Home Metamath Proof ExplorerTheorem List (p. 338 of 437) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28347) Hilbert Space Explorer (28348-29872) Users' Mathboxes (29873-43661)

Theorem List for Metamath Proof Explorer - 33701-33800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-bj-minfty 33701 Definition of "minus infinity". (Contributed by BJ, 27-Jun-2019.)
-∞ = (+∞ei‘π)

Theorembj-minftyccb 33702 The class -∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
-∞ ∈ ℂ̅

Theorembj-minftynrr 33703 The extended complex number -∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
¬ -∞ ∈ ℂ

Theorembj-pinftynminfty 33704 The extended complex numbers +∞ and -∞ are different. (Contributed by BJ, 27-Jun-2019.)
+∞ ≠ -∞

Syntaxcrrbar 33705 Syntax for the set of extended real numbers.
class ℝ̅

Definitiondf-bj-rrbar 33706 Definition of the set of extended real numbers. This aims to replace df-xr 10415. (Contributed by BJ, 29-Jun-2019.)
ℝ̅ = (ℝ ∪ {-∞, +∞})

Syntaxcinfty 33707 Syntax for .
class

Definitiondf-bj-infty 33708 Definition of , the point at infinity of the real or complex projective line. (Contributed by BJ, 27-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
∞ = 𝒫

Syntaxccchat 33709 Syntax for ℂ̂.
class ℂ̂

Definitiondf-bj-cchat 33710 Define the complex projective line, or Riemann sphere. (Contributed by BJ, 27-Jun-2019.)
ℂ̂ = (ℂ ∪ {∞})

Syntaxcrrhat 33711 Syntax for ℝ̂.
class ℝ̂

Definitiondf-bj-rrhat 33712 Define the real projective line. (Contributed by BJ, 27-Jun-2019.)
ℝ̂ = (ℝ ∪ {∞})

Theorembj-rrhatsscchat 33713 The real projective line is included in the complex projective line. (Contributed by BJ, 27-Jun-2019.)
ℝ̂ ⊆ ℂ̂

We define the operations of addition and opposite on the extended complex numbers and on the complex projective line (Riemann sphere) simultaneously, thus "overloading" the operations.

class +ℂ̅

Definitiondf-bj-addc 33715 Define the additions on the extended complex numbers (on the subset of (ℂ̅ × ℂ̅) where it makes sense) and on the complex projective line (Riemann sphere). We use the plural in "additions" since these are two different operations, even though +ℂ̅ is overloaded. (Contributed by BJ, 22-Jun-2019.)
+ℂ̅ = (𝑥 ∈ (((ℂ × ℂ̅) ∪ (ℂ̅ × ℂ)) ∪ ((ℂ̂ × ℂ̂) ∪ (Diag‘ℂ))) ↦ if(((1st𝑥) = ∞ ∨ (2nd𝑥) = ∞), ∞, if((1st𝑥) ∈ ℂ, if((2nd𝑥) ∈ ℂ, ⟨((1st ‘(1st𝑥)) +R (1st ‘(2nd𝑥))), ((2nd ‘(1st𝑥)) +R (2nd ‘(2nd𝑥)))⟩, (2nd𝑥)), (1st𝑥))))

Syntaxcoppcc 33716 Syntax for negation on the set of extended complex numbers and the complex projective line (Riemann sphere).
class -ℂ̅

Definitiondf-bj-oppc 33717* Define the negation (operation giving the opposite) on the set of extended complex numbers and the complex projective line (Riemann sphere). For infinite extended complex numbers, the argument of +∞e is the opposite of the argument temporary real. (Contributed by BJ, 22-Jun-2019.)
-ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = ∞, ∞, if(𝑥 ∈ ℂ, (𝑦 ∈ ℂ (𝑥 +ℂ̅ 𝑦) = 0), (+∞e‘(𝑥 +ℂ̅ 1/2)))))

20.14.6.5  Order relation on the extended reals

In this section, we redefine df-ltxr 10416 without the intermediate step of df-lt 10285.

