Theorem List for Intuitionistic Logic Explorer - 9701-9800 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Definition | df-xmul 9701* |
Define multiplication over extended real numbers. (Contributed by Mario
Carneiro, 20-Aug-2015.)
|
⊢ ·e = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ if((𝑥 = 0 ∨
𝑦 = 0), 0, if((((0 <
𝑦 ∧ 𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))), +∞, if((((0 < 𝑦 ∧ 𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦))))) |
|
Theorem | ltxr 9702 |
The 'less than' binary relation on the set of extended reals.
Definition 12-3.1 of [Gleason] p. 173.
(Contributed by NM,
14-Oct-2005.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝐴 < 𝐵 ↔ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵) ∨ (𝐴 = -∞ ∧ 𝐵 = +∞)) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))))) |
|
Theorem | elxr 9703 |
Membership in the set of extended reals. (Contributed by NM,
14-Oct-2005.)
|
⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) |
|
Theorem | xrnemnf 9704 |
An extended real other than minus infinity is real or positive infinite.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
|
Theorem | xrnepnf 9705 |
An extended real other than plus infinity is real or negative infinite.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞)) |
|
Theorem | xrltnr 9706 |
The extended real 'less than' is irreflexive. (Contributed by NM,
14-Oct-2005.)
|
⊢ (𝐴 ∈ ℝ* → ¬
𝐴 < 𝐴) |
|
Theorem | ltpnf 9707 |
Any (finite) real is less than plus infinity. (Contributed by NM,
14-Oct-2005.)
|
⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) |
|
Theorem | ltpnfd 9708 |
Any (finite) real is less than plus infinity. (Contributed by Glauco
Siliprandi, 11-Dec-2019.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ)
⇒ ⊢ (𝜑 → 𝐴 < +∞) |
|
Theorem | 0ltpnf 9709 |
Zero is less than plus infinity (common case). (Contributed by David A.
Wheeler, 8-Dec-2018.)
|
⊢ 0 < +∞ |
|
Theorem | mnflt 9710 |
Minus infinity is less than any (finite) real. (Contributed by NM,
14-Oct-2005.)
|
⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) |
|
Theorem | mnflt0 9711 |
Minus infinity is less than 0 (common case). (Contributed by David A.
Wheeler, 8-Dec-2018.)
|
⊢ -∞ < 0 |
|
Theorem | mnfltpnf 9712 |
Minus infinity is less than plus infinity. (Contributed by NM,
14-Oct-2005.)
|
⊢ -∞ < +∞ |
|
Theorem | mnfltxr 9713 |
Minus infinity is less than an extended real that is either real or plus
infinity. (Contributed by NM, 2-Feb-2006.)
|
⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴) |
|
Theorem | pnfnlt 9714 |
No extended real is greater than plus infinity. (Contributed by NM,
15-Oct-2005.)
|
⊢ (𝐴 ∈ ℝ* → ¬
+∞ < 𝐴) |
|
Theorem | nltmnf 9715 |
No extended real is less than minus infinity. (Contributed by NM,
15-Oct-2005.)
|
⊢ (𝐴 ∈ ℝ* → ¬
𝐴 <
-∞) |
|
Theorem | pnfge 9716 |
Plus infinity is an upper bound for extended reals. (Contributed by NM,
30-Jan-2006.)
|
⊢ (𝐴 ∈ ℝ* → 𝐴 ≤
+∞) |
|
Theorem | 0lepnf 9717 |
0 less than or equal to positive infinity. (Contributed by David A.
Wheeler, 8-Dec-2018.)
|
⊢ 0 ≤ +∞ |
|
Theorem | nn0pnfge0 9718 |
If a number is a nonnegative integer or positive infinity, it is greater
than or equal to 0. (Contributed by Alexander van der Vekens,
6-Jan-2018.)
|
⊢ ((𝑁 ∈ ℕ0 ∨ 𝑁 = +∞) → 0 ≤
𝑁) |
|
Theorem | mnfle 9719 |
Minus infinity is less than or equal to any extended real. (Contributed
by NM, 19-Jan-2006.)
|
⊢ (𝐴 ∈ ℝ* → -∞
≤ 𝐴) |
|
Theorem | xrltnsym 9720 |
Ordering on the extended reals is not symmetric. (Contributed by NM,
15-Oct-2005.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)) |
|
Theorem | xrltnsym2 9721 |
'Less than' is antisymmetric and irreflexive for extended reals.
