P. 374 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37301-37400   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-bj-mulc 37301	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 37303). 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 37302	Syntax for the inverse of nonzero extended complex numbers.
class -1ℂ̅
Definition	df-bj-invc 37303*	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 37301, 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)))
21.19.6.9  The canonical bijection from the finite ordinals
Syntax	ciomnn 37304	Syntax for the canonical bijection from (ω ∪ {ω}) onto (ℕ0 ∪ {+∞}).
class iω↪ℕ
Definition	df-bj-iomnn 37305*	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 37252 but we did not use the present definition there since we wanted to have defined +∞ first. See bj-iomnnom 37314 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 37306	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 37307	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 37308	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 37309	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 37310	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 37311	The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7126. (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 37312*	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 37313*	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 37314	The canonical bijection from (ω ∪ {ω}) onto (ℕ0 ∪ {+∞}) maps ω to +∞. (Contributed by BJ, 18-Feb-2023.)
⊢ (iω↪ℕ‘ω) = +∞
21.19.6.10  Divisibility
Syntax	cnnbar 37315	Syntax for the extended natural numbers.
class ℕ̅
Definition	df-bj-nnbar 37316	Definition of the extended natural numbers. (Contributed by BJ, 28-Jul-2023.)
⊢ ℕ̅ = (ℕ0 ∪ {+∞})
Syntax	czzbar 37317	Syntax for the extended integers.
class ℤ̅
Definition	df-bj-zzbar 37318	Definition of the extended integers. (Contributed by BJ, 28-Jul-2023.)
⊢ ℤ̅ = (ℤ ∪ {-∞, +∞})
Syntax	czzhat 37319	Syntax for the one-point-compactified integers.
class ℤ̂
Definition	df-bj-zzhat 37320	Definition of the one-point-compactified. (Contributed by BJ, 28-Jul-2023.)
⊢ ℤ̂ = (ℤ ∪ {∞})
Syntax	cdivc 37321	Syntax for the divisibility relation.
class ∥ℂ
Definition	df-bj-divc 37322*	Definition of the divisibility relation (compare df-dvds 16174). 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)}
21.19.7  Monoids See ccmn 19702 and subsequents. The first few statements of this subsection can be put very early after ccmn 19702. Proposal: in the main part, make separate subsections of commutative monoids and abelian groups. Relabel cabl 19703 to "cabl" or, preferably, other labels containing "abl" to "abel", for consistency.
Theorem	bj-smgrpssmgm 37323	Semigroups are magmas. (Contributed by BJ, 12-Apr-2024.) (Proof modification is discouraged.)
⊢ Smgrp ⊆ Mgm
Theorem	bj-smgrpssmgmel 37324	Semigroups are magmas (elemental version). (Contributed by BJ, 12-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
Theorem	bj-mndsssmgrp 37325	Monoids are semigroups. (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.)
⊢ Mnd ⊆ Smgrp
Theorem	bj-mndsssmgrpel 37326	Monoids are semigroups (elemental version). (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)
Theorem	bj-cmnssmnd 37327	Commutative monoids are monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ CMnd ⊆ Mnd
Theorem	bj-cmnssmndel 37328	Commutative monoids are monoids (elemental version). This is a more direct proof of cmnmnd 19719, which relies on iscmn 19711. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ CMnd → 𝐴 ∈ Mnd)
Theorem	bj-grpssmnd 37329	Groups are monoids. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.)
⊢ Grp ⊆ Mnd
Theorem	bj-grpssmndel 37330	Groups are monoids (elemental version). Shorter proof of grpmnd 18863. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ Grp → 𝐴 ∈ Mnd)
Theorem	bj-ablssgrp 37331	Abelian groups are groups. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ Abel ⊆ Grp
Theorem	bj-ablssgrpel 37332	Abelian groups are groups (elemental version). This is a shorter proof of ablgrp 19707. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ Abel → 𝐴 ∈ Grp)
Theorem	bj-ablsscmn 37333	Abelian groups are commutative monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ Abel ⊆ CMnd
Theorem	bj-ablsscmnel 37334	Abelian groups are commutative monoids (elemental version). This is a shorter proof of ablcmn 19709. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ Abel → 𝐴 ∈ CMnd)
Theorem	bj-modssabl 37335	(The additive groups of) modules are abelian groups. (The elemental version is lmodabl 20852; see also lmodgrp 20810 and lmodcmn 20853.) (Contributed by BJ, 9-Jun-2019.)
⊢ LMod ⊆ Abel
Theorem	bj-vecssmod 37336	Vector spaces are modules. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ LVec ⊆ LMod
Theorem	bj-vecssmodel 37337	Vector spaces are modules (elemental version). This is a shorter proof of lveclmod 21050. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ LVec → 𝐴 ∈ LMod)
21.19.7.1  Finite sums in monoids UPDATE: a similar summation is already defined as df-gsum 17356 (although it mixes finite and infinite sums, which makes it harder to understand).
