Theorem List for Intuitionistic Logic Explorer - 10901-11000 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | absrei 10901 |
Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)
|
⊢ 𝐴 ∈ ℝ
⇒ ⊢ (abs‘𝐴) = (√‘(𝐴↑2)) |
|
Theorem | sqrtpclii 10902 |
The square root of a positive real is a real. (Contributed by Mario
Carneiro, 6-Sep-2013.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 0 < 𝐴 ⇒ ⊢ (√‘𝐴) ∈
ℝ |
|
Theorem | sqrtgt0ii 10903 |
The square root of a positive real is positive. (Contributed by NM,
26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 0 < 𝐴 ⇒ ⊢ 0 < (√‘𝐴) |
|
Theorem | sqrt11i 10904 |
The square root function is one-to-one. (Contributed by NM,
27-Jul-1999.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈
ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((√‘𝐴) = (√‘𝐵) ↔ 𝐴 = 𝐵)) |
|
Theorem | sqrtmuli 10905 |
Square root distributes over multiplication. (Contributed by NM,
30-Jul-1999.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈
ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵))) |
|
Theorem | sqrtmulii 10906 |
Square root distributes over multiplication. (Contributed by NM,
30-Jul-1999.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 0 ≤ 𝐴 & ⊢ 0 ≤ 𝐵 ⇒ ⊢ (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵)) |
|
Theorem | sqrtmsq2i 10907 |
Relationship between square root and squares. (Contributed by NM,
31-Jul-1999.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈
ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((√‘𝐴) = 𝐵 ↔ 𝐴 = (𝐵 · 𝐵))) |
|
Theorem | sqrtlei 10908 |
Square root is monotonic. (Contributed by NM, 3-Aug-1999.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈
ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 ≤ 𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵))) |
|
Theorem | sqrtlti 10909 |
Square root is strictly monotonic. (Contributed by Roy F. Longton,
8-Aug-2005.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈
ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 < 𝐵 ↔ (√‘𝐴) < (√‘𝐵))) |
|
Theorem | abslti 10910 |
Absolute value and 'less than' relation. (Contributed by NM,
6-Apr-2005.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈
ℝ ⇒ ⊢ ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴 ∧ 𝐴 < 𝐵)) |
|
Theorem | abslei 10911 |
Absolute value and 'less than or equal to' relation. (Contributed by
NM, 6-Apr-2005.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈
ℝ ⇒ ⊢ ((abs‘𝐴) ≤ 𝐵 ↔ (-𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) |
|
Theorem | absvalsqi 10912 |
Square of value of absolute value function. (Contributed by NM,
2-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)) |
|
Theorem | absvalsq2i 10913 |
Square of value of absolute value function. (Contributed by NM,
2-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ ((abs‘𝐴)↑2) = (((ℜ‘𝐴)↑2) +
((ℑ‘𝐴)↑2)) |
|
Theorem | abscli 10914 |
Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (abs‘𝐴) ∈ ℝ |
|
Theorem | absge0i 10915 |
Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ 0 ≤ (abs‘𝐴) |
|
Theorem | absval2i 10916 |
Value of absolute value function. Definition 10.36 of [Gleason] p. 133.
(Contributed by NM, 2-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (abs‘𝐴) = (√‘(((ℜ‘𝐴)↑2) +
((ℑ‘𝐴)↑2))) |
|
Theorem | abs00i 10917 |
The absolute value of a number is zero iff the number is zero.
Proposition 10-3.7(c) of [Gleason] p.
133. (Contributed by NM,
28-Jul-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ ((abs‘𝐴) = 0 ↔ 𝐴 = 0) |
|
Theorem | absgt0api 10918 |
The absolute value of a nonzero number is positive. Remark in [Apostol]
p. 363. (Contributed by NM, 1-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 # 0 ↔ 0 < (abs‘𝐴)) |
|
Theorem | absnegi 10919 |
Absolute value of negative. (Contributed by NM, 2-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (abs‘-𝐴) = (abs‘𝐴) |
|
Theorem | abscji 10920 |
The absolute value of a number and its conjugate are the same.
Proposition 10-3.7(b) of [Gleason] p.
133. (Contributed by NM,
2-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢
(abs‘(∗‘𝐴)) = (abs‘𝐴) |
|
Theorem | releabsi 10921 |
The real part of a number is less than or equal to its absolute value.
Proposition 10-3.7(d) of [Gleason] p.
