Home Metamath Proof ExplorerTheorem List (p. 347 of 454) < 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 (1-28705) Hilbert Space Explorer (28706-30228) Users' Mathboxes (30229-45330)

Theorem List for Metamath Proof Explorer - 34601-34700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-imdiridlem 34601* Lemma for bj-imdirid 34602 and bj-iminvid 34611. (Contributed by BJ, 26-May-2024.)
((𝑥𝐴𝑦𝐴) → (𝜑𝑥 = 𝑦))       {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ 𝜑)} = ( I ↾ 𝒫 𝐴)

Theorembj-imdirid 34602 Functorial property of the direct image: the direct image by the identity on a set is the identity on the powerset. (Contributed by BJ, 24-Dec-2023.)
(𝜑𝐴𝑈)       (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))

Theorembj-opelopabid 34603* Membership in an ordered-pair class abstraction. One can remove the DV condition on 𝑥, 𝑦 by using opabid 5381 in place of opabidw 5380. (Contributed by BJ, 22-May-2024.)
(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝜑)

Theorembj-opabco 34604* Composition of ordered-pair class abstractions. (Contributed by BJ, 22-May-2024.)
({⟨𝑦, 𝑧⟩ ∣ 𝜓} ∘ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝜑𝜓)}

Theorembj-xpcossxp 34605 The composition of two Cartesian products is included in the expected Cartesian product. There is equality if (𝐵𝐶) ≠ ∅, see xpcogend 14329. (Contributed by BJ, 22-May-2024.)
((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐷)

Theorembj-imdirco 34606 Functorial property of the direct image: the direct image by a composition is the composition of the direct images. (Contributed by BJ, 23-May-2024.)
(𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑𝑅 ⊆ (𝐴 × 𝐵))    &   (𝜑𝑆 ⊆ (𝐵 × 𝐶))       (𝜑 → ((𝐴𝒫*𝐶)‘(𝑆𝑅)) = (((𝐵𝒫*𝐶)‘𝑆) ∘ ((𝐴𝒫*𝐵)‘𝑅)))

Syntaxciminv 34607 Syntax for the functionalized inverse image.
class 𝒫*

Definitiondf-iminv 34608* Definition of the functionalized inverse image, which maps a binary relation between two given sets to its associated inverse image relation. (Contributed by BJ, 23-Dec-2023.)
𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ 𝑥 = (𝑟𝑦))}))

Theorembj-iminvval 34609* Value of the functionalized inverse image. (Contributed by BJ, 23-May-2024.)
(𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)       (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑟𝑦))}))

Theorembj-iminvval2 34610* Value of the functionalized inverse image. (Contributed by BJ, 23-May-2024.)
(𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝑅 ⊆ (𝐴 × 𝐵))       (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑅𝑦))})

Theorembj-iminvid 34611 Functorial property of the inverse image: the inverse image by the identity on a set is the identity on the powerset. (Contributed by BJ, 26-May-2024.)
(𝜑𝐴𝑈)       (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))

20.15.6.5  Extended numbers and projective lines as sets

We parameterize the set of infinite extended complex numbers (df-bj-ccinfty 34628) using the real numbers (df-r 10540) via the function +∞e. Since at that point, we have only defined the set of real numbers but no operations on it, we define a temporary "fractional part" function, which is more convenient to define on the temporary reals R (df-nr 10471) since we can use operations on the latter. We also define the temporary real "one-half" in order to define minus infinity (df-bj-minfty 34640) and then we can define the sets of extended real numbers and of extended complex numbers, and the projective real and complex lines, as well as addition and negation on these, and also the order relation on the extended reals (which bypasses the intermediate definition of a temporary order on the real numbers and then a superseding one on the extended real numbers).

Syntaxcfractemp 34612 Syntax for the fractional part of a tempopary real.
class {R

Definitiondf-bj-fractemp 34613* Temporary definition: fractional part of a temporary real.