Syntaxcltxr 33718 Syntax for the standard (strict) order on the extended reals.
class <R

Definitiondf-bj-lt 33719* Define the standard (strict) order on the extended reals. (Contributed by BJ, 4-Feb-2023.)
<R = ({𝑥 ∈ (ℝ̅ × ℝ̅) ∣ ∃𝑦𝑧(((1st𝑥) = ⟨𝑦, 0R⟩ ∧ (2nd𝑥) = ⟨𝑧, 0R⟩) ∧ 𝑦 <R 𝑧)} ∪ ((({-∞} × ℝ) ∪ (ℝ × {+∞})) ∪ ({-∞} × {+∞})))

20.14.6.6  Argument, multiplication and inverse

Since one needs arguments in order to define multiplication in ℂ̅, and one needs complex multiplication in order to define arguments, it would be contrieved to construct a whole theory for a temporary multiplication (and temporary powers, then temporary logarithm, and finally temporary argument) before redefining the extended complex multiplication. Therefore, we adopt a two-step process, see df-bj-mulc 33723

Syntaxcarg 33720 Syntax for the argument of a nonzero extended complex number.
class Arg

Definitiondf-bj-arg 33721 Define the argument of a nonzero extended complex number. By convention, it has values in (-π, π]. Another convention chooses [0, 2π) but the present one simplifies formulas giving the argument as an arctangent. (Contributed by BJ, 22-Jun-2019.)
Arg = (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), (1st𝑥)))

Syntaxcmulc 33722 Syntax for the multiplication of extended complex numbers.
class ·ℂ̅

Definitiondf-bj-mulc 33723 Define the multiplication of extended complex numbers and of the complex projective line (Riemann sphere). In our convention, a product with 0 is 0, even when the other factor is infinite. An alternate convention leaves products of 0 with an infinite number undefined since the multiplication is not continuous at these points. Note that our convention entails (0 / 0) = 0 (given df-bj-invc 33725).

Note that this definition uses · and Arg and /. Indeed, it would be contrieved to bypass ordinary complex multiplication, and the present two-step definition looks like a good compromise. (Contributed by BJ, 22-Jun-2019.)

·ℂ̅ = (𝑥 ∈ ((ℂ̅ × ℂ̅) ∪ (ℂ̂ × ℂ̂)) ↦ if(((1st𝑥) = 0 ∨ (2nd𝑥) = 0), 0, if(((1st𝑥) = ∞ ∨ (2nd𝑥) = ∞), ∞, if(𝑥 ∈ (ℂ × ℂ), ((1st𝑥) · (2nd𝑥)), (+∞e‘(((Arg‘(1st𝑥)) +ℂ̅ (Arg‘(2nd𝑥))) / τ))))))

Syntaxcinvc 33724 Syntax for the inverse of nonzero extended complex numbers.
class -1ℂ̅

Definitiondf-bj-invc 33725* Define inversion, which maps a nonzero extended complex number or element of the complex projective line (Riemann sphere) to its inverse. Beware of the overloading: the equality (-1ℂ̅‘0) = ∞ is to be understood in the complex projective line, but 0 as an extended complex number does not have an inverse, which we can state as (-1ℂ̅‘0) ∉ ℂ̅. Note that this definition relies on df-bj-mulc 33723, which does not bypass ordinary complex multiplication, but defines extended complex multiplication on top of it. Therefore, we could have used directly / instead of (... ·ℂ̅ ...). (Contributed by BJ, 22-Jun-2019.)
-1ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = 0, ∞, if(𝑥 ∈ ℂ, (𝑦 ∈ ℂ (𝑥 ·ℂ̅ 𝑦) = 1), 0)))

20.14.6.7  The canonical bijection from the finite ordinals

Syntaxciomnn 33726 Syntax for the canonical bijection from _om onto NN0.
class iω↪ℕ

Definitiondf-bj-iomnn 33727* Definition of the canonical bijection from (ω ∪ {ω}) onto (ℕ0 ∪ {+∞}).