133. (Contributed by NM,
2-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (ℜ‘𝐴) ≤ (abs‘𝐴) |
|
Theorem | abssubi 10922 |
Swapping order of subtraction doesn't change the absolute value.
Example of [Apostol] p. 363.
(Contributed by NM, 1-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴)) |
|
Theorem | absmuli 10923 |
Absolute value distributes over multiplication. Proposition 10-3.7(f)
of [Gleason] p. 133. (Contributed by
NM, 1-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (abs‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (abs‘𝐵)) |
|
Theorem | sqabsaddi 10924 |
Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason]
p. 133. (Contributed by NM, 2-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ ((abs‘(𝐴 + 𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) + (2 ·
(ℜ‘(𝐴 ·
(∗‘𝐵))))) |
|
Theorem | sqabssubi 10925 |
Square of absolute value of difference. (Contributed by Steve
Rodriguez, 20-Jan-2007.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ ((abs‘(𝐴 − 𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) − (2 ·
(ℜ‘(𝐴 ·
(∗‘𝐵))))) |
|
Theorem | absdivapzi 10926 |
Absolute value distributes over division. (Contributed by Jim Kingdon,
13-Aug-2021.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵))) |
|
Theorem | abstrii 10927 |
Triangle inequality for absolute value. Proposition 10-3.7(h) of
[Gleason] p. 133. This is Metamath 100
proof #91. (Contributed by NM,
2-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (abs‘(𝐴 + 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)) |
|
Theorem | abs3difi 10928 |
Absolute value of differences around common element. (Contributed by
NM, 2-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ (abs‘(𝐴 − 𝐵)) ≤ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))) |
|
Theorem | abs3lemi 10929 |
Lemma involving absolute value of differences. (Contributed by NM,
2-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈
ℝ ⇒ ⊢ (((abs‘(𝐴 − 𝐶)) < (𝐷 / 2) ∧ (abs‘(𝐶 − 𝐵)) < (𝐷 / 2)) → (abs‘(𝐴 − 𝐵)) < 𝐷) |
|
Theorem | rpsqrtcld 10930 |
The square root of a positive real is positive. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈
ℝ+) ⇒ ⊢ (𝜑 → (√‘𝐴) ∈
ℝ+) |
|
Theorem | sqrtgt0d 10931 |
The square root of a positive real is positive. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈
ℝ+) ⇒ ⊢ (𝜑 → 0 < (√‘𝐴)) |
|
Theorem | absnidd 10932 |
A negative number is the negative of its own absolute value.
(Contributed by Mario Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 0) ⇒ ⊢ (𝜑 → (abs‘𝐴) = -𝐴) |
|
Theorem | leabsd 10933 |
A real number is less than or equal to its absolute value. (Contributed
by Mario Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ)
⇒ ⊢ (𝜑 → 𝐴 ≤ (abs‘𝐴)) |
|
Theorem | absred 10934 |
Absolute value of a real number. (Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ)
⇒ ⊢ (𝜑 → (abs‘𝐴) = (√‘(𝐴↑2))) |
|
Theorem | resqrtcld 10935 |
The square root of a nonnegative real is a real. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (√‘𝐴) ∈ ℝ) |
|
Theorem | sqrtmsqd 10936 |
Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (√‘(𝐴 · 𝐴)) = 𝐴) |
|
Theorem | sqrtsqd 10937 |
Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (√‘(𝐴↑2)) = 𝐴) |
|
Theorem | sqrtge0d 10938 |
The square root of a nonnegative real is nonnegative. (Contributed by
Mario Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → 0 ≤ (√‘𝐴)) |
|
Theorem | absidd 10939 |
A nonnegative number is its own absolute value. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (abs‘𝐴) = 𝐴) |
|
Theorem | sqrtdivd 10940 |
Square root distributes over division. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴)
& ⊢ (𝜑 → 𝐵 ∈
ℝ+) ⇒ ⊢ (𝜑 → (√‘(𝐴 / 𝐵)) = ((√‘𝐴) / (√‘𝐵))) |
|
Theorem | sqrtmuld 10941 |
Square root distributes over multiplication. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴)
& ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵))) |
|
Theorem | sqrtsq2d 10942 |
Relationship between square root and squares. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴)
& ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → ((√‘𝐴) = 𝐵 ↔ 𝐴 = (𝐵↑2))) |
|
Theorem | sqrtled 10943 |
Square root is monotonic. (Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴)
& ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵))) |
|
Theorem | sqrtltd 10944 |
Square root is strictly monotonic. (Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴)
& ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (√‘𝐴) < (√‘𝐵))) |
|
Theorem | sqr11d 10945 |
The square root function is one-to-one. (Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴)
& ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐵)
& ⊢ (𝜑 → (√‘𝐴) = (√‘𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) |
|
Theorem | absltd 10946 |
Absolute value and 'less than' relation. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ)
⇒ ⊢ (𝜑 → ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴 ∧ 𝐴 < 𝐵))) |
|
Theorem | absled 10947 |
Absolute value and 'less than or equal to' relation. (Contributed by
Mario Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ)
⇒ ⊢ (𝜑 → ((abs‘𝐴) ≤ 𝐵 ↔ (-𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))) |
|
Theorem | abssubge0d 10948 |
Absolute value of a nonnegative difference. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) = (𝐵 − 𝐴)) |
|
Theorem | abssuble0d 10949 |
Absolute value of a nonpositive difference. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (𝐵 − 𝐴)) |
|
Theorem | absdifltd 10950 |
The absolute value of a difference and 'less than' relation.
