theorem | sqeq0 8971 |
A number is zero iff its square is zero. (Contributed by NM,
11-Mar-2006.)
|
⊢ (A ∈ ℂ → ((A↑2) = 0 ↔ A = 0)) |
|
Theorem | sqdivap 8972 |
Distribution of square over division. (Contributed by Jim Kingdon,
11-Jun-2020.)
|
⊢ ((A ∈ ℂ ∧
B ∈
ℂ ∧ B # 0) → ((A / B)↑2)
= ((A↑2) / (B↑2))) |
|
Theorem | sqne0 8973 |
A number is nonzero iff its square is nonzero. (Contributed by NM,
11-Mar-2006.)
|
⊢ (A ∈ ℂ → ((A↑2) ≠ 0 ↔ A ≠ 0)) |
|
Theorem | resqcl 8974 |
Closure of the square of a real number. (Contributed by NM,
18-Oct-1999.)
|
⊢ (A ∈ ℝ → (A↑2) ∈
ℝ) |
|
Theorem | sqgt0ap 8975 |
The square of a nonzero real is positive. (Contributed by Jim Kingdon,
11-Jun-2020.)
|
⊢ ((A ∈ ℝ ∧
A # 0) → 0 < (A↑2)) |
|
Theorem | nnsqcl 8976 |
The naturals are closed under squaring. (Contributed by Scott Fenton,
29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|
⊢ (A ∈ ℕ → (A↑2) ∈
ℕ) |
|
Theorem | zsqcl 8977 |
Integers are closed under squaring. (Contributed by Scott Fenton,
18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|
⊢ (A ∈ ℤ → (A↑2) ∈
ℤ) |
|
Theorem | qsqcl 8978 |
The square of a rational is rational. (Contributed by Stefan O'Rear,
15-Sep-2014.)
|
⊢ (A ∈ ℚ → (A↑2) ∈
ℚ) |
|
Theorem | sq11 8979 |
The square function is one-to-one for nonnegative reals. (Contributed by
NM, 8-Apr-2001.) (Proof shortened by Mario Carneiro, 28-May-2016.)
|
⊢ (((A ∈ ℝ ∧ 0
≤ A) ∧
(B ∈
ℝ ∧ 0 ≤ B)) → ((A↑2) = (B↑2) ↔ A = B)) |
|
Theorem | lt2sq 8980 |
The square function on nonnegative reals is strictly monotonic.
(Contributed by NM, 24-Feb-2006.)
|
⊢ (((A ∈ ℝ ∧ 0
≤ A) ∧
(B ∈
ℝ ∧ 0 ≤ B)) → (A
< B ↔ (A↑2) < (B↑2))) |
|
Theorem | le2sq 8981 |
The square function on nonnegative reals is monotonic. (Contributed by
NM, 18-Oct-1999.)
|
⊢ (((A ∈ ℝ ∧ 0
≤ A) ∧
(B ∈
ℝ ∧ 0 ≤ B)) → (A
≤ B ↔ (A↑2) ≤ (B↑2))) |
|
Theorem | le2sq2 8982 |
The square of a 'less than or equal to' ordering. (Contributed by NM,
21-Mar-2008.)
|
⊢ (((A ∈ ℝ ∧ 0
≤ A) ∧
(B ∈
ℝ ∧ A ≤ B))
→ (A↑2) ≤ (B↑2)) |
|
Theorem | sqge0 8983 |
A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
|
⊢ (A ∈ ℝ → 0 ≤ (A↑2)) |
|
Theorem | zsqcl2 8984 |
The square of an integer is a nonnegative integer. (Contributed by Mario
Carneiro, 18-Apr-2014.) (Revised by Mario Carneiro, 14-Jul-2014.)
|
⊢ (A ∈ ℤ → (A↑2) ∈
ℕ0) |
|
Theorem | sumsqeq0 8985 |
Two real numbers are equal to 0 iff their Euclidean norm is. (Contributed
by NM, 29-Apr-2005.) (Revised by Stefan O'Rear, 5-Oct-2014.) (Proof
shortened by Mario Carneiro, 28-May-2016.)