To understand this definition, recall the canonical injection ω⟶R, 𝑛 ↦ [{𝑥Q𝑥 <Q ⟨suc 𝑛, 1o⟩}, 1P] ~R where we successively take the successor of 𝑛 to land in positive integers, then take the couple with 1o as second component to land in positive rationals, then take the Dedekind cut that positive rational forms, and finally take the equivalence class of the couple with 1P as second component. Adding one at the beginning and subtracting it at the end is necessary since the constructions used in set.mm use the positive integers, positive rationals, and positive reals as intermediate number systems. (Contributed by BJ, 22-Jan-2023.) The precise definition is irrelevant and should generally not be used. One could even inline it. The definitive fractional part of an extended or projective complex number will be defined later. (New usage is discouraged.)

{R = (𝑥R ↦ (𝑦R ((𝑦 = 0R ∨ (0R <R 𝑦𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧Q𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥)))

Syntaxcinftyexpitau 34614 Syntax for the function +∞e parameterizing .
class +∞e

Definitiondf-bj-inftyexpitau 34615 Definition of the auxiliary function +∞e parameterizing the circle at infinity in ℂ̅. We use coupling with {R} to simplify the proof of bj-inftyexpitaudisj 34621. (Contributed by BJ, 22-Jan-2023.) The precise definition is irrelevant and should generally not be used. TODO: prove only the necessary lemmas to prove (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((+∞e𝐴) = (+∞e𝐵) ↔ (𝐴𝐵) ∈ ℤ)). (New usage is discouraged.)
+∞e = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st𝑥)), {R}⟩)

SyntaxcccinftyN 34616 Syntax for the circle at infinity ∞N.
class ∞N

Definitiondf-bj-ccinftyN 34617 Definition of the circle at infinity ∞N. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
∞N = ran +∞e

Theorembj-inftyexpitaufo 34618 The function +∞e written as a surjection with domain and range. (Contributed by BJ, 4-Feb-2023.)
+∞e:ℝ–onto→ℂ∞N

Syntaxchalf 34619 Syntax for the temporary one-half.
class 1/2

Definitiondf-bj-onehalf 34620 Define the temporary real "one-half". Once the machinery is developed, the real number "one-half" is commonly denoted by (1 / 2). (Contributed by BJ, 4-Feb-2023.) (New usage is discouraged.)

TODO:

\$p |- 1/2 e. R. \$= ? \$. (riotacl 7114)

\$p |- -. 0R = 1/2 \$= ? \$. (since -. ( 0R +R 0R ) = 1R )

\$p |- 0R <R 1/2 \$= ? \$.

\$p |- 1/2 <R 1R \$= ? \$.

\$p |- ( {R ` 0R ) = 0R \$= ? \$.

\$p |- ( {R ` 1/2 ) = 1/2 \$= ? \$.

df-minfty \$a |- minfty = ( inftyexpitau ` <. 1/2 , 0R >. ) \$.

1/2 = (𝑥R (𝑥 +R 𝑥) = 1R)

Theorembj-inftyexpitaudisj 34621 An element of the circle at infinity is not a complex number. (Contributed by BJ, 4-Feb-2023.)
¬ (+∞e𝐴) ∈ ℂ

Syntaxcinftyexpi 34622 Syntax for the function +∞ei parameterizing .
class +∞ei

Definitiondf-bj-inftyexpi 34623 Definition of the auxiliary function +∞ei parameterizing the circle at infinity in ℂ̅. We use coupling with to simplify the proof of bj-ccinftydisj 34629. It could seem more natural to define +∞ei on all of , but we want to use only basic functions in the definition of ℂ̅. TODO: transition to df-bj-inftyexpitau 34615 instead. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
+∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)

Theorembj-inftyexpiinv 34624 Utility theorem for the inverse of +∞ei. (Contributed by BJ, 22-Jun-2019.) This utility theorem is irrelevant and should generally not be used. (New usage is discouraged.)
(𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei𝐴)) = 𝐴)

Theorembj-inftyexpiinj 34625 Injectivity of the parameterization +∞ei. Remark: a more conceptual proof would use bj-inftyexpiinv 34624 and the fact that a function with a retraction is injective. (Contributed by BJ, 22-Jun-2019.)
((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (+∞ei𝐴) = (+∞ei𝐵)))