To understand this definition, recall that set.mm constructs reals as couples whose first component is a prereal and second component is the zero prereal (in order that one have ℝ ⊆ ℂ), that prereals are equivalence classes of couples of positive reals, the latter are Dedekind cuts of positive rationals, which are equivalence classes of positive ordinals. In partiular, we take the successor ordinal at the beginning and subtract one at the end since the intermediate systems contain only (strictly) positive numbers.

Note the similarity with df-bj-fractemp 33673 but we did not use the present definition there since we wanted to have defined +∞ first.

See bj-iomnnom 33737 for its value at +∞.

TODO: define 0 = (iω↪ℕ “ ω) (and then ℕ = (ℕ0 ∖ ℕ)) and prove (iω↪ℕ ↾ ω):ω–1-1-onto→ℕ0 and that this bijection is an ordered rig isomorphism. Prove (iω↪ℕ ↾ ω) = rec((𝑥 ∈ ℝ ↦ (𝑥 + 1)), 0) and (iω↪ℕ‘∅) = 0. (Contributed by BJ, 18-Feb-2023.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)

iω↪ℕ = ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟Q𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})

Theorembj-imafv 33728 If the direct image of a singleton under any of two functions is the same, then the values of these function at the corresponding point agree. (Contributed by BJ, 18-Mar-2023.)
((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → (𝐹𝐴) = (𝐺𝐴))

Theorembj-funun 33729 Value of a function expressed as a union of two functions at a point not in the domain of one of them. (Contributed by BJ, 18-Mar-2023.)
(𝜑𝐹 = (𝐺𝐻))    &   (𝜑 → ¬ 𝐴 ∈ dom 𝐻)       (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Theorembj-fununsn1 33730 Value of a function expressed as a union of a function and a singleton on a couple (with disjoint domain) at a point not equal to the first component of that couple. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.)
(𝜑𝐹 = (𝐺 ∪ {⟨𝐵, 𝐶⟩}))    &   (𝜑 → ¬ 𝐴 = 𝐵)       (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Theorembj-fununsn2 33731 Value of a function expressed as a union of a function and a singleton on a couple (with disjoint domain) at the first component of that couple. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.)
(𝜑𝐹 = (𝐺 ∪ {⟨𝐵, 𝐶⟩}))    &   (𝜑 → ¬ 𝐵 ∈ dom 𝐺)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → (𝐹𝐵) = 𝐶)

Theoremeldifsnneq 33732 An element of a difference with a singleton is not equal to the element of that singleton. Note that 𝐴 ∈ {𝐶} → ¬ 𝐴 = 𝐶) need not hold if 𝐴 is a proper class. (Contributed by BJ, 18-Mar-2023.)
(𝐴 ∈ (𝐵 ∖ {𝐶}) → ¬ 𝐴 = 𝐶)

Theorembj-fvsnun1 33733 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. (Contributed by NM, 23-Sep-2007.) Put in deduction form and remove two sethood hypotheses. (Revised by BJ, 18-Mar-2023.)
(𝜑𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))    &   (𝜑𝐷 ∈ (𝐶 ∖ {𝐴}))       (𝜑 → (𝐺𝐷) = (𝐹𝐷))

Theorembj-fvsnun2 33734 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 6718. (Contributed by NM, 23-Sep-2007.) Put in deduction form. (Revised by BJ, 18-Mar-2023.) (Proof modification is discouraged.)
(𝜑𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐺𝐴) = 𝐵)

Theorembj-fvmptunsn1 33735* Value of a function expressed as a union of a mapsto expression and a singleton on a couple (with disjoint domain) at the first component of that couple. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.)
(𝜑𝐹 = ((𝑥𝐴𝐵) ∪ {⟨𝐶, 𝐷⟩}))    &   (𝜑 → ¬ 𝐶𝐴)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)       (𝜑 → (𝐹𝐶) = 𝐷)