(Contributed by NM, 6-Feb-2007.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ ¬ (𝐴 < 𝐵 ∧ 𝐵 < 𝐴)) |
|
Theorem | xrlttr 9722 |
Ordering on the extended reals is transitive. (Contributed by NM,
15-Oct-2005.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
|
Theorem | xrltso 9723 |
'Less than' is a weakly linear ordering on the extended reals.
(Contributed by NM, 15-Oct-2005.)
|
⊢ < Or
ℝ* |
|
Theorem | xrlttri3 9724 |
Extended real version of lttri3 7969. (Contributed by NM, 9-Feb-2006.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) |
|
Theorem | xrltle 9725 |
'Less than' implies 'less than or equal' for extended reals. (Contributed
by NM, 19-Jan-2006.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
|
Theorem | xrltled 9726 |
'Less than' implies 'less than or equal to' for extended reals.
Deduction form of xrltle 9725. (Contributed by Glauco Siliprandi,
11-Dec-2019.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
|
Theorem | xrleid 9727 |
'Less than or equal to' is reflexive for extended reals. (Contributed by
NM, 7-Feb-2007.)
|
⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) |
|
Theorem | xrleidd 9728 |
'Less than or equal to' is reflexive for extended reals. Deduction form
of xrleid 9727. (Contributed by Glauco Siliprandi,
26-Jun-2021.)
|
⊢ (𝜑 → 𝐴 ∈
ℝ*) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
|
Theorem | xnn0dcle 9729 |
Decidability of ≤ for extended nonnegative integers.
(Contributed by
Jim Kingdon, 13-Oct-2024.)
|
⊢ ((𝐴 ∈ ℕ0*
∧ 𝐵 ∈
ℕ0*) → DECID 𝐴 ≤ 𝐵) |
|
Theorem | xnn0letri 9730 |
Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon,
13-Oct-2024.)
|
⊢ ((𝐴 ∈ ℕ0*
∧ 𝐵 ∈
ℕ0*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) |
|
Theorem | xrletri3 9731 |
Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
|
Theorem | xrletrid 9732 |
Trichotomy law for extended reals. (Contributed by Glauco Siliprandi,
17-Aug-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ≤ 𝐵)
& ⊢ (𝜑 → 𝐵 ≤ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) |
|
Theorem | xrlelttr 9733 |
Transitive law for ordering on extended reals. (Contributed by NM,
19-Jan-2006.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
|
Theorem | xrltletr 9734 |
Transitive law for ordering on extended reals. (Contributed by NM,
19-Jan-2006.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) |
|
Theorem | xrletr 9735 |
Transitive law for ordering on extended reals. (Contributed by NM,
9-Feb-2006.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
|
Theorem | xrlttrd 9736 |
Transitive law for ordering on extended reals. (Contributed by Mario
Carneiro, 23-Aug-2015.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) |
|
Theorem | xrlelttrd 9737 |
Transitive law for ordering on extended reals. (Contributed by Mario
Carneiro, 23-Aug-2015.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ≤ 𝐵)
& ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) |
|
Theorem | xrltletrd 9738 |
Transitive law for ordering on extended reals. (Contributed by Mario
Carneiro, 23-Aug-2015.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) |
|
Theorem | xrletrd 9739 |
Transitive law for ordering on extended reals. (Contributed by Mario
Carneiro, 23-Aug-2015.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ≤ 𝐵)
& ⊢ (𝜑 → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
|
Theorem | xrltne 9740 |
'Less than' implies not equal for extended reals. (Contributed by NM,
20-Jan-2006.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) |
|
Theorem | nltpnft 9741 |
An extended real is not less than plus infinity iff they are equal.
(Contributed by NM, 30-Jan-2006.)
|
⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 <
+∞)) |
|
Theorem | npnflt 9742 |
An extended real is less than plus infinity iff they are not equal.
(Contributed by Jim Kingdon, 17-Apr-2023.)
|
⊢ (𝐴 ∈ ℝ* → (𝐴 < +∞ ↔ 𝐴 ≠
+∞)) |
|
Theorem | xgepnf 9743 |
An extended real which is greater than plus infinity is plus infinity.
