Theorem List for Intuitionistic Logic Explorer - 11101-11200 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | sqr11d 11101 |
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 11102 |
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 11103 |
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 11104 |
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 11105 |
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 11106 |
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 11107 |
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 11108 |
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 11109 |
Real closure of absolute value. (Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ) |
|
Theorem | absvalsqd 11110 |
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 11111 |
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 11112 |
Absolute value is nonnegative. (Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → 0 ≤ (abs‘𝐴)) |
|
Theorem | absval2d 11113 |
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 11114 |
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 11115 |
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 11116 |
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 11117 |
Absolute value of negative. (Contributed by Mario Carneiro,
29-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (abs‘-𝐴) = (abs‘𝐴)) |
|
Theorem | abscjd 11118 |
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 11119 |
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 11120 |
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 11121 |
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 11122 |
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 11123 |
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 11124 |
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 11125 |
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 11126 |
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 11127 |
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 11128 |
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 11129 |
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 11130* |
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 10182. (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 11131 |
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 11132 |
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 11133 |
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 11134 |
Lemma for maxabs 11137. A variation of maxleim 11133- 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 11135 |
Lemma for maxabs 11137. 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 11136* |
Lemma for maxabs 11137. 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 11137 |
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 11138 |
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 11139 |
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 11140 |
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 11141 |
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 11142 |
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 11143 |
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 11144 |
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 11145 |
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 11146 |
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 11147 |
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 11148* |
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 11149* |
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 11150 |
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 9226 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 11151 |
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 11152 |
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 11153 |
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 11154* |
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 11155* |
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 11156 |
The minimum of two reals is commutative. (Contributed by Jim Kingdon,
8-Feb-2021.)
|
⊢ inf({𝐴, 𝐵}, ℝ, < ) = inf({𝐵, 𝐴}, ℝ, < ) |
|
Theorem | minmax 11157 |
Minimum expressed in terms of maximum. (Contributed by Jim Kingdon,
8-Feb-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = -sup({-𝐴, -𝐵}, ℝ, < )) |
|
Theorem | mincl 11158 |
The minumum of two real numbers is a real number. (Contributed by Jim
Kingdon, 25-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) ∈
ℝ) |
|
Theorem | min1inf 11159 |
The minimum of two numbers is less than or equal to the first.
(Contributed by Jim Kingdon, 8-Feb-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) ≤ 𝐴) |
|
Theorem | min2inf 11160 |
The minimum of two numbers is less than or equal to the second.
(Contributed by Jim Kingdon, 9-Feb-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) ≤ 𝐵) |
|
Theorem | lemininf 11161 |
Two ways of saying a number is less than or equal to the minimum of two
others. (Contributed by NM, 3-Aug-2007.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ, < ) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶))) |
|
Theorem | ltmininf 11162 |
Two ways of saying a number is less than the minimum of two others.
(Contributed by Jim Kingdon, 10-Feb-2022.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < inf({𝐵, 𝐶}, ℝ, < ) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶))) |
|
Theorem | minabs 11163 |
The minimum of two real numbers in terms of absolute value. (Contributed
by Jim Kingdon, 15-May-2023.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = (((𝐴 + 𝐵) − (abs‘(𝐴 − 𝐵))) / 2)) |
|
Theorem | minclpr 11164 |
The minimum 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 9226 if one is dealing with integers, but real
number
dichotomy in general does not follow from our axioms. (Contributed by Jim
Kingdon, 23-May-2023.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (inf({𝐴, 𝐵}, ℝ, < ) ∈ {𝐴, 𝐵} ↔ (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))) |
|
Theorem | rpmincl 11165 |
The minumum of two positive real numbers is a positive real number.
(Contributed by Jim Kingdon, 25-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+)
→ inf({𝐴, 𝐵}, ℝ, < ) ∈
ℝ+) |
|
Theorem | bdtrilem 11166 |
Lemma for bdtri 11167. (Contributed by Steven Nguyen and Jim
Kingdon,
17-May-2023.)
|
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) →
((abs‘(𝐴 −
𝐶)) + (abs‘(𝐵 − 𝐶))) ≤ (𝐶 + (abs‘((𝐴 + 𝐵) − 𝐶)))) |
|
Theorem | bdtri 11167 |
Triangle inequality for bounded values. (Contributed by Jim Kingdon,
15-May-2023.)
|
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) →
inf({(𝐴 + 𝐵), 𝐶}, ℝ, < ) ≤ (inf({𝐴, 𝐶}, ℝ, < ) + inf({𝐵, 𝐶}, ℝ, < ))) |
|
Theorem | mul0inf 11168 |
Equality of a product with zero. A bit of a curiosity, in the sense that
theorems like abs00ap 10990 and mulap0bd 8545 may better express the ideas behind
it. (Contributed by Jim Kingdon, 31-Jul-2023.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = 0 ↔ inf({(abs‘𝐴), (abs‘𝐵)}, ℝ, < ) = 0)) |
|
Theorem | mingeb 11169 |
Equivalence of ≤ and being equal to the minimum of
two reals.