Theorembj-fvmptunsn2 33736* Value of a function expressed as a union of a mapsto expression and a singleton on a couple (with disjoint domain) at a point in the domain of the mapsto construction. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.)
(𝜑𝐹 = ((𝑥𝐴𝐵) ∪ {⟨𝐶, 𝐷⟩}))    &   (𝜑 → ¬ 𝐶𝐴)    &   (𝜑𝐸𝐴)    &   (𝜑𝐺𝑉)    &   ((𝜑𝑥 = 𝐸) → 𝐵 = 𝐺)       (𝜑 → (𝐹𝐸) = 𝐺)

Theorembj-iomnnom 33737 The canonical bijection from (ω ∪ {ω}) onto (ℕ0 ∪ {+∞}) maps ω to +∞. (Contributed by BJ, 18-Feb-2023.)
(iω↪ℕ‘ω) = +∞

20.14.7  Monoids

See ccmn 18579 and subsequents. The first few statements of this subsection can be put very early after ccmn 18579. Proposal: in the main part, make separate subsections of commutative monoids and abelian groups.

Relabel cabl 18580 to "cabl" or, preferably, other labels containing "abl" to "abel", for consistency.

Theorembj-cmnssmnd 33738 Commutative monoids are monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
CMnd ⊆ Mnd

Theorembj-cmnssmndel 33739 Commutative monoids are monoids (elemental version). This is a more direct proof of cmnmnd 18594, which relies on iscmn 18586. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ CMnd → 𝐴 ∈ Mnd)

Theorembj-ablssgrp 33740 Abelian groups are groups. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
Abel ⊆ Grp

Theorembj-ablssgrpel 33741 Abelian groups are groups (elemental version). This is a shorter proof of ablgrp 18584. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ Abel → 𝐴 ∈ Grp)

Theorembj-ablsscmn 33742 Abelian groups are commutative monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
Abel ⊆ CMnd

Theorembj-ablsscmnel 33743 Abelian groups are commutative monoids (elemental version). This is a shorter proof of ablcmn 18585. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ Abel → 𝐴 ∈ CMnd)

Theorembj-modssabl 33744 (The additive groups of) modules are abelian groups. (The elemental version is lmodabl 19302; see also lmodgrp 19262 and lmodcmn 19303.) (Contributed by BJ, 9-Jun-2019.)
LMod ⊆ Abel

Theorembj-vecssmod 33745 Vector spaces are modules. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
LVec ⊆ LMod

Theorembj-vecssmodel 33746 Vector spaces are modules (elemental version). This is a shorter proof of lveclmod 19501. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ LVec → 𝐴 ∈ LMod)

20.14.7.1  Finite sums in monoids

UPDATE: a similar summation is already defined as df-gsum 16489 (although it mixes finite and infinite sums, which makes it harder to understand).

Syntaxcfinsum 33747 Syntax for the class "finite summation in monoids".
class FinSum

Definitiondf-bj-finsum 33748* Finite summation in commutative monoids. This finite summation function can be extended to pairs 𝑦, 𝑧 where 𝑦 is a left-unital magma and 𝑧 is defined on a totally ordered set (choosing left-associative composition), or dropping unitality and requiring nonempty families, or on any monoids for families of permutable elements, etc. We use the term "summation", even though the definition stands for any unital, commutative and associative composition law. (Contributed by BJ, 9-Jun-2019.)
FinSum = (𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↦ (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚))))

Theorembj-finsumval0 33749* Value of a finite sum. (Contributed by BJ, 9-Jun-2019.) (Proof shortened by AV, 5-May-2021.)
(𝜑𝐴 ∈ CMnd)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐵:𝐼⟶(Base‘𝐴))       (𝜑 → (𝐴 FinSum 𝐵) = (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))))

20.14.8  Affine, Euclidean, and Cartesian geometry

A few basic theorems to start affine, Euclidean, and Cartesian geometry.

