P. 362 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36101-36200   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-bj-pinfty 36101	Definition of "plus infinity". (Contributed by BJ, 27-Jun-2019.)
⊢ +∞ = (+∞ei‘0)
Theorem	bj-pinftyccb 36102	The class +∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ +∞ ∈ ℂ̅
Theorem	bj-pinftynrr 36103	The extended complex number +∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ ¬ +∞ ∈ ℂ
Syntax	cminfty 36104	Syntax for "minus infinity".
class -∞
Definition	df-bj-minfty 36105	Definition of "minus infinity". (Contributed by BJ, 27-Jun-2019.)
⊢ -∞ = (+∞ei‘π)
Theorem	bj-minftyccb 36106	The class -∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ -∞ ∈ ℂ̅
Theorem	bj-minftynrr 36107	The extended complex number -∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ ¬ -∞ ∈ ℂ
Theorem	bj-pinftynminfty 36108	The extended complex numbers +∞ and -∞ are different. (Contributed by BJ, 27-Jun-2019.)
⊢ +∞ ≠ -∞
Syntax	crrbar 36109	Syntax for the set of extended real numbers.
class ℝ̅
Definition	df-bj-rrbar 36110	Definition of the set of extended real numbers. This aims to replace df-xr 11252. (Contributed by BJ, 29-Jun-2019.)
⊢ ℝ̅ = (ℝ ∪ {-∞, +∞})
Syntax	cinfty 36111	Syntax for ∞.
class ∞
Definition	df-bj-infty 36112	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 36113	Syntax for ℂ̂.
class ℂ̂
Definition	df-bj-cchat 36114	Define the complex projective line, or Riemann sphere. (Contributed by BJ, 27-Jun-2019.)
⊢ ℂ̂ = (ℂ ∪ {∞})
Syntax	crrhat 36115	Syntax for ℝ̂.
class ℝ̂
Definition	df-bj-rrhat 36116	Define the real projective line. (Contributed by BJ, 27-Jun-2019.)
⊢ ℝ̂ = (ℝ ∪ {∞})
Theorem	bj-rrhatsscchat 36117	The real projective line is included in the complex projective line. (Contributed by BJ, 27-Jun-2019.)
⊢ ℝ̂ ⊆ ℂ̂
21.17.6.6  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 36118	Syntax for the addition on extended complex numbers.
class +ℂ̅
Definition	df-bj-addc 36119	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 36120	Syntax for negation on the set of extended complex numbers and the complex projective line (Riemann sphere).
class -ℂ̅
Definition	df-bj-oppc 36121*	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⟩)))))
21.17.6.7  Order relation on the extended reals In this section, we redefine df-ltxr 11253 without the intermediate step of df-lt 11123.
Syntax	cltxr 36122	Syntax for the standard (strict) order on the extended reals.
class <ℝ̅
Definition	df-bj-lt 36123*	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 × {+∞})) ∪ ({-∞} × {+∞})))
21.17.6.8  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 36127.
Syntax	carg 36124	Syntax for the argument of a nonzero extended complex number.
class Arg
Definition	df-bj-arg 36125	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 36080), 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 36126	Syntax for the multiplication of extended complex numbers.
class ·ℂ̅
Definition	df-bj-mulc 36127	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 36129). 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 36128	Syntax for the inverse of nonzero extended complex numbers.
class -1ℂ̅
Definition	df-bj-invc 36129*	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 36127, 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)))
21.17.6.9  The canonical bijection from the finite ordinals
Syntax	ciomnn 36130	Syntax for the canonical bijection from (ω ∪ {ω}) onto (ℕ0 ∪ {+∞}).
class iω↪ℕ
Definition	df-bj-iomnn 36131*	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 36078 but we did not use the present definition there since we wanted to have defined +∞ first. See bj-iomnnom 36140 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 36132	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 36133	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 36134	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 36135	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 36136	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 36137	The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7181. (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 36138*	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 36139*	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 36140	The canonical bijection from (ω ∪ {ω}) onto (ℕ0 ∪ {+∞}) maps ω to +∞. (Contributed by BJ, 18-Feb-2023.)
