P. 355 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35401-35500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-pinftyccb 35401	The class +∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ +∞ ∈ ℂ̅
Theorem	bj-pinftynrr 35402	The extended complex number +∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ ¬ +∞ ∈ ℂ
Syntax	cminfty 35403	Syntax for "minus infinity".
class -∞
Definition	df-bj-minfty 35404	Definition of "minus infinity". (Contributed by BJ, 27-Jun-2019.)
⊢ -∞ = (+∞ei‘π)
Theorem	bj-minftyccb 35405	The class -∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ -∞ ∈ ℂ̅
Theorem	bj-minftynrr 35406	The extended complex number -∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ ¬ -∞ ∈ ℂ
Theorem	bj-pinftynminfty 35407	The extended complex numbers +∞ and -∞ are different. (Contributed by BJ, 27-Jun-2019.)
⊢ +∞ ≠ -∞
Syntax	crrbar 35408	Syntax for the set of extended real numbers.
class ℝ̅
Definition	df-bj-rrbar 35409	Definition of the set of extended real numbers. This aims to replace df-xr 11022. (Contributed by BJ, 29-Jun-2019.)
⊢ ℝ̅ = (ℝ ∪ {-∞, +∞})
Syntax	cinfty 35410	Syntax for ∞.
class ∞
Definition	df-bj-infty 35411	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 ∪ ℂ
Syntax	ccchat 35412	Syntax for ℂ̂.
class ℂ̂
Definition	df-bj-cchat 35413	Define the complex projective line, or Riemann sphere. (Contributed by BJ, 27-Jun-2019.)
⊢ ℂ̂ = (ℂ ∪ {∞})
Syntax	crrhat 35414	Syntax for ℝ̂.
class ℝ̂
Definition	df-bj-rrhat 35415	Define the real projective line. (Contributed by BJ, 27-Jun-2019.)
⊢ ℝ̂ = (ℝ ∪ {∞})
Theorem	bj-rrhatsscchat 35416	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.
Syntax	caddcc 35417	Syntax for the addition on extended complex numbers.
class +ℂ̅
Definition	df-bj-addc 35418	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(((1st ‘𝑥) = ∞ ∨ (2nd ‘𝑥) = ∞), ∞, if((1st ‘𝑥) ∈ ℂ, if((2nd ‘𝑥) ∈ ℂ, ⟨((1st ‘(1st ‘𝑥)) +R (1st ‘(2nd ‘𝑥))), ((2nd ‘(1st ‘𝑥)) +R (2nd ‘(2nd ‘𝑥)))⟩, (2nd ‘𝑥)), (1st ‘𝑥))))
Syntax	coppcc 35419	Syntax for negation on the set of extended complex numbers and the complex projective line (Riemann sphere).
class -ℂ̅
Definition	df-bj-oppc 35420*	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), (+∞eiτ‘(𝑥 +ℂ̅ ⟨1/2, 0R⟩)))))
20.15.6.7  Order relation on the extended reals In this section, we redefine df-ltxr 11023 without the intermediate step of df-lt 10893.
Syntax	cltxr 35421	Syntax for the standard (strict) order on the extended reals.
class <ℝ̅
Definition	df-bj-lt 35422*	Define the standard (strict) order on the extended reals. (Contributed by BJ, 4-Feb-2023.)
⊢ <ℝ̅ = ({𝑥 ∈ (ℝ̅ × ℝ̅) ∣ ∃𝑦∃𝑧(((1st ‘𝑥) = ⟨𝑦, 0R⟩ ∧ (2nd ‘𝑥) = ⟨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 35426.