20.14.8.1  Convex hull in real vector spaces

A few basic definitions and theorems about convex hulls in real vector spaces. TODO: prove inclusion in the class of subcomplex vector spaces.

Syntaxcrrvec 33750 Syntax for the class of real vector spaces.
class ℝ-Vec

Definitiondf-bj-rrvec 33751 Definition of the class of real vector spaces. (Contributed by BJ, 9-Jun-2019.)
ℝ-Vec = {𝑥 ∈ LVec ∣ (Scalar‘𝑥) = ℝfld}

Theorembj-rrvecssvec 33752 Real vector spaces are vector spaces. (Contributed by BJ, 9-Jun-2019.)
ℝ-Vec ⊆ LVec

Theorembj-rrvecssvecel 33753 Real vector spaces are vector spaces (elemental version). (Contributed by BJ, 9-Jun-2019.)
(𝐴 ∈ ℝ-Vec → 𝐴 ∈ LVec)

Theorembj-rrvecsscmn 33754 (The additive groups of) real vector spaces are commutative monoids. (Contributed by BJ, 9-Jun-2019.)
ℝ-Vec ⊆ CMnd

Theorembj-rrvecsscmnel 33755 (The additive groups of) real vector spaces are commutative monoids (elemental version). (Contributed by BJ, 9-Jun-2019.)
(𝐴 ∈ ℝ-Vec → 𝐴 ∈ CMnd)

20.14.8.2  Complex numbers (supplements)

Some lemmas to ease algebraic manipulations.

Theorembj-subcom 33756 A consequence of commutativity of multiplication. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) − (𝐵 · 𝐴)) = 0)

Theorembj-lineqi 33757 Solution of a (scalar) linear equation. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑌 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑 → ((𝐴 · 𝑋) + 𝐵) = 𝑌)       (𝜑𝑋 = ((𝑌𝐵) / 𝐴))

20.14.8.3  Barycentric coordinates

Lemmas about barycentric coordinates. For the moment, this is limited to the one-dimensional case (complex line), where existence and uniqueness of barycentric coordinates are proved by bj-bary1 33760 (which computes them). It would be nice to prove the two-dimensional case (is it easier to use ad hoc computations, or Cramer formulas?), in order to do some planar geometry.

Theorembj-bary1lem 33758 A lemma for barycentric coordinates in one dimension. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑𝑋 = ((((𝐵𝑋) / (𝐵𝐴)) · 𝐴) + (((𝑋𝐴) / (𝐵𝐴)) · 𝐵)))

Theorembj-bary1lem1 33759 Existence and uniqueness (and actual computation) of barycentric coordinates in dimension 1 (complex line). (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑆 ∈ ℂ)    &   (𝜑𝑇 ∈ ℂ)       (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) → 𝑇 = ((𝑋𝐴) / (𝐵𝐴))))

Theorembj-bary1 33760 Barycentric coordinates in one dimension. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑆 ∈ ℂ)    &   (𝜑𝑇 ∈ ℂ)       (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) ↔ (𝑆 = ((𝐵𝑋) / (𝐵𝐴)) ∧ 𝑇 = ((𝑋𝐴) / (𝐵𝐴)))))

20.15  Mathbox for Jim Kingdon

20.15.0.1  Circle constant

Theoremtaupilem3 33761 Lemma for tau-related theorems. (Contributed by Jim Kingdon, 16-Feb-2019.)
(𝐴 ∈ (ℝ+ ∩ (cos “ {1})) ↔ (𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1))

Theoremtaupilemrplb 33762* A set of positive reals has (in the reals) a lower bound. (Contributed by Jim Kingdon, 19-Feb-2019.)
𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ+𝐴)𝑥𝑦

Theoremtaupilem1 33763 Lemma for taupi 33765. A positive real whose cosine is one is at least 2 · π. (Contributed by Jim Kingdon, 19-Feb-2019.)
((𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1) → (2 · π) ≤ 𝐴)