(Contributed by Mario Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ)
⇒ ⊢ (𝜑 → ((abs‘(𝐴 − 𝐵)) < 𝐶 ↔ ((𝐵 − 𝐶) < 𝐴 ∧ 𝐴 < (𝐵 + 𝐶)))) |
|
Theorem | absdifled 10951 |
The absolute value of a difference and 'less than or equal to' relation.
(Contributed by Mario Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ)
⇒ ⊢ (𝜑 → ((abs‘(𝐴 − 𝐵)) ≤ 𝐶 ↔ ((𝐵 − 𝐶) ≤ 𝐴 ∧ 𝐴 ≤ (𝐵 + 𝐶)))) |
|
Theorem | icodiamlt 10952 |
Two elements in a half-open interval have separation strictly less than
the difference between the endpoints. (Contributed by Stefan O'Rear,
12-Sep-2014.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵))) → (abs‘(𝐶 − 𝐷)) < (𝐵 − 𝐴)) |
|
Theorem | abscld 10953 |
Real closure of absolute value. (Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ) |
|
Theorem | absvalsqd 10954 |
Square of value of absolute value function. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) |
|
Theorem | absvalsq2d 10955 |
Square of value of absolute value function. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → ((abs‘𝐴)↑2) = (((ℜ‘𝐴)↑2) +
((ℑ‘𝐴)↑2))) |
|
Theorem | absge0d 10956 |
Absolute value is nonnegative. (Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → 0 ≤ (abs‘𝐴)) |
|
Theorem | absval2d 10957 |
Value of absolute value function. Definition 10.36 of [Gleason] p. 133.
(Contributed by Mario Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘𝐴) = (√‘(((ℜ‘𝐴)↑2) +
((ℑ‘𝐴)↑2)))) |
|
Theorem | abs00d 10958 |
The absolute value of a number is zero iff the number is zero.
Proposition 10-3.7(c) of [Gleason] p.
133. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝐴) = 0)
⇒ ⊢ (𝜑 → 𝐴 = 0) |
|
Theorem | absne0d 10959 |
The absolute value of a number is zero iff the number is zero.
Proposition 10-3.7(c) of [Gleason] p.
133. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (abs‘𝐴) ≠ 0) |
|
Theorem | absrpclapd 10960 |
The absolute value of a complex number apart from zero is a positive
real. (Contributed by Jim Kingdon, 13-Aug-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (abs‘𝐴) ∈
ℝ+) |
|
Theorem | absnegd 10961 |
Absolute value of negative. (Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘-𝐴) = (abs‘𝐴)) |
|
Theorem | abscjd 10962 |
The absolute value of a number and its conjugate are the same.
Proposition 10-3.7(b) of [Gleason] p.
133. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘(∗‘𝐴)) = (abs‘𝐴)) |
|
Theorem | releabsd 10963 |
The real part of a number is less than or equal to its absolute value.
Proposition 10-3.7(d) of [Gleason] p.
133. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (ℜ‘𝐴) ≤ (abs‘𝐴)) |
|
Theorem | absexpd 10964 |
Absolute value of positive integer exponentiation. (Contributed by
Mario Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈
ℕ0) ⇒ ⊢ (𝜑 → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁)) |
|
Theorem | abssubd 10965 |
Swapping order of subtraction doesn't change the absolute value.
Example of [Apostol] p. 363.
(Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) |
|
Theorem | absmuld 10966 |
Absolute value distributes over multiplication. Proposition 10-3.7(f)
of [Gleason] p. 133. (Contributed by
Mario Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (abs‘𝐵))) |
|
Theorem | absdivapd 10967 |
Absolute value distributes over division. (Contributed by Jim
Kingdon, 13-Aug-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵))) |
|
Theorem | abstrid 10968 |
Triangle inequality for absolute value. Proposition 10-3.7(h) of
[Gleason] p. 133. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘(𝐴 + 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵))) |
|
Theorem | abs2difd 10969 |
Difference of absolute values. (Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵))) |
|
Theorem | abs2dif2d 10970 |
Difference of absolute values. (Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵))) |
|
Theorem | abs2difabsd 10971 |
Absolute value of difference of absolute values. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴 − 𝐵))) |
|
Theorem | abs3difd 10972 |
Absolute value of differences around common element. (Contributed by
Mario Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ≤ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵)))) |
|
Theorem | abs3lemd 10973 |
Lemma involving absolute value of differences. (Contributed by Mario
Carneiro, 29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (abs‘(𝐴 − 𝐶)) < (𝐷 / 2)) & ⊢ (𝜑 → (abs‘(𝐶 − 𝐵)) < (𝐷 / 2)) ⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) < 𝐷) |
|
Theorem | qdenre 10974* |
The rational numbers are dense in ℝ: any real
number can be
approximated with arbitrary precision by a rational number. For order
theoretic density, see qbtwnre 10034. (Contributed by BJ, 15-Oct-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) →
∃𝑥 ∈ ℚ
(abs‘(𝑥 −
𝐴)) < 𝐵) |
|
4.7.5 The maximum of two real
numbers
|
|
Theorem | maxcom 10975 |
The maximum of two reals is commutative. Lemma 3.9 of [Geuvers], p. 10.
(Contributed by Jim Kingdon, 21-Dec-2021.)
|
⊢ sup({𝐴, 𝐵}, ℝ, < ) = sup({𝐵, 𝐴}, ℝ, < ) |
|
Theorem | maxabsle 10976 |
An upper bound for {𝐴, 𝐵}. (Contributed by Jim Kingdon,
20-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) |
|
Theorem | maxleim 10977 |
Value of maximum when we know which number is larger. (Contributed by
Jim Kingdon, 21-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 → sup({𝐴, 𝐵}, ℝ, < ) = 𝐵)) |
|
Theorem | maxabslemab 10978 |
Lemma for maxabs 10981. A variation of maxleim 10977- that is, if we know
which of two real numbers is larger, we know the maximum of the two.
(Contributed by Jim Kingdon, 21-Dec-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) = 𝐵) |
|
Theorem | maxabslemlub 10979 |
Lemma for maxabs 10981. A least upper bound for {𝐴, 𝐵}.
(Contributed by Jim Kingdon, 20-Dec-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐶 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) ⇒ ⊢ (𝜑 → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵)) |
|
Theorem | maxabslemval 10980* |
Lemma for maxabs 10981. Value of the supremum. (Contributed by
Jim
Kingdon, 22-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) ∈ ℝ ∧ ∀𝑥 ∈ {𝐴, 𝐵} ¬ (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) < 𝑥 ∧ ∀𝑥 ∈ ℝ (𝑥 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) → ∃𝑧 ∈ {𝐴, 𝐵}𝑥 < 𝑧))) |
|
Theorem | maxabs 10981 |
Maximum of two real numbers in terms of absolute value. (Contributed by
Jim Kingdon, 20-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({𝐴, 𝐵}, ℝ, < ) = (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) |
|
Theorem | maxcl 10982 |
The maximum of two real numbers is a real number. (Contributed by Jim
Kingdon, 22-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({𝐴, 𝐵}, ℝ, < ) ∈
ℝ) |
|
Theorem | maxle1 10983 |
The maximum of two reals is no smaller than the first real. Lemma 3.10 of
[Geuvers], p. 10. (Contributed by Jim
Kingdon, 21-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ sup({𝐴, 𝐵}, ℝ, < )) |
|
Theorem | maxle2 10984 |
The maximum of two reals is no smaller than the second real. Lemma 3.10
of [Geuvers], p. 10. (Contributed by Jim
Kingdon, 21-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ sup({𝐴, 𝐵}, ℝ, < )) |
|
Theorem | maxleast 10985 |
The maximum of two reals is a least upper bound. Lemma 3.11 of
[Geuvers], p. 10. (Contributed by Jim
Kingdon, 22-Dec-2021.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶)) → sup({𝐴, 𝐵}, ℝ, < ) ≤ 𝐶) |
|
Theorem | maxleastb 10986 |
Two ways of saying the maximum of two numbers is less than or equal to a
third. (Contributed by Jim Kingdon, 31-Jan-2022.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (sup({𝐴, 𝐵}, ℝ, < ) ≤ 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶))) |
|
Theorem | maxleastlt 10987 |
The maximum as a least upper bound, in terms of less than. (Contributed
by Jim Kingdon, 9-Feb-2022.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ, < ))) → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵)) |
|
Theorem | maxleb 10988 |
Equivalence of ≤ and being equal to the maximum of
two reals. Lemma
3.12 of [Geuvers], p. 10. (Contributed by
Jim Kingdon, 21-Dec-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ sup({𝐴, 𝐵}, ℝ, < ) = 𝐵)) |
|
Theorem | dfabsmax 10989 |
Absolute value of a real number in terms of maximum. Definition 3.13 of
[Geuvers], p. 11. (Contributed by BJ and
Jim Kingdon, 21-Dec-2021.)