Syntax	carg 35423	Syntax for the argument of a nonzero extended complex number.
class Arg
Definition	df-bj-arg 35424	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 35379), 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, (((1st ‘𝑥) / (2 · π)) − π))))
Syntax	cmulc 35425	Syntax for the multiplication of extended complex numbers.
class ·ℂ̅
Definition	df-bj-mulc 35426	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 35428). 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(((1st ‘𝑥) = 0 ∨ (2nd ‘𝑥) = 0), 0, if(((1st ‘𝑥) = ∞ ∨ (2nd ‘𝑥) = ∞), ∞, if(𝑥 ∈ (ℂ × ℂ), ((1st ‘𝑥) · (2nd ‘𝑥)), (+∞eiτ‘(((Arg‘(1st ‘𝑥)) +ℂ̅ (Arg‘(2nd ‘𝑥))) / τ))))))
Syntax	cinvc 35427	Syntax for the inverse of nonzero extended complex numbers.
class -1ℂ̅
Definition	df-bj-invc 35428*	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 35426, 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
Syntax	ciomnn 35429	Syntax for the canonical bijection from (ω ∪ {ω}) onto (ℕ0 ∪ {+∞}).
class iω↪ℕ
Definition	df-bj-iomnn 35430*	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 35377 but we did not use the present definition there since we wanted to have defined +∞ first. See bj-iomnnom 35439 for its value at +∞. TODO: Prove ⊢ (iω↪ℕ‘∅) = 0. Define ⊢ ℕ0 = (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 ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})
Theorem	bj-imafv 35431	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}) → (𝐹‘𝐴) = (𝐺‘𝐴))
Theorem	bj-funun 35432	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.)
⊢ (𝜑 → 𝐹 = (𝐺 ∪ 𝐻))    &   ⊢ (𝜑 → ¬ 𝐴 ∈ dom 𝐻)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))
Theorem	bj-fununsn1 35433	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.)
⊢ (𝜑 → 𝐹 = (𝐺 ∪ {⟨𝐵, 𝐶⟩}))    &   ⊢ (𝜑 → ¬ 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))
Theorem	bj-fununsn2 35434	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.)
⊢ (𝜑 → 𝐹 = (𝐺 ∪ {⟨𝐵, 𝐶⟩}))    &   ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐺)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐹‘𝐵) = 𝐶)
Theorem	bj-fvsnun1 35435	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.)
⊢ (𝜑 → 𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))    &   ⊢ (𝜑 → 𝐷 ∈ (𝐶 ∖ {𝐴}))    ⇒   ⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷))
Theorem	bj-fvsnun2 35436	The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7064. (Contributed by NM, 23-Sep-2007.) Put in deduction form. (Revised by BJ, 18-Mar-2023.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵)
Theorem	bj-fvmptunsn1 35437*	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.)
⊢ (𝜑 → 𝐹 = ((𝑥 ∈ 𝐴 ↦ 𝐵) ∪ {⟨𝐶, 𝐷⟩}))    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐹‘𝐶) = 𝐷)
Theorem	bj-fvmptunsn2 35438*	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.)
⊢ (𝜑 → 𝐹 = ((𝑥 ∈ 𝐴 ↦ 𝐵) ∪ {⟨𝐶, 𝐷⟩}))    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐸) → 𝐵 = 𝐺)    ⇒   ⊢ (𝜑 → (𝐹‘𝐸) = 𝐺)
Theorem	bj-iomnnom 35439	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
Syntax	cnnbar 35440	Syntax for the extended natural numbers.
class ℕ̅
Definition	df-bj-nnbar 35441	Definition of the extended natural numbers. (Contributed by BJ, 28-Jul-2023.)
⊢ ℕ̅ = (ℕ0 ∪ {+∞})
Syntax	czzbar 35442	Syntax for the extended integers.
class ℤ̅
Definition	df-bj-zzbar 35443	Definition of the extended integers. (Contributed by BJ, 28-Jul-2023.)
⊢ ℤ̅ = (ℤ ∪ {-∞, +∞})
Syntax	czzhat 35444	Syntax for the one-point-compactified integers.
class ℤ̂
Definition	df-bj-zzhat 35445	Definition of the one-point-compactified. (Contributed by BJ, 28-Jul-2023.)