Syntax	cfinsum 37338	Syntax for the class "finite summation in monoids".
class FinSum
Definition	df-bj-finsum 37339*	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 37340*	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)))))
21.19.8  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.
21.19.8.1  Real vector spaces In this section, we introduce real vector spaces.
Theorem	bj-fvimacnv0 37341	Variant of fvimacnv 6995 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 47234. (Contributed by BJ, 7-Jan-2024.)
⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))
Theorem	bj-isvec 37342	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 37343	Fields are division rings. (Contributed by BJ, 6-Jan-2024.)
⊢ Field ⊆ DivRing
Theorem	bj-flddrng 37344	Fields are division rings (elemental version). (Contributed by BJ, 9-Nov-2024.)
⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing)
Theorem	bj-rrdrg 37345	The field of real numbers is a division ring. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝfld ∈ DivRing
Theorem	bj-isclm 37346	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 37347	Syntax for the class of real vector spaces.
class ℝ-Vec
Definition	df-bj-rvec 37348	Definition of the class of real vector spaces. The previous definition, ⊢ ℝ-Vec = {𝑥 ∈ LMod ∣ (Scalar‘𝑥) = ℝfld}, can be recovered using bj-isrvec 37349. 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 37355. (Contributed by BJ, 9-Jun-2019.)
⊢ ℝ-Vec = (LMod ∩ (◡Scalar “ {ℝfld}))
Theorem	bj-isrvec 37349	The predicate "is a real vector space". Using df-sca 17187 instead of scaid 17229 would shorten the proof by two syntactic steps, but it is preferable not to rely on the precise definition df-sca 17187. (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LMod ∧ (Scalar‘𝑉) = ℝfld))
Theorem	bj-rvecmod 37350	Real vector spaces are modules (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ LMod)
Theorem	bj-rvecssmod 37351	Real vector spaces are modules. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ LMod
Theorem	bj-rvecrr 37352	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 37353	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 37354	Real vector spaces are vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ LVec)
Theorem	bj-isrvec2 37355	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 37356	Real vector spaces are vector spaces. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ LVec
Theorem	bj-rveccmod 37357	Real vector spaces are subcomplex modules (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂMod)
Theorem	bj-rvecsscmod 37358	Real vector spaces are subcomplex modules. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ ℂMod
Theorem	bj-rvecsscvec 37359	Real vector spaces are subcomplex vector spaces. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ ℂVec
Theorem	bj-rveccvec 37360	Real vector spaces are subcomplex vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂVec)
Theorem	bj-rvecssabl 37361	(The additive groups of) real vector spaces are commutative groups. (Contributed by BJ, 9-Jun-2019.)
⊢ ℝ-Vec ⊆ Abel
Theorem	bj-rvecabl 37362	(The additive groups of) real vector spaces are commutative groups (elemental version). (Contributed by BJ, 9-Jun-2019.)
⊢ (𝐴 ∈ ℝ-Vec → 𝐴 ∈ Abel)
21.19.8.2  Complex numbers (supplements) Some lemmas to ease algebraic manipulations.
Theorem	bj-subcom 37363	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 37364	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))
21.19.8.3  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 37367 (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 37365	Lemma for bj-bary1 37367: 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 37366	Lemma for bj-bary1 37367: 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 37367	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)))))
21.19.9  Monoid of endomorphisms
Syntax	cend 37368	Token for the monoid of endomorphisms.
class End
Definition	df-bj-end 37369*	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 37370	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 37371	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 37372	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 37373	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)
21.20  Mathbox for Jim Kingdon
21.20.1  Circle constant
Theorem	taupilem3 37374	Lemma for tau-related theorems. (Contributed by Jim Kingdon, 16-Feb-2019.)
⊢ (𝐴 ∈ (ℝ+ ∩ (◡cos “ {1})) ↔ (𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1))
Theorem	taupilemrplb 37375*	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
Theorem	taupilem1 37376	Lemma for taupi 37378. A positive real whose cosine is one is at least 2 · π. (Contributed by Jim Kingdon, 19-Feb-2019.)
⊢ ((𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1) → (2 · π) ≤ 𝐴)
Theorem	taupilem2 37377	Lemma for taupi 37378. The smallest positive real whose cosine is one is at most 2 · π. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.)
⊢ τ ≤ (2 · π)
Theorem	taupi 37378	Relationship between τ and π. This can be seen as connecting the ratio of a circle's circumference to its radius and the ratio of a circle's circumference to its diameter. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.)