Theorembj-inftyexpidisj 34626 An element of the circle at infinity is not a complex number. (Contributed by BJ, 22-Jun-2019.) This utility theorem is irrelevant and should generally not be used. (New usage is discouraged.)
¬ (+∞ei𝐴) ∈ ℂ

Syntaxcccinfty 34627 Syntax for the circle at infinity .
class

Definitiondf-bj-ccinfty 34628 Definition of the circle at infinity . (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
= ran +∞ei

Theorembj-ccinftydisj 34629 The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.)
(ℂ ∩ ℂ) = ∅

Theorembj-elccinfty 34630 A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
(𝐴 ∈ (-π(,]π) → (+∞ei𝐴) ∈ ℂ)

Syntaxcccbar 34631 Syntax for the set of extended complex numbers ℂ̅.
class ℂ̅

Definitiondf-bj-ccbar 34632 Definition of the set of extended complex numbers ℂ̅. (Contributed by BJ, 22-Jun-2019.)
ℂ̅ = (ℂ ∪ ℂ)

Theorembj-ccssccbar 34633 Complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
ℂ ⊆ ℂ̅

Theorembj-ccinftyssccbar 34634 Infinite extended complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊆ ℂ̅

Syntaxcpinfty 34635 Syntax for "plus infinity".
class +∞

Definitiondf-bj-pinfty 34636 Definition of "plus infinity". (Contributed by BJ, 27-Jun-2019.)
+∞ = (+∞ei‘0)

Theorembj-pinftyccb 34637 The class +∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
+∞ ∈ ℂ̅

Theorembj-pinftynrr 34638 The extended complex number +∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
¬ +∞ ∈ ℂ

Syntaxcminfty 34639 Syntax for "minus infinity".
class -∞

Definitiondf-bj-minfty 34640 Definition of "minus infinity". (Contributed by BJ, 27-Jun-2019.)
-∞ = (+∞ei‘π)

Theorembj-minftyccb 34641 The class -∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
-∞ ∈ ℂ̅

Theorembj-minftynrr 34642 The extended complex number -∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
¬ -∞ ∈ ℂ

Theorembj-pinftynminfty 34643 The extended complex numbers +∞ and -∞ are different. (Contributed by BJ, 27-Jun-2019.)
+∞ ≠ -∞

Syntaxcrrbar 34644 Syntax for the set of extended real numbers.
class ℝ̅

Definitiondf-bj-rrbar 34645 Definition of the set of extended real numbers. This aims to replace df-xr 10672. (Contributed by BJ, 29-Jun-2019.)
ℝ̅ = (ℝ ∪ {-∞, +∞})

Syntaxcinfty 34646 Syntax for .
class

Definitiondf-bj-infty 34647 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 34648 Syntax for ℂ̂.
class ℂ̂

Definitiondf-bj-cchat 34649 Define the complex projective line, or Riemann sphere. (Contributed by BJ, 27-Jun-2019.)
ℂ̂ = (ℂ ∪ {∞})

Syntaxcrrhat 34650 Syntax for ℝ̂.
class ℝ̂

Definitiondf-bj-rrhat 34651 Define the real projective line. (Contributed by BJ, 27-Jun-2019.)
ℝ̂ = (ℝ ∪ {∞})

Theorembj-rrhatsscchat 34652 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 34654 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) ∪ ( I ↾ ℂ))) ↦ if(((1st𝑥) = ∞ ∨ (2nd𝑥) = ∞), ∞, if((1st𝑥) ∈ ℂ, if((2nd𝑥) ∈ ℂ, ⟨((1st ‘(1st𝑥)) +R (1st ‘(2nd𝑥))), ((2nd ‘(1st𝑥)) +R (2nd ‘(2nd𝑥)))⟩, (2nd𝑥)), (1st𝑥))))

Syntaxcoppcc 34655 Syntax for negation on the set of extended complex numbers and the complex projective line (Riemann sphere).
class -ℂ̅

Definitiondf-bj-oppc 34656* Define the negation (operation giving the opposite) on the set of extended complex numbers and the complex projective line (Riemann sphere). (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, 0R⟩)))))

20.15.6.7  Order relation on the extended reals

In this section, we redefine df-ltxr 10673 without the intermediate step of df-lt 10543.