Theoremtaupilem2 33764 Lemma for taupi 33765. The smallest positive real whose cosine is one is at most 2 · π. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.)
τ ≤ (2 · π)

Theoremtaupi 33765 Relationship between τ and π. This can be seen as connecting the ratio of a circle's circumference to its radius and the ratio of a circle's circumference to its diameter. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.)
τ = (2 · π)

20.15.0.2  Number theory

Theoremdfgcd3 33766* Alternate definition of the gcd operator. (Contributed by Jim Kingdon, 31-Dec-2021.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (𝑑 ∈ ℕ0𝑧 ∈ ℤ (𝑧𝑑 ↔ (𝑧𝑀𝑧𝑁))))

20.16  Mathbox for ML

Theoremcsbdif 33767 Distribution of class substitution over difference of two classes. (Contributed by ML, 14-Jul-2020.)
𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)

Theoremcsbpredg 33768 Move class substitution in and out of the predecessor class of a relationship. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑋) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑋))

Theoremcsbwrecsg 33769 Move class substitution in and out of the well-founded recursive function generator. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥wrecs(𝑅, 𝐷, 𝐹) = wrecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹))

Theoremcsbrecsg 33770 Move class substitution in and out of recs. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥recs(𝐹) = recs(𝐴 / 𝑥𝐹))

Theoremcsbrdgg 33771 Move class substitution in and out of the recursive function generator. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥rec(𝐹, 𝐼) = rec(𝐴 / 𝑥𝐹, 𝐴 / 𝑥𝐼))

Theoremcsboprabg 33772* Move class substitution in and out of class abstractions of nested ordered pairs. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]𝜑})

Theoremcsbmpt22g 33773* Move class substitution in and out of maps-to notation for operations. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥(𝑦𝑌, 𝑧𝑍𝐷) = (𝑦𝐴 / 𝑥𝑌, 𝑧𝐴 / 𝑥𝑍𝐴 / 𝑥𝐷))

Theoremmpnanrd 33774 Eliminate the right side of a negated conjunction in an implication. (Contributed by ML, 17-Oct-2020.)
(𝜑𝜓)    &   (𝜑 → ¬ (𝜓𝜒))       (𝜑 → ¬ 𝜒)

Theoremcon1bii2 33775 A contraposition inference. (Contributed by ML, 18-Oct-2020.)
𝜑𝜓)       (𝜑 ↔ ¬ 𝜓)

Theoremcon2bii2 33776 A contraposition inference. (Contributed by ML, 18-Oct-2020.)
(𝜑 ↔ ¬ 𝜓)       𝜑𝜓)

Theoremvtoclefex 33777* Implicit substitution of a class for a setvar variable. (Contributed by ML, 17-Oct-2020.)
𝑥𝜑    &   (𝑥 = 𝐴𝜑)       (𝐴𝑉𝜑)

Theoremrnmptsn 33778* The range of a function mapping to singletons. (Contributed by ML, 15-Jul-2020.)
ran (𝑥𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}

Theoremf1omptsnlem 33779* This is the core of the proof of f1omptsn 33780, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 15-Jul-2020.)
𝐹 = (𝑥𝐴 ↦ {𝑥})    &   𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       𝐹:𝐴1-1-onto𝑅

Theoremf1omptsn 33780* A function mapping to singletons is bijective onto a set of singletons. (Contributed by ML, 16-Jul-2020.)
𝐹 = (𝑥𝐴 ↦ {𝑥})    &   𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       𝐹:𝐴1-1-onto𝑅

Theoremmptsnunlem 33781* This is the core of the proof of mptsnun 33782, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.)
𝐹 = (𝑥𝐴 ↦ {𝑥})    &   𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       (𝐵𝐴𝐵 = (𝐹𝐵))