⊢ (iω↪ℕ‘ω) = +∞
21.17.6.10  Divisibility
Syntax	cnnbar 36141	Syntax for the extended natural numbers.
class ℕ̅
Definition	df-bj-nnbar 36142	Definition of the extended natural numbers. (Contributed by BJ, 28-Jul-2023.)
⊢ ℕ̅ = (ℕ0 ∪ {+∞})
Syntax	czzbar 36143	Syntax for the extended integers.
class ℤ̅
Definition	df-bj-zzbar 36144	Definition of the extended integers. (Contributed by BJ, 28-Jul-2023.)
⊢ ℤ̅ = (ℤ ∪ {-∞, +∞})
Syntax	czzhat 36145	Syntax for the one-point-compactified integers.
class ℤ̂
Definition	df-bj-zzhat 36146	Definition of the one-point-compactified. (Contributed by BJ, 28-Jul-2023.)
⊢ ℤ̂ = (ℤ ∪ {∞})
Syntax	cdivc 36147	Syntax for the divisibility relation.
class ∥ℂ
Definition	df-bj-divc 36148*	Definition of the divisibility relation (compare df-dvds 16198). 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)}
21.17.7  Monoids See ccmn 19648 and subsequents. The first few statements of this subsection can be put very early after ccmn 19648. Proposal: in the main part, make separate subsections of commutative monoids and abelian groups. Relabel cabl 19649 to "cabl" or, preferably, other labels containing "abl" to "abel", for consistency.
Theorem	bj-smgrpssmgm 36149	Semigroups are magmas. (Contributed by BJ, 12-Apr-2024.) (Proof modification is discouraged.)
⊢ Smgrp ⊆ Mgm
Theorem	bj-smgrpssmgmel 36150	Semigroups are magmas (elemental version). (Contributed by BJ, 12-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
Theorem	bj-mndsssmgrp 36151	Monoids are semigroups. (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.)
⊢ Mnd ⊆ Smgrp
Theorem	bj-mndsssmgrpel 36152	Monoids are semigroups (elemental version). (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)
Theorem	bj-cmnssmnd 36153	Commutative monoids are monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ CMnd ⊆ Mnd
Theorem	bj-cmnssmndel 36154	Commutative monoids are monoids (elemental version). This is a more direct proof of cmnmnd 19665, which relies on iscmn 19657. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ CMnd → 𝐴 ∈ Mnd)
Theorem	bj-grpssmnd 36155	Groups are monoids. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.)
⊢ Grp ⊆ Mnd
Theorem	bj-grpssmndel 36156	Groups are monoids (elemental version). Shorter proof of grpmnd 18826. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ Grp → 𝐴 ∈ Mnd)
Theorem	bj-ablssgrp 36157	Abelian groups are groups. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ Abel ⊆ Grp
Theorem	bj-ablssgrpel 36158	Abelian groups are groups (elemental version). This is a shorter proof of ablgrp 19653. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ Abel → 𝐴 ∈ Grp)
Theorem	bj-ablsscmn 36159	Abelian groups are commutative monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ Abel ⊆ CMnd
Theorem	bj-ablsscmnel 36160	Abelian groups are commutative monoids (elemental version). This is a shorter proof of ablcmn 19655. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ Abel → 𝐴 ∈ CMnd)
Theorem	bj-modssabl 36161	(The additive groups of) modules are abelian groups. (The elemental version is lmodabl 20519; see also lmodgrp 20478 and lmodcmn 20520.) (Contributed by BJ, 9-Jun-2019.)
⊢ LMod ⊆ Abel
Theorem	bj-vecssmod 36162	Vector spaces are modules. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ LVec ⊆ LMod
Theorem	bj-vecssmodel 36163	Vector spaces are modules (elemental version). This is a shorter proof of lveclmod 20717. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ LVec → 𝐴 ∈ LMod)
21.17.7.1  Finite sums in monoids UPDATE: a similar summation is already defined as df-gsum 17388 (although it mixes finite and infinite sums, which makes it harder to understand).