Syntaxcltxr 34657 Syntax for the standard (strict) order on the extended reals.
class <ℝ̅

Definitiondf-bj-lt 34658* Define the standard (strict) order on the extended reals. (Contributed by BJ, 4-Feb-2023.)
<ℝ̅ = ({𝑥 ∈ (ℝ̅ × ℝ̅) ∣ ∃𝑦𝑧(((1st𝑥) = ⟨𝑦, 0R⟩ ∧ (2nd𝑥) = ⟨𝑧, 0R⟩) ∧ 𝑦 <R 𝑧)} ∪ ((({-∞} × ℝ) ∪ (ℝ × {+∞})) ∪ ({-∞} × {+∞})))

20.15.6.8  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 contrived 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 34662.

Syntaxcarg 34659 Syntax for the argument of a nonzero extended complex number.
class Arg

Definitiondf-bj-arg 34660 Define the argument of a nonzero extended complex number. By convention, it has values in (-π, π]. Another convention chooses values in [0, 2π) but the present convention simplifies formulas giving the argument as an arctangent. (Contributed by BJ, 22-Jun-2019.) The "else" case of the second conditional operator, corresponding to infinite extended complex numbers other than -∞, gives a definition depending on the specific definition chosen for these numbers (df-bj-inftyexpitau 34615), and therefore should not be relied upon. (New usage is discouraged.)
Arg = (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), if(𝑥<ℝ̅0, π, (((1st𝑥) / (2 · π)) − π))))

Syntaxcmulc 34661 Syntax for the multiplication of extended complex numbers.
class ·ℂ̅

Definitiondf-bj-mulc 34662 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 34664).

Note that this definition uses · and Arg and /. Indeed, it would be contrived 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 34663 Syntax for the inverse of nonzero extended complex numbers.
class -1ℂ̅

Definitiondf-bj-invc 34664* 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 34662, 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)))

20.15.6.9  The canonical bijection from the finite ordinals

Syntaxciomnn 34665 Syntax for the canonical bijection from (ω ∪ {ω}) onto (ℕ0 ∪ {+∞}).
class iω↪ℕ

Definitiondf-bj-iomnn 34666* 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 1 at the end since the intermediate systems contain only (strictly) positive numbers.

Note the similarity with df-bj-fractemp 34613 but we did not use the present definition there since we wanted to have defined +∞ first.

See bj-iomnnom 34675 for its value at +∞.

TODO:

Prove (iω↪ℕ‘∅) = 0.

Define 0 = (iω↪ℕ “ ω) and ℕ = (ℕ0 ∖ {0}).

Prove iω↪ℕ:(ω ∪ {ω})–1-1-onto→(ℕ0 ∪ {+∞}) and (iω↪ℕ ↾ ω):ω–1-1-onto→ℕ0.

Prove that these bijections are respectively an isomorphism of ordered "extended rigs" and of ordered rigs.

Prove (iω↪ℕ ↾ ω) = rec((𝑥 ∈ ℝ ↦ (𝑥 + 1)), 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 34667 If the direct image of a singleton under any of two functions is the same, then the values of these functions at the corresponding point agree. (Contributed by BJ, 18-Mar-2023.)
((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → (𝐹𝐴) = (𝐺𝐴))

Theorembj-funun 34668 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 34669 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 34670 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)

Theorembj-fvsnun1 34671 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 34672 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 6926. (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 34673* 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 34674* 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 34675 The canonical bijection from (ω ∪ {ω}) onto (ℕ0 ∪ {+∞}) maps ω to +∞. (Contributed by BJ, 18-Feb-2023.)
(iω↪ℕ‘ω) = +∞

20.15.6.10  Divisibility

Syntaxcnnbar 34676 Syntax for the extended natural numbers.
class ℕ̅

Definitiondf-bj-nnbar 34677 Definition of the extended natural numbers. (Contributed by BJ, 28-Jul-2023.)
ℕ̅ = (ℕ0 ∪ {+∞})

Syntaxczzbar 34678 Syntax for the extended integers.
class ℤ̅

Definitiondf-bj-zzbar 34679 Definition of the extended integers. (Contributed by BJ, 28-Jul-2023.)
ℤ̅ = (ℤ ∪ {-∞, +∞})

Syntaxczzhat 34680 Syntax for the one-point-compactified integers.
class ℤ̂

Definitiondf-bj-zzhat 34681 Definition of the one-point-compactified. (Contributed by BJ, 28-Jul-2023.)
ℤ̂ = (ℤ ∪ {∞})

Syntaxcdivc 34682 Syntax for the divisibility relation.
class

Definitiondf-bj-divc 34683* Definition of the divisibility relation (compare df-dvds 15604).