⊢ τ = (2 · π)
21.20.2  Number theory
Theorem	dfgcd3 37379*	Alternate definition of the gcd operator. (Contributed by Jim Kingdon, 31-Dec-2021.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (℩𝑑 ∈ ℕ0 ∀𝑧 ∈ ℤ (𝑧 ∥ 𝑑 ↔ (𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁))))
21.20.3  Real numbers
Theorem	irrdifflemf 37380	Lemma for irrdiff 37381. The forward direction. (Contributed by Jim Kingdon, 20-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ ℚ)    &   ⊢ (𝜑 → 𝑄 ∈ ℚ)    &   ⊢ (𝜑 → 𝑅 ∈ ℚ)    &   ⊢ (𝜑 → 𝑄 ≠ 𝑅)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) ≠ (abs‘(𝐴 − 𝑅)))
Theorem	irrdiff 37381*	The irrationals are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 19-May-2024.)
⊢ (𝐴 ∈ ℝ → (¬ 𝐴 ∈ ℚ ↔ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))))
Theorem	iccioo01 37382	The closed unit interval is equinumerous to the open unit interval. Based on a Mastodon post by Michael Kinyon. (Contributed by Jim Kingdon, 4-Jun-2024.)
⊢ (0[,]1) ≈ (0(,)1)
21.21  Mathbox for ML
21.21.1  Miscellaneous
Theorem	csbrecsg 37383	Move class substitution in and out of recs. (Contributed by ML, 25-Oct-2020.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌recs(𝐹) = recs(⦋𝐴 / 𝑥⦌𝐹))
Theorem	csbrdgg 37384	Move class substitution in and out of the recursive function generator. (Contributed by ML, 25-Oct-2020.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌rec(𝐹, 𝐼) = rec(⦋𝐴 / 𝑥⦌𝐹, ⦋𝐴 / 𝑥⦌𝐼))
Theorem	csboprabg 37385*	Move class substitution in and out of class abstractions of nested ordered pairs. (Contributed by ML, 25-Oct-2020.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]𝜑})
Theorem	csbmpo123 37386*	Move class substitution in and out of maps-to notation for operations. (Contributed by ML, 25-Oct-2020.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐷) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌, 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍 ↦ ⦋𝐴 / 𝑥⦌𝐷))
Theorem	con1bii2 37387	A contraposition inference. (Contributed by ML, 18-Oct-2020.)
⊢ (¬ 𝜑 ↔ 𝜓)    ⇒   ⊢ (𝜑 ↔ ¬ 𝜓)
Theorem	con2bii2 37388	A contraposition inference. (Contributed by ML, 18-Oct-2020.)
⊢ (𝜑 ↔ ¬ 𝜓)    ⇒   ⊢ (¬ 𝜑 ↔ 𝜓)
Theorem	vtoclefex 37389*	Implicit substitution of a class for a setvar variable. (Contributed by ML, 17-Oct-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑥 = 𝐴 → 𝜑)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜑)
Theorem	rnmptsn 37390*	The range of a function mapping to singletons. (Contributed by ML, 15-Jul-2020.)
⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
Theorem	f1omptsnlem 37391*	This is the core of the proof of f1omptsn 37392, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 15-Jul-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})    &   ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}    ⇒   ⊢ 𝐹:𝐴–1-1-onto→𝑅
Theorem	f1omptsn 37392*	A function mapping to singletons is bijective onto a set of singletons. (Contributed by ML, 16-Jul-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})    &   ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}    ⇒   ⊢ 𝐹:𝐴–1-1-onto→𝑅
Theorem	mptsnunlem 37393*	This is the core of the proof of mptsnun 37394, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})    &   ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}    ⇒   ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵))
Theorem	mptsnun 37394*	A class 𝐵 is equal to the union of the class of all singletons of elements of 𝐵. (Contributed by ML, 16-Jul-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})    &   ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}    ⇒   ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵))
Theorem	dissneqlem 37395*	This is the core of the proof of dissneq 37396, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.)
⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}    ⇒   ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
Theorem	dissneq 37396*	Any topology that contains every single-point set is the discrete topology. (Contributed by ML, 16-Jul-2020.)
⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}    ⇒   ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
Theorem	exlimim 37397*	Closed form of exlimimd 37398. (Contributed by ML, 17-Jul-2020.)
⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → 𝜓)
Theorem	exlimimd 37398*	Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.)
⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	exellim 37399*	Closed form of exellimddv 37400. See also exlimim 37397 for a more general theorem. (Contributed by ML, 17-Jul-2020.)
⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑)
Theorem	exellimddv 37400*	Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim 37399 for the closed form, which requires the use of a universal quantifier. (Contributed by ML, 17-Jul-2020.)
⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))    ⇒   ⊢ (𝜑 → 𝜓)