|
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = sup({𝐴, -𝐴}, ℝ, < )) |
|
Theorem | maxltsup 10990 |
Two ways of saying the maximum of two numbers is less than a third.
(Contributed by Jim Kingdon, 10-Feb-2022.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (sup({𝐴, 𝐵}, ℝ, < ) < 𝐶 ↔ (𝐴 < 𝐶 ∧ 𝐵 < 𝐶))) |
|
Theorem | max0addsup 10991 |
The sum of the positive and negative part functions is the absolute value
function over the reals. (Contributed by Jim Kingdon, 30-Jan-2022.)
|
⊢ (𝐴 ∈ ℝ → (sup({𝐴, 0}, ℝ, < ) +
sup({-𝐴, 0}, ℝ, <
)) = (abs‘𝐴)) |
|
Theorem | rexanre 10992* |
Combine two different upper real properties into one. (Contributed by
Mario Carneiro, 8-May-2016.)
|
⊢ (𝐴 ⊆ ℝ → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓)) ↔ (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜓)))) |
|
Theorem | rexico 10993* |
Restrict the base of an upper real quantifier to an upper real set.
(Contributed by Mario Carneiro, 12-May-2016.)
|
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ↔ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑))) |
|
Theorem | maxclpr 10994 |
The maximum of two real numbers is one of those numbers if and only if
dichotomy (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴) holds. For example, this can be
combined with zletric 9098 if one is dealing with integers, but real
number
dichotomy in general does not follow from our axioms. (Contributed by Jim
Kingdon, 1-Feb-2022.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (sup({𝐴, 𝐵}, ℝ, < ) ∈ {𝐴, 𝐵} ↔ (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))) |
|
Theorem | rpmaxcl 10995 |
The maximum of two positive real numbers is a positive real number.
(Contributed by Jim Kingdon, 10-Nov-2023.)
|
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+)
→ sup({𝐴, 𝐵}, ℝ, < ) ∈
ℝ+) |
|
Theorem | zmaxcl 10996 |
The maximum of two integers is an integer. (Contributed by Jim Kingdon,
27-Sep-2022.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → sup({𝐴, 𝐵}, ℝ, < ) ∈
ℤ) |
|
Theorem | 2zsupmax 10997 |
Two ways to express the maximum of two integers. Because order of
integers is decidable, we have more flexibility than for real numbers.
(Contributed by Jim Kingdon, 22-Jan-2023.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → sup({𝐴, 𝐵}, ℝ, < ) = if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
|
Theorem | fimaxre2 10998* |
A nonempty finite set of real numbers has an upper bound. (Contributed
by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro,
13-Feb-2014.)
|
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
|
Theorem | negfi 10999* |
The negation of a finite set of real numbers is finite. (Contributed by
AV, 9-Aug-2020.)
|
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → {𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴} ∈ Fin) |
|
4.7.6 The minimum of two real
numbers
|
|
Theorem | mincom 11000 |
The minimum of two reals is commutative. (Contributed by Jim Kingdon,
8-Feb-2021.)
|
⊢ inf({𝐴, 𝐵}, ℝ, < ) = inf({𝐵, 𝐴}, ℝ, < ) |