Since 0 is absorbing, (𝐴 ∈ (ℂ̅ ∪ ℂ̂) → (𝐴 0)) and ((0 ∥ 𝐴) ↔ 𝐴 = 0).

(Contributed by BJ, 28-Jul-2023.)

= {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ ((ℂ̅ × ℂ̅) ∪ (ℂ̂ × ℂ̂)) ∧ ∃𝑛 ∈ (ℤ̅ ∪ ℤ̂)(𝑛 ·ℂ̅ 𝑥) = 𝑦)}

20.15.7  Monoids

See ccmn 18902 and subsequents. The first few statements of this subsection can be put very early after ccmn 18902. Proposal: in the main part, make separate subsections of commutative monoids and abelian groups.

Relabel cabl 18903 to "cabl" or, preferably, other labels containing "abl" to "abel", for consistency.

Theorembj-smgrpssmgm 34684 Semigroups are magmas. (Contributed by BJ, 12-Apr-2024.) (Proof modification is discouraged.)
Smgrp ⊆ Mgm

Theorembj-smgrpssmgmel 34685 Semigroups are magmas (elemental version). (Contributed by BJ, 12-Apr-2024.) (Proof modification is discouraged.)
(𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)

Theorembj-mndsssmgrp 34686 Monoids are semigroups. (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.)
Mnd ⊆ Smgrp

Theorembj-mndsssmgrpel 34687 Monoids are semigroups (elemental version). (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.)
(𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)

Theorembj-cmnssmnd 34688 Commutative monoids are monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
CMnd ⊆ Mnd

Theorembj-cmnssmndel 34689 Commutative monoids are monoids (elemental version). This is a more direct proof of cmnmnd 18918, which relies on iscmn 18910. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ CMnd → 𝐴 ∈ Mnd)

Theorembj-grpssmnd 34690 Groups are monoids. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.)
Grp ⊆ Mnd

Theorembj-grpssmndel 34691 Groups are monoids (elemental version). Shorter proof of grpmnd 18106. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.)
(𝐴 ∈ Grp → 𝐴 ∈ Mnd)

Theorembj-ablssgrp 34692 Abelian groups are groups. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
Abel ⊆ Grp

Theorembj-ablssgrpel 34693 Abelian groups are groups (elemental version). This is a shorter proof of ablgrp 18907. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ Abel → 𝐴 ∈ Grp)

Theorembj-ablsscmn 34694 Abelian groups are commutative monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
Abel ⊆ CMnd

Theorembj-ablsscmnel 34695 Abelian groups are commutative monoids (elemental version). This is a shorter proof of ablcmn 18909. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ Abel → 𝐴 ∈ CMnd)

Theorembj-modssabl 34696 (The additive groups of) modules are abelian groups. (The elemental version is lmodabl 19678; see also lmodgrp 19638 and lmodcmn 19679.) (Contributed by BJ, 9-Jun-2019.)
LMod ⊆ Abel

Theorembj-vecssmod 34697 Vector spaces are modules. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
LVec ⊆ LMod

Theorembj-vecssmodel 34698 Vector spaces are modules (elemental version). This is a shorter proof of lveclmod 19875. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ LVec → 𝐴 ∈ LMod)

20.15.7.1  Finite sums in monoids

UPDATE: a similar summation is already defined as df-gsum 16712 (although it mixes finite and infinite sums, which makes it harder to understand).

Syntaxcfinsum 34699 Syntax for the class "finite summation in monoids".
class FinSum

Definitiondf-bj-finsum 34700* 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))))

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 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45330
 Copyright terms: Public domain < Previous  Next >