Theoremmptsnun 33782* A class 𝐵 is equal to the union of the class of all singletons of elements of 𝐵. (Contributed by ML, 16-Jul-2020.)
𝐹 = (𝑥𝐴 ↦ {𝑥})    &   𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       (𝐵𝐴𝐵 = (𝐹𝐵))

Theoremdissneqlem 33783* This is the core of the proof of dissneq 33784, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.)
𝐶 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)

Theoremdissneq 33784* Any topology that contains every single-point set is the discrete topology. (Contributed by ML, 16-Jul-2020.)
𝐶 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)

Theoremexlimim 33785* Closed form of exlimimd 33786. (Contributed by ML, 17-Jul-2020.)
((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → 𝜓)

Theoremexlimimd 33786* Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.)
(𝜑 → ∃𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremexellim 33787* Closed form of exellimddv 33788. See also exlimim 33785 for a more general theorem. (Contributed by ML, 17-Jul-2020.)
((∃𝑥 𝑥𝐴 ∧ ∀𝑥(𝑥𝐴𝜑)) → 𝜑)

Theoremexellimddv 33788* Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim 33787 for the closed form, which requires the use of a universal quantifier. (Contributed by ML, 17-Jul-2020.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → (𝑥𝐴𝜓))       (𝜑𝜓)

Theoremtopdifinfindis 33789* Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is the trivial topology when 𝐴 is finite. (Contributed by ML, 14-Jul-2020.)
𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}       (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴})

Theoremtopdifinffinlem 33790* This is the core of the proof of topdifinffin 33791, but to avoid the distinct variables on the definition, we need to split this proof into two. (Contributed by ML, 17-Jul-2020.)
𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}       (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)

Theoremtopdifinffin 33791* Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is a topology only if 𝐴 is finite. (Contributed by ML, 17-Jul-2020.)
𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}       (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)

Theoremtopdifinf 33792* Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is a topology if and only if 𝐴 is finite, in which case it is the trivial topology. (Contributed by ML, 17-Jul-2020.)
𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}       ((𝑇 ∈ (TopOn‘𝐴) ↔ 𝐴 ∈ Fin) ∧ (𝑇 ∈ (TopOn‘𝐴) → 𝑇 = {∅, 𝐴}))

Theoremtopdifinfeq 33793* Two different ways of defining the collection from Exercise 3 of [Munkres] p. 83. (Contributed by ML, 18-Jul-2020.)
{𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴))} = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}

Theoremicorempt2 33794* Closed-below, open-above intervals of reals. (Contributed by ML, 26-Jul-2020.)
𝐹 = ([,) ↾ (ℝ × ℝ))       𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})

Theoremicoreresf 33795 Closed-below, open-above intervals of reals map to subsets of reals. (Contributed by ML, 25-Jul-2020.)
([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ

Theoremicoreval 33796* Value of the closed-below, open-above interval function on reals. (Contributed by ML, 26-Jul-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) = {𝑧 ∈ ℝ ∣ (𝐴𝑧𝑧 < 𝐵)})

Theoremicoreelrnab 33797* Elementhood in the set of closed-below, open-above intervals of reals. (Contributed by ML, 27-Jul-2020.)
𝐼 = ([,) “ (ℝ × ℝ))       (𝑋𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})

Theoremisbasisrelowllem1 33798* Lemma for isbasisrelowl 33801. (Contributed by ML, 27-Jul-2020.)
𝐼 = ([,) “ (ℝ × ℝ))       ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑎𝑐𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)

Theoremisbasisrelowllem2 33799* Lemma for isbasisrelowl 33801. (Contributed by ML, 27-Jul-2020.)
𝐼 = ([,) “ (ℝ × ℝ))       ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑎𝑐𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)

Theoremicoreclin 33800* The set of closed-below, open-above intervals of reals is closed under finite intersection. (Contributed by ML, 27-Jul-2020.)
𝐼 = ([,) “ (ℝ × ℝ))       ((𝑥𝐼𝑦𝐼) → (𝑥𝑦) ∈ 𝐼)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43661
 Copyright terms: Public domain < Previous  Next >