(Contributed by Thierry Arnoux, 18-Dec-2016.)
|
⊢ (𝐴 ∈ ℝ* →
(+∞ ≤ 𝐴 ↔
𝐴 =
+∞)) |
|
Theorem | ngtmnft 9744 |
An extended real is not greater than minus infinity iff they are equal.
(Contributed by NM, 2-Feb-2006.)
|
⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬
-∞ < 𝐴)) |
|
Theorem | nmnfgt 9745 |
An extended real is greater than minus infinite iff they are not equal.
(Contributed by Jim Kingdon, 17-Apr-2023.)
|
⊢ (𝐴 ∈ ℝ* →
(-∞ < 𝐴 ↔
𝐴 ≠
-∞)) |
|
Theorem | xrrebnd 9746 |
An extended real is real iff it is strictly bounded by infinities.
(Contributed by NM, 2-Feb-2006.)
|
⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔
(-∞ < 𝐴 ∧
𝐴 <
+∞))) |
|
Theorem | xrre 9747 |
A way of proving that an extended real is real. (Contributed by NM,
9-Mar-2006.)
|
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧
(-∞ < 𝐴 ∧
𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ) |
|
Theorem | xrre2 9748 |
An extended real between two others is real. (Contributed by NM,
6-Feb-2007.)
|
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ) |
|
Theorem | xrre3 9749 |
A way of proving that an extended real is real. (Contributed by FL,
29-May-2014.)
|
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → 𝐴 ∈ ℝ) |
|
Theorem | ge0gtmnf 9750 |
A nonnegative extended real is greater than negative infinity.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤
𝐴) → -∞ <
𝐴) |
|
Theorem | ge0nemnf 9751 |
A nonnegative extended real is greater than negative infinity.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤
𝐴) → 𝐴 ≠
-∞) |
|
Theorem | xrrege0 9752 |
A nonnegative extended real that is less than a real bound is real.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤
𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ) |
|
Theorem | z2ge 9753* |
There exists an integer greater than or equal to any two others.
(Contributed by NM, 28-Aug-2005.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ∃𝑘 ∈ ℤ (𝑀 ≤ 𝑘 ∧ 𝑁 ≤ 𝑘)) |
|
Theorem | xnegeq 9754 |
Equality of two extended numbers with -𝑒 in front of them.
(Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro,
20-Aug-2015.)
|
⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵) |
|
Theorem | xnegpnf 9755 |
Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL,
26-Dec-2011.)
|
⊢ -𝑒+∞ =
-∞ |
|
Theorem | xnegmnf 9756 |
Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL,
26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
|
⊢ -𝑒-∞ =
+∞ |
|
Theorem | rexneg 9757 |
Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by
FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
|
⊢ (𝐴 ∈ ℝ →
-𝑒𝐴 =
-𝐴) |
|
Theorem | xneg0 9758 |
The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
|
⊢ -𝑒0 = 0 |
|
Theorem | xnegcl 9759 |
Closure of extended real negative. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ (𝐴 ∈ ℝ* →
-𝑒𝐴
∈ ℝ*) |
|
Theorem | xnegneg 9760 |
Extended real version of negneg 8139. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ (𝐴 ∈ ℝ* →
-𝑒-𝑒𝐴 = 𝐴) |
|
Theorem | xneg11 9761 |
Extended real version of neg11 8140. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (-𝑒𝐴 = -𝑒𝐵 ↔ 𝐴 = 𝐵)) |
|
Theorem | xltnegi 9762 |
Forward direction of xltneg 9763. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐴 < 𝐵) →
-𝑒𝐵
< -𝑒𝐴) |
|
Theorem | xltneg 9763 |
Extended real version of ltneg 8351. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝐴 < 𝐵 ↔
-𝑒𝐵
< -𝑒𝐴)) |
|
Theorem | xleneg 9764 |
Extended real version of leneg 8354. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝐴 ≤ 𝐵 ↔
-𝑒𝐵
≤ -𝑒𝐴)) |
|
Theorem | xlt0neg1 9765 |
Extended real version of lt0neg1 8357. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ (𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 <
-𝑒𝐴)) |
|
Theorem | xlt0neg2 9766 |
Extended real version of lt0neg2 8358. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ (𝐴 ∈ ℝ* → (0 <
𝐴 ↔
-𝑒𝐴
< 0)) |
|
Theorem | xle0neg1 9767 |
Extended real version of le0neg1 8359. (Contributed by Mario Carneiro,
9-Sep-2015.)