|
⊢ ((A ∈ ℝ ∧
B ∈
ℝ) → ((A = 0 ∧ B = 0) ↔
((A↑2) + (B↑2)) = 0)) |
|
Theorem | sqvali 8986 |
Value of square. Inference version. (Contributed by NM,
1-Aug-1999.)
|
⊢ A ∈ ℂ ⇒ ⊢ (A↑2) = (A
· A) |
|
Theorem | sqcli 8987 |
Closure of square. (Contributed by NM, 2-Aug-1999.)
|
⊢ A ∈ ℂ ⇒ ⊢ (A↑2) ∈
ℂ |
|
Theorem | sqeq0i 8988 |
A number is zero iff its square is zero. (Contributed by NM,
2-Oct-1999.)
|
⊢ A ∈ ℂ ⇒ ⊢ ((A↑2) = 0 ↔ A = 0) |
|
Theorem | sqmuli 8989 |
Distribution of square over multiplication. (Contributed by NM,
3-Sep-1999.)
|
⊢ A ∈ ℂ & ⊢ B ∈
ℂ ⇒ ⊢ ((A · B)↑2) = ((A↑2) · (B↑2)) |
|
Theorem | sqdivapi 8990 |
Distribution of square over division. (Contributed by Jim Kingdon,
12-Jun-2020.)
|
⊢ A ∈ ℂ & ⊢ B ∈
ℂ
& ⊢ B #
0 ⇒ ⊢ ((A / B)↑2)
= ((A↑2) / (B↑2)) |
|
Theorem | resqcli 8991 |
Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
|
⊢ A ∈ ℝ ⇒ ⊢ (A↑2) ∈
ℝ |
|
Theorem | sqgt0api 8992 |
The square of a nonzero real is positive. (Contributed by Jim Kingdon,
12-Jun-2020.)
|
⊢ A ∈ ℝ ⇒ ⊢ (A # 0 → 0 < (A↑2)) |
|
Theorem | sqge0i 8993 |
A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
|
⊢ A ∈ ℝ ⇒ ⊢ 0 ≤ (A↑2) |
|
Theorem | lt2sqi 8994 |
The square function on nonnegative reals is strictly monotonic.
(Contributed by NM, 12-Sep-1999.)
|
⊢ A ∈ ℝ & ⊢ B ∈
ℝ ⇒ ⊢ ((0 ≤ A ∧ 0 ≤
B) → (A < B ↔
(A↑2) < (B↑2))) |
|
Theorem | le2sqi 8995 |
The square function on nonnegative reals is monotonic. (Contributed by
NM, 12-Sep-1999.)
|
⊢ A ∈ ℝ & ⊢ B ∈
ℝ ⇒ ⊢ ((0 ≤ A ∧ 0 ≤
B) → (A ≤ B ↔
(A↑2) ≤ (B↑2))) |
|
Theorem | sq11i 8996 |
The square function is one-to-one for nonnegative reals. (Contributed
by NM, 27-Oct-1999.)
|
⊢ A ∈ ℝ & ⊢ B ∈
ℝ ⇒ ⊢ ((0 ≤ A ∧ 0 ≤
B) → ((A↑2) = (B↑2) ↔ A = B)) |
|
Theorem | sq0 8997 |
The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
|
⊢ (0↑2) = 0 |
|
Theorem | sq0i 8998 |
If a number is zero, its square is zero. (Contributed by FL,
10-Dec-2006.)
|
⊢ (A = 0
→ (A↑2) = 0) |
|
Theorem | sq0id 8999 |
If a number is zero, its square is zero. Deduction form of sq0i 8998.
Converse of sqeq0d 9033. (Contributed by David Moews, 28-Feb-2017.)
|
⊢ (φ
→ A = 0)
⇒ ⊢ (φ → (A↑2) = 0) |
|
Theorem | sq1 9000 |
The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
|
⊢ (1↑2) = 1 |