⊢ ℤ̂ = (ℤ ∪ {∞})
Syntax	cdivc 35446	Syntax for the divisibility relation.
class ∥ℂ
Definition	df-bj-divc 35447*	Definition of the divisibility relation (compare df-dvds 15973). Since 0 is absorbing, ⊢ (𝐴 ∈ (ℂ̅ ∪ ℂ̂) → (𝐴 ∥ℂ 0)) and ⊢ ((0 ∥ℂ 𝐴) ↔ 𝐴 = 0). (Contributed by BJ, 28-Jul-2023.)
⊢ ∥ℂ = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ ((ℂ̅ × ℂ̅) ∪ (ℂ̂ × ℂ̂)) ∧ ∃𝑛 ∈ (ℤ̅ ∪ ℤ̂)(𝑛 ·ℂ̅ 𝑥) = 𝑦)}
20.15.7  Monoids See ccmn 19395 and subsequents. The first few statements of this subsection can be put very early after ccmn 19395. Proposal: in the main part, make separate subsections of commutative monoids and abelian groups. Relabel cabl 19396 to "cabl" or, preferably, other labels containing "abl" to "abel", for consistency.
Theorem	bj-smgrpssmgm 35448	Semigroups are magmas. (Contributed by BJ, 12-Apr-2024.) (Proof modification is discouraged.)
⊢ Smgrp ⊆ Mgm
Theorem	bj-smgrpssmgmel 35449	Semigroups are magmas (elemental version). (Contributed by BJ, 12-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
Theorem	bj-mndsssmgrp 35450	Monoids are semigroups. (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.)
⊢ Mnd ⊆ Smgrp
Theorem	bj-mndsssmgrpel 35451	Monoids are semigroups (elemental version). (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)
Theorem	bj-cmnssmnd 35452	Commutative monoids are monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ CMnd ⊆ Mnd
Theorem	bj-cmnssmndel 35453	Commutative monoids are monoids (elemental version). This is a more direct proof of cmnmnd 19411, which relies on iscmn 19403. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ CMnd → 𝐴 ∈ Mnd)
Theorem	bj-grpssmnd 35454	Groups are monoids. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.)
⊢ Grp ⊆ Mnd
Theorem	bj-grpssmndel 35455	Groups are monoids (elemental version). Shorter proof of grpmnd 18593. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ Grp → 𝐴 ∈ Mnd)
Theorem	bj-ablssgrp 35456	Abelian groups are groups. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ Abel ⊆ Grp
Theorem	bj-ablssgrpel 35457	Abelian groups are groups (elemental version). This is a shorter proof of ablgrp 19400. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ Abel → 𝐴 ∈ Grp)
Theorem	bj-ablsscmn 35458	Abelian groups are commutative monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ Abel ⊆ CMnd
Theorem	bj-ablsscmnel 35459	Abelian groups are commutative monoids (elemental version). This is a shorter proof of ablcmn 19402. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ Abel → 𝐴 ∈ CMnd)
Theorem	bj-modssabl 35460	(The additive groups of) modules are abelian groups. (The elemental version is lmodabl 20179; see also lmodgrp 20139 and lmodcmn 20180.) (Contributed by BJ, 9-Jun-2019.)
⊢ LMod ⊆ Abel
Theorem	bj-vecssmod 35461	Vector spaces are modules. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ LVec ⊆ LMod
Theorem	bj-vecssmodel 35462	Vector spaces are modules (elemental version). This is a shorter proof of lveclmod 20377. (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 17162 (although it mixes finite and infinite sums, which makes it harder to understand).
Syntax	cfinsum 35463	Syntax for the class "finite summation in monoids".
class FinSum
Definition	df-bj-finsum 35464*	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))} ↦ (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚))))
Theorem	bj-finsumval0 35465*	Value of a finite sum. (Contributed by BJ, 9-Jun-2019.) (Proof shortened by AV, 5-May-2021.)