|
⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤
-𝑒𝐴)) |
|
Theorem | xle0neg2 9768 |
Extended real version of le0neg2 8360. (Contributed by Mario Carneiro,
9-Sep-2015.)
|
⊢ (𝐴 ∈ ℝ* → (0 ≤
𝐴 ↔
-𝑒𝐴
≤ 0)) |
|
Theorem | xrpnfdc 9769 |
An extended real is or is not plus infinity. (Contributed by Jim Kingdon,
13-Apr-2023.)
|
⊢ (𝐴 ∈ ℝ* →
DECID 𝐴 =
+∞) |
|
Theorem | xrmnfdc 9770 |
An extended real is or is not minus infinity. (Contributed by Jim
Kingdon, 13-Apr-2023.)
|
⊢ (𝐴 ∈ ℝ* →
DECID 𝐴 =
-∞) |
|
Theorem | xaddf 9771 |
The extended real addition operation is closed in extended reals.
(Contributed by Mario Carneiro, 21-Aug-2015.)
|
⊢ +𝑒 :(ℝ*
× ℝ*)⟶ℝ* |
|
Theorem | xaddval 9772 |
Value of the extended real addition operation. (Contributed by Mario
Carneiro, 20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝐴
+𝑒 𝐵) =
if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞),
if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞),
if(𝐵 = +∞, +∞,
if(𝐵 = -∞, -∞,
(𝐴 + 𝐵)))))) |
|
Theorem | xaddpnf1 9773 |
Addition of positive infinity on the right. (Contributed by Mario
Carneiro, 20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒
+∞) = +∞) |
|
Theorem | xaddpnf2 9774 |
Addition of positive infinity on the left. (Contributed by Mario
Carneiro, 20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) →
(+∞ +𝑒 𝐴) = +∞) |
|
Theorem | xaddmnf1 9775 |
Addition of negative infinity on the right. (Contributed by Mario
Carneiro, 20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 +𝑒
-∞) = -∞) |
|
Theorem | xaddmnf2 9776 |
Addition of negative infinity on the left. (Contributed by Mario
Carneiro, 20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) →
(-∞ +𝑒 𝐴) = -∞) |
|
Theorem | pnfaddmnf 9777 |
Addition of positive and negative infinity. This is often taken to be a
"null" value or out of the domain, but we define it (somewhat
arbitrarily)
to be zero so that the resulting function is total, which simplifies
proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)
|
⊢ (+∞ +𝑒 -∞) =
0 |
|
Theorem | mnfaddpnf 9778 |
Addition of negative and positive infinity. This is often taken to be a
"null" value or out of the domain, but we define it (somewhat
arbitrarily)
to be zero so that the resulting function is total, which simplifies
proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)
|
⊢ (-∞ +𝑒 +∞) =
0 |
|
Theorem | rexadd 9779 |
The extended real addition operation when both arguments are real.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) |
|
Theorem | rexsub 9780 |
Extended real subtraction when both arguments are real. (Contributed by
Mario Carneiro, 23-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒
-𝑒𝐵) =
(𝐴 − 𝐵)) |
|
Theorem | rexaddd 9781 |
The extended real addition operation when both arguments are real.
Deduction version of rexadd 9779. (Contributed by Glauco Siliprandi,
24-Dec-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ)
⇒ ⊢ (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) |
|
Theorem | xnegcld 9782 |
Closure of extended real negative. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈
ℝ*) ⇒ ⊢ (𝜑 → -𝑒𝐴 ∈
ℝ*) |
|
Theorem | xrex 9783 |
The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
|
⊢ ℝ* ∈
V |
|
Theorem | xaddnemnf 9784 |
Closure of extended real addition in the subset ℝ* / {-∞}.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ (𝐵 ∈ ℝ*
∧ 𝐵 ≠ -∞))
→ (𝐴
+𝑒 𝐵)
≠ -∞) |
|
Theorem | xaddnepnf 9785 |
Closure of extended real addition in the subset ℝ* / {+∞}.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ (𝐵 ∈ ℝ*
∧ 𝐵 ≠ +∞))
→ (𝐴
+𝑒 𝐵)
≠ +∞) |
|
Theorem | xnegid 9786 |
Extended real version of negid 8136. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒
-𝑒𝐴) =
0) |
|
Theorem | xaddcl 9787 |
The extended real addition operation is closed in extended reals.