Syntax	cfinsum 36164	Syntax for the class "finite summation in monoids".
class FinSum
Definition	df-bj-finsum 36165*	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 36166*	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)))))
21.17.8  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.
21.17.8.1  Real vector spaces In this section, we introduce real vector spaces.
Theorem	bj-fvimacnv0 36167	Variant of fvimacnv 7055 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 45828. (Contributed by BJ, 7-Jan-2024.)
⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))
Theorem	bj-isvec 36168	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 36169	Fields are division rings. (Contributed by BJ, 6-Jan-2024.)
⊢ Field ⊆ DivRing
Theorem	bj-flddrng 36170	Fields are division rings (elemental version). (Contributed by BJ, 9-Nov-2024.)
⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing)
Theorem	bj-rrdrg 36171	The field of real numbers is a division ring. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝfld ∈ DivRing
Theorem	bj-isclm 36172	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 36173	Syntax for the class of real vector spaces.
class ℝ-Vec
Definition	df-bj-rvec 36174	Definition of the class of real vector spaces. The previous definition, ⊢ ℝ-Vec = {𝑥 ∈ LMod ∣ (Scalar‘𝑥) = ℝfld}, can be recovered using bj-isrvec 36175. 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 36181. (Contributed by BJ, 9-Jun-2019.)
⊢ ℝ-Vec = (LMod ∩ (◡Scalar “ {ℝfld}))
Theorem	bj-isrvec 36175	The predicate "is a real vector space". Using df-sca 17213 instead of scaid 17260 would shorten the proof by two syntactic steps, but it is preferable not to rely on the precise definition df-sca 17213. (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LMod ∧ (Scalar‘𝑉) = ℝfld))
Theorem	bj-rvecmod 36176	Real vector spaces are modules (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ LMod)
Theorem	bj-rvecssmod 36177	Real vector spaces are modules. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ LMod
Theorem	bj-rvecrr 36178	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 36179	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 36180	Real vector spaces are vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ LVec)
Theorem	bj-isrvec2 36181	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 36182	Real vector spaces are vector spaces. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ LVec
Theorem	bj-rveccmod 36183	Real vector spaces are subcomplex modules (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂMod)
Theorem	bj-rvecsscmod 36184	Real vector spaces are subcomplex modules. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ ℂMod
Theorem	bj-rvecsscvec 36185	Real vector spaces are subcomplex vector spaces. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ ℂVec
Theorem	bj-rveccvec 36186	Real vector spaces are subcomplex vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂVec)
Theorem	bj-rvecssabl 36187	(The additive groups of) real vector spaces are commutative groups. (Contributed by BJ, 9-Jun-2019.)
⊢ ℝ-Vec ⊆ Abel
Theorem	bj-rvecabl 36188	(The additive groups of) real vector spaces are commutative groups (elemental version). (Contributed by BJ, 9-Jun-2019.)
⊢ (𝐴 ∈ ℝ-Vec → 𝐴 ∈ Abel)
21.17.8.2  Complex numbers (supplements) Some lemmas to ease algebraic manipulations.
Theorem	bj-subcom 36189	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 36190	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))
21.17.8.3  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 36193 (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 36191	Lemma for bj-bary1 36193: 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 36192	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 36193	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)))))
21.17.9  Monoid of endomorphisms
Syntax	cend 36194	Token for the monoid of endomorphisms.
class End
Definition	df-bj-end 36195*	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 36196	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 36197	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 36198	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 36199	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)
21.18  Mathbox for Jim Kingdon
21.18.0.1  Circle constant
Theorem	taupilem3 36200	Lemma for tau-related theorems. (Contributed by Jim Kingdon, 16-Feb-2019.)
⊢ (𝐴 ∈ (ℝ+ ∩ (◡cos “ {1})) ↔ (𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1))