⊢ (𝜑 → 𝐴 ∈ CMnd)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝐵:𝐼⟶(Base‘𝐴))    ⇒   ⊢ (𝜑 → (𝐴 FinSum 𝐵) = (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)))))
20.15.8  Affine, Euclidean, and Cartesian geometry A few basic theorems to start affine, Euclidean, and Cartesian geometry. The first step is to define real vector spaces, then barycentric coordinates and convex hulls.
20.15.8.1  Real vector spaces In this section, we introduce real vector spaces.
Theorem	bj-fvimacnv0 35466	Variant of fvimacnv 6939 where membership of 𝐴 in the domain is not needed provided the containing class 𝐵 does not contain the empty set. Note that this antecedent would not be needed with Definition df-afv 44623. (Contributed by BJ, 7-Jan-2024.)
⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))
Theorem	bj-isvec 35467	The predicate "is a vector space". (Contributed by BJ, 6-Jan-2024.)
⊢ (𝜑 → 𝐾 = (Scalar‘𝑉))    ⇒   ⊢ (𝜑 → (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing)))
Theorem	bj-fldssdrng 35468	Fields are division rings. (Contributed by BJ, 6-Jan-2024.)
⊢ Field ⊆ DivRing
Theorem	bj-flddrng 35469	Fields are division rings (elemental version). (Contributed by BJ, 9-Nov-2024.)
⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing)
Theorem	bj-rrdrg 35470	The field of real numbers is a division ring. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝfld ∈ DivRing
Theorem	bj-isclm 35471	The predicate "is a subcomplex module". (Contributed by BJ, 6-Jan-2024.)
⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))    &   ⊢ (𝜑 → 𝐾 = (Base‘𝐹))    ⇒   ⊢ (𝜑 → (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
Syntax	crrvec 35472	Syntax for the class of real vector spaces.
class ℝ-Vec
Definition	df-bj-rvec 35473	Definition of the class of real vector spaces. The previous definition, ⊢ ℝ-Vec = {𝑥 ∈ LMod ∣ (Scalar‘𝑥) = ℝfld}, can be recovered using bj-isrvec 35474. The present one is preferred since it does not use any dummy variable. That ℝ-Vec could be defined with LVec in place of LMod is a consequence of bj-isrvec2 35480. (Contributed by BJ, 9-Jun-2019.)
⊢ ℝ-Vec = (LMod ∩ (◡Scalar “ {ℝfld}))
Theorem	bj-isrvec 35474	The predicate "is a real vector space". Using df-sca 16987 instead of scaid 17034 would shorten the proof by two syntactic steps, but it is preferable not to rely on the precise definition df-sca 16987. (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LMod ∧ (Scalar‘𝑉) = ℝfld))
Theorem	bj-rvecmod 35475	Real vector spaces are modules (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ LMod)
Theorem	bj-rvecssmod 35476	Real vector spaces are modules. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ LMod
Theorem	bj-rvecrr 35477	The field of scalars of a real vector space is the field of real numbers. (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → (Scalar‘𝑉) = ℝfld)
Theorem	bj-isrvecd 35478	The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.)
⊢ (𝜑 → (Scalar‘𝑉) = 𝐾)    ⇒   ⊢ (𝜑 → (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LMod ∧ 𝐾 = ℝfld)))
Theorem	bj-rvecvec 35479	Real vector spaces are vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ LVec)
Theorem	bj-isrvec2 35480	The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.)
⊢ (𝜑 → (Scalar‘𝑉) = 𝐾)    ⇒   ⊢ (𝜑 → (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LVec ∧ 𝐾 = ℝfld)))
Theorem	bj-rvecssvec 35481	Real vector spaces are vector spaces. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ LVec
Theorem	bj-rveccmod 35482	Real vector spaces are subcomplex modules (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂMod)
Theorem	bj-rvecsscmod 35483	Real vector spaces are subcomplex modules. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ ℂMod
Theorem	bj-rvecsscvec 35484	Real vector spaces are subcomplex vector spaces. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ ℂVec
Theorem	bj-rveccvec 35485	Real vector spaces are subcomplex vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂVec)
Theorem	bj-rvecssabl 35486	(The additive groups of) real vector spaces are commutative groups. (Contributed by BJ, 9-Jun-2019.)