(Contributed by Jim Kingdon, 14-Oct-2024.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ inf({𝐴, 𝐵}, ℝ, < ) = 𝐴)) |
|
Theorem | 2zinfmin 11170 |
Two ways to express the minimum of two integers. Because order of
integers is decidable, we have more flexibility than for real numbers.
(Contributed by Jim Kingdon, 14-Oct-2024.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → inf({𝐴, 𝐵}, ℝ, < ) = if(𝐴 ≤ 𝐵, 𝐴, 𝐵)) |
|
4.7.7 The maximum of two extended
reals
|
|
Theorem | xrmaxleim 11171 |
Value of maximum when we know which extended real is larger.
(Contributed by Jim Kingdon, 25-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝐴 ≤ 𝐵 → sup({𝐴, 𝐵}, ℝ*, < ) = 𝐵)) |
|
Theorem | xrmaxiflemcl 11172 |
Lemma for xrmaxif 11178. Closure. (Contributed by Jim Kingdon,
29-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ if(𝐵 = +∞,
+∞, if(𝐵 = -∞,
𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) ∈
ℝ*) |
|
Theorem | xrmaxifle 11173 |
An upper bound for {𝐴, 𝐵} in the extended reals.
(Contributed by
Jim Kingdon, 26-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ 𝐴 ≤ if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) |
|
Theorem | xrmaxiflemab 11174 |
Lemma for xrmaxif 11178. A variation of xrmaxleim 11171- that is, if we know
which of two real numbers is larger, we know the maximum of the two.
(Contributed by Jim Kingdon, 26-Apr-2023.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = 𝐵) |
|
Theorem | xrmaxiflemlub 11175 |
Lemma for xrmaxif 11178. A least upper bound for {𝐴, 𝐵}.
(Contributed by Jim Kingdon, 28-Apr-2023.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, <
)))))) ⇒ ⊢ (𝜑 → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵)) |
|
Theorem | xrmaxiflemcom 11176 |
Lemma for xrmaxif 11178. Commutativity of an expression which we
will
later show to be the supremum. (Contributed by Jim Kingdon,
29-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ if(𝐵 = +∞,
+∞, if(𝐵 = -∞,
𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < )))))) |
|
Theorem | xrmaxiflemval 11177* |
Lemma for xrmaxif 11178. Value of the supremum. (Contributed by
Jim
Kingdon, 29-Apr-2023.)
|
⊢ 𝑀 = if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, <
))))) ⇒ ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝑀 ∈
ℝ* ∧ ∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝑀 < 𝑥 ∧ ∀𝑥 ∈ ℝ* (𝑥 < 𝑀 → ∃𝑧 ∈ {𝐴, 𝐵}𝑥 < 𝑧))) |
|
Theorem | xrmaxif 11178 |
Maximum of two extended reals in terms of if
expressions.
(Contributed by Jim Kingdon, 26-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ sup({𝐴, 𝐵}, ℝ*, < )
= if(𝐵 = +∞,
+∞, if(𝐵 = -∞,
𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) |
|
Theorem | xrmaxcl 11179 |
The maximum of two extended reals is an extended real. (Contributed by
Jim Kingdon, 29-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ sup({𝐴, 𝐵}, ℝ*, < )
∈ ℝ*) |
|
Theorem | xrmax1sup 11180 |
An extended real is less than or equal to the maximum of it and another.
(Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon,
30-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ 𝐴 ≤ sup({𝐴, 𝐵}, ℝ*, <
)) |
|
Theorem | xrmax2sup 11181 |
An extended real is less than or equal to the maximum of it and another.
(Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon,
30-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ 𝐵 ≤ sup({𝐴, 𝐵}, ℝ*, <
)) |
|
Theorem | xrmaxrecl 11182 |
The maximum of two real numbers is the same when taken as extended reals
or as reals. (Contributed by Jim Kingdon, 30-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({𝐴, 𝐵}, ℝ*, < ) = sup({𝐴, 𝐵}, ℝ, < )) |
|
Theorem | xrmaxleastlt 11183 |
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 | xrltmaxsup 11184 |
The maximum as a least upper bound. (Contributed by Jim Kingdon,
10-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (𝐶 < sup({𝐴, 𝐵}, ℝ*, < ) ↔
(𝐶 < 𝐴 ∨ 𝐶 < 𝐵))) |
|
Theorem | xrmaxltsup 11185 |
Two ways of saying the maximum of two numbers is less than a third.
