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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
 
20.15.6.6  Addition and opposite

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.

 
Syntaxcaddcc 34653 Syntax for the addition on extended complex numbers.
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))))
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