⊢ ℝ-Vec ⊆ Abel
Theorem	bj-rvecabl 35487	(The additive groups of) real vector spaces are commutative groups (elemental version). (Contributed by BJ, 9-Jun-2019.)
⊢ (𝐴 ∈ ℝ-Vec → 𝐴 ∈ Abel)
20.15.8.2  Complex numbers (supplements) Some lemmas to ease algebraic manipulations.
Theorem	bj-subcom 35488	A consequence of commutativity of multiplication. (Contributed by BJ, 6-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) − (𝐵 · 𝐴)) = 0)
Theorem	bj-lineqi 35489	Solution of a (scalar) linear equation. (Contributed by BJ, 6-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑌 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → ((𝐴 · 𝑋) + 𝐵) = 𝑌)    ⇒   ⊢ (𝜑 → 𝑋 = ((𝑌 − 𝐵) / 𝐴))
20.15.8.3  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 35492 (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.
Theorem	bj-bary1lem 35490	Lemma for bj-bary1 35492: expression for a barycenter of two points in one dimension (complex line). (Contributed by BJ, 6-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 = ((((𝐵 − 𝑋) / (𝐵 − 𝐴)) · 𝐴) + (((𝑋 − 𝐴) / (𝐵 − 𝐴)) · 𝐵)))
Theorem	bj-bary1lem1 35491	Lemma for bj-bary1: computation of one of the two barycentric coordinates of a barycenter of two points in one dimension (complex line). (Contributed by BJ, 6-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ ℂ)    &   ⊢ (𝜑 → 𝑇 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) → 𝑇 = ((𝑋 − 𝐴) / (𝐵 − 𝐴))))
Theorem	bj-bary1 35492	Barycentric coordinates in one dimension (complex line). In the statement, 𝑋 is the barycenter of the two points 𝐴, 𝐵 with respective normalized coefficients 𝑆, 𝑇. (Contributed by BJ, 6-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ ℂ)    &   ⊢ (𝜑 → 𝑇 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) ↔ (𝑆 = ((𝐵 − 𝑋) / (𝐵 − 𝐴)) ∧ 𝑇 = ((𝑋 − 𝐴) / (𝐵 − 𝐴)))))
20.15.9  Monoid of endomorphisms
Syntax	cend 35493	Token for the monoid of endomorphisms.
class End
Definition	df-bj-end 35494*	The monoid of endomorphisms on an object of a category. (Contributed by BJ, 4-Apr-2024.)
⊢ End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩}))
Theorem	bj-endval 35495	Value of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) = {⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩, ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩})
Theorem	bj-endbase 35496	Base set of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → (Base‘((End ‘𝐶)‘𝑋)) = (𝑋(Hom ‘𝐶)𝑋))
Theorem	bj-endcomp 35497	Composition law of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → (+g‘((End ‘𝐶)‘𝑋)) = (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋))
Theorem	bj-endmnd 35498	The monoid of endomorphisms on an object of a category is a monoid. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) ∈ Mnd)
20.16  Mathbox for Jim Kingdon
20.16.0.1  Circle constant
Theorem	taupilem3 35499	Lemma for tau-related theorems. (Contributed by Jim Kingdon, 16-Feb-2019.)
⊢ (𝐴 ∈ (ℝ+ ∩ (◡cos “ {1})) ↔ (𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1))
Theorem	taupilemrplb 35500*	A set of positive reals has (in the reals) a lower bound. (Contributed by Jim Kingdon, 19-Feb-2019.)
⊢ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)𝑥 ≤ 𝑦