(Contributed by Mario Carneiro, 20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝐴
+𝑒 𝐵)
∈ ℝ*) |
|
Theorem | xaddcom 9788 |
The extended real addition operation is commutative. (Contributed by NM,
26-Dec-2011.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝐴
+𝑒 𝐵) =
(𝐵 +𝑒
𝐴)) |
|
Theorem | xaddid1 9789 |
Extended real version of addid1 8027. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) =
𝐴) |
|
Theorem | xaddid2 9790 |
Extended real version of addid2 8028. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ (𝐴 ∈ ℝ* → (0
+𝑒 𝐴) =
𝐴) |
|
Theorem | xaddid1d 9791 |
0 is a right identity for extended real addition.
(Contributed by
Glauco Siliprandi, 17-Aug-2020.)
|
⊢ (𝜑 → 𝐴 ∈
ℝ*) ⇒ ⊢ (𝜑 → (𝐴 +𝑒 0) = 𝐴) |
|
Theorem | xnn0lenn0nn0 9792 |
An extended nonnegative integer which is less than or equal to a
nonnegative integer is a nonnegative integer. (Contributed by AV,
24-Nov-2021.)
|
⊢ ((𝑀 ∈ ℕ0*
∧ 𝑁 ∈
ℕ0 ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈
ℕ0) |
|
Theorem | xnn0le2is012 9793 |
An extended nonnegative integer which is less than or equal to 2 is either
0 or 1 or 2. (Contributed by AV, 24-Nov-2021.)
|
⊢ ((𝑁 ∈ ℕ0*
∧ 𝑁 ≤ 2) →
(𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2)) |
|
Theorem | xnn0xadd0 9794 |
The sum of two extended nonnegative integers is 0 iff
each of the two
extended nonnegative integers is 0. (Contributed
by AV,
14-Dec-2020.)
|
⊢ ((𝐴 ∈ ℕ0*
∧ 𝐵 ∈
ℕ0*) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) |
|
Theorem | xnegdi 9795 |
Extended real version of negdi 8146. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ -𝑒(𝐴 +𝑒 𝐵) = (-𝑒𝐴 +𝑒
-𝑒𝐵)) |
|
Theorem | xaddass 9796 |
Associativity of extended real addition. The correct condition here is
"it is not the case that both +∞ and
-∞ appear as one of
𝐴,
𝐵, 𝐶, i.e. ¬
{+∞, -∞} ⊆ {𝐴, 𝐵, 𝐶}", but this
condition is difficult to work with, so we break the theorem into two
parts: this one, where -∞ is not present in
𝐴, 𝐵, 𝐶, and
xaddass2 9797, where +∞ is
not present. (Contributed by Mario
Carneiro, 20-Aug-2015.)
|
⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ (𝐵 ∈ ℝ*
∧ 𝐵 ≠ -∞)
∧ (𝐶 ∈
ℝ* ∧ 𝐶 ≠ -∞)) → ((𝐴 +𝑒 𝐵) +𝑒 𝐶) = (𝐴 +𝑒 (𝐵 +𝑒 𝐶))) |
|
Theorem | xaddass2 9797 |
Associativity of extended real addition. See xaddass 9796 for notes on the
hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.)
|
⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ (𝐵 ∈ ℝ*
∧ 𝐵 ≠ +∞)
∧ (𝐶 ∈
ℝ* ∧ 𝐶 ≠ +∞)) → ((𝐴 +𝑒 𝐵) +𝑒 𝐶) = (𝐴 +𝑒 (𝐵 +𝑒 𝐶))) |
|
Theorem | xpncan 9798 |
Extended real version of pncan 8095. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 𝐵) +𝑒
-𝑒𝐵) =
𝐴) |
|
Theorem | xnpcan 9799 |
Extended real version of npcan 8098. (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒
-𝑒𝐵)
+𝑒 𝐵) =
𝐴) |
|
Theorem | xleadd1a 9800 |
Extended real version of leadd1 8319; note that the converse implication is
not true, unlike the real version (for example 0 <
1 but
(1 +𝑒 +∞) ≤ (0
+𝑒 +∞)). (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) ∧ 𝐴 ≤ 𝐵) → (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶)) |