(Contributed by Jim Kingdon, 30-Apr-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (sup({𝐴, 𝐵}, ℝ*, < ) < 𝐶 ↔ (𝐴 < 𝐶 ∧ 𝐵 < 𝐶))) |
|
Theorem | xrmaxlesup 11186 |
Two ways of saying the maximum of two numbers is less than or equal to a
third. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim
Kingdon, 10-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (sup({𝐴, 𝐵}, ℝ*, < ) ≤ 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶))) |
|
Theorem | xrmaxaddlem 11187 |
Lemma for xrmaxadd 11188. The case where 𝐴 is real. (Contributed
by
Jim Kingdon, 11-May-2023.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)
→ sup({(𝐴
+𝑒 𝐵),
(𝐴 +𝑒
𝐶)}, ℝ*,
< ) = (𝐴
+𝑒 sup({𝐵, 𝐶}, ℝ*, <
))) |
|
Theorem | xrmaxadd 11188 |
Distributing addition over maximum. (Contributed by Jim Kingdon,
11-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → sup({(𝐴 +𝑒 𝐵), (𝐴 +𝑒 𝐶)}, ℝ*, < ) = (𝐴 +𝑒
sup({𝐵, 𝐶}, ℝ*, <
))) |
|
4.7.8 The minimum of two extended
reals
|
|
Theorem | xrnegiso 11189 |
Negation is an order anti-isomorphism of the extended reals, which is
its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
|
⊢ 𝐹 = (𝑥 ∈ ℝ* ↦
-𝑒𝑥) ⇒ ⊢ (𝐹 Isom < , ◡ < (ℝ*,
ℝ*) ∧ ◡𝐹 = 𝐹) |
|
Theorem | infxrnegsupex 11190* |
The infimum of a set of extended reals 𝐴 is the negative of the
supremum of the negatives of its elements. (Contributed by Jim Kingdon,
2-May-2023.)
|
⊢ (𝜑 → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) & ⊢ (𝜑 → 𝐴 ⊆
ℝ*) ⇒ ⊢ (𝜑 → inf(𝐴, ℝ*, < ) =
-𝑒sup({𝑧 ∈ ℝ* ∣
-𝑒𝑧
∈ 𝐴},
ℝ*, < )) |
|
Theorem | xrnegcon1d 11191 |
Contraposition law for extended real unary minus. (Contributed by Jim
Kingdon, 2-May-2023.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈
ℝ*) ⇒ ⊢ (𝜑 → (-𝑒𝐴 = 𝐵 ↔ -𝑒𝐵 = 𝐴)) |
|
Theorem | xrminmax 11192 |
Minimum expressed in terms of maximum. (Contributed by Jim Kingdon,
2-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ inf({𝐴, 𝐵}, ℝ*, < )
= -𝑒sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, <
)) |
|
Theorem | xrmincl 11193 |
The minumum of two extended reals is an extended real. (Contributed by
Jim Kingdon, 3-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ inf({𝐴, 𝐵}, ℝ*, < )
∈ ℝ*) |
|
Theorem | xrmin1inf 11194 |
The minimum of two extended reals is less than or equal to the first.
(Contributed by Jim Kingdon, 3-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ inf({𝐴, 𝐵}, ℝ*, < )
≤ 𝐴) |
|
Theorem | xrmin2inf 11195 |
The minimum of two extended reals is less than or equal to the second.
(Contributed by Jim Kingdon, 3-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ inf({𝐴, 𝐵}, ℝ*, < )
≤ 𝐵) |
|
Theorem | xrmineqinf 11196 |
The minimum of two extended reals is equal to the second if the first is
bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) (Revised by Jim
Kingdon, 3-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐵 ≤ 𝐴) → inf({𝐴, 𝐵}, ℝ*, < ) = 𝐵) |
|
Theorem | xrltmininf 11197 |
Two ways of saying an extended real is less than the minimum of two
others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon,
3-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (𝐴 < inf({𝐵, 𝐶}, ℝ*, < ) ↔
(𝐴 < 𝐵 ∧ 𝐴 < 𝐶))) |
|
Theorem | xrlemininf 11198 |
Two ways of saying a number is less than or equal to the minimum of two
others. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim
Kingdon, 4-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ*, < ) ↔
(𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶))) |
|
Theorem | xrminltinf 11199 |
Two ways of saying an extended real is greater than the minimum of two
others. (Contributed by Jim Kingdon, 19-May-2023.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (inf({𝐵, 𝐶}, ℝ*, < ) < 𝐴 ↔ (𝐵 < 𝐴 ∨ 𝐶 < 𝐴))) |
|
Theorem | xrminrecl 11200 |
The minimum of two real numbers is the same when taken as extended reals
or as reals. (Contributed by Jim Kingdon, 18-May-2023.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ*, < ) = inf({𝐴, 𝐵}, ℝ, < )) |