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Theorem List for Metamath Proof Explorer - 33201-33300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-bj-sethom 33201* Define the set of functions (morphisms of sets) between two sets. Same as df-map 7901 with arguments swapped. TODO: prove the same staple lemmas as for 𝑚.

Remark: one may define Set⟶ = (𝑥 ∈ dom Struct , 𝑦 ∈ dom Struct ↦ {𝑓𝑓:(Base‘𝑥)⟶(Base‘𝑦)}) so that for morphisms between other structures, one could write ... = {𝑓 ∈ (𝑥 Set𝑦) ∣ ...}.

(Contributed by BJ, 11-Apr-2020.)

Set⟶ = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑥𝑦})

Syntaxctophom 33202 Syntax for the set of topological morphisms.
class Top

Definitiondf-bj-tophom 33203* Define the set of continuous functions (morphisms of topological spaces) between two topological spaces. Similar to df-cn 21079 (which is in terms of topologies instead of topological spaces). (Contributed by BJ, 10-Feb-2022.)
Top⟶ = (𝑥 ∈ TopSp, 𝑦 ∈ TopSp ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (TopOpen‘𝑦)(𝑓𝑢) ∈ (TopOpen‘𝑥)})

Syntaxcmgmhom 33204 Syntax for the set of magma morphisms.
class Mgm

Definitiondf-bj-mgmhom 33205* Define the set of magma morphisms between two magmas. If domain and codomain are semigroups, monoids, or groups, then one obtains the set of morphisms of these structures. (Contributed by BJ, 10-Feb-2022.)
Mgm⟶ = (𝑥 ∈ Mgm, 𝑦 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (Base‘𝑥)∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g𝑥)𝑣)) = ((𝑓𝑢)(+g𝑦)(𝑓𝑣))})

Syntaxctopmgmhom 33206 Syntax for the set of topological magma morphisms.
class TopMgm

Definitiondf-bj-topmgmhom 33207* Define the set of topological magma morphisms (continuous magma morphisms) between two topological magmas. If domain and codomain are topological semigroups, monoids, or groups, then one obtains the set of morphisms of these structures. This definition is currently stated with topological monoid domain and codomain, since topological magmas are currently not defined in set.mm. (Contributed by BJ, 10-Feb-2022.)
TopMgm⟶ = (𝑥 ∈ TopMnd, 𝑦 ∈ TopMnd ↦ ((𝑥 Top𝑦) ∩ (𝑥 Mgm𝑦)))

Syntaxccur- 33208 Syntax for the parameterized currying function.
class curry_

Definitiondf-bj-cur 33209* Define currying. See also df-cur 7438. (Contributed by BJ, 11-Apr-2020.)
curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set𝑧) ↦ (𝑎𝑥 ↦ (𝑏𝑦 ↦ (𝑓‘⟨𝑎, 𝑏⟩)))))

Syntaxcunc- 33210 Notation for the parameterized uncurrying function.
class uncurry_

Definitiondf-bj-unc 33211* Define uncurrying. See also df-unc 7439. (Contributed by BJ, 11-Apr-2020.)
uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set𝑧)) ↦ (𝑎𝑥, 𝑏𝑦 ↦ ((𝑓𝑎)‘𝑏))))

20.14.5.21  Setting components of extensible structures

Groundwork for changing the definition, syntax and token for component-setting in extensible structures. See https://github.com/metamath/set.mm/issues/2401

Syntaxcstrset 33212 Syntax for component-setting in extensible structures.
class [𝐵 / 𝐴]struct𝑆

Definitiondf-strset 33213 Component-setting in extensible structures. Define the extensible structure [𝐵 / 𝐴]struct𝑆, which is like the extensible structure 𝑆 except that the value 𝐵 has been put in the slot 𝐴 (replacing the current value if there was already one). In such expressions, 𝐴 is generally substituted for slot mnemonics like Base or +g or dist. (Contributed by BJ, 13-Feb-2022.)
[𝐵 / 𝐴]struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {⟨(𝐴‘ndx), 𝐵⟩})

Theoremsetsstrset 33214 Relation between df-sets 15911 and df-strset 33213. Temporary theorem kept during the transition from the former to the latter. (Contributed by BJ, 13-Feb-2022.)
((𝑆𝑉𝐵𝑊) → [𝐵 / 𝐴]struct𝑆 = (𝑆 sSet ⟨(𝐴‘ndx), 𝐵⟩))

20.14.6  Extended real and complex numbers, real and complex projective lines

In this section, we indroduce several supersets of the set of real numbers and the set of complex numbers.

Once they are given their usual topologies, which are locally compact, both topological spaces have a one-point compactification. They are denoted by ℝ̂ and ℂ̂ respectively, defined in df-bj-cchat 33250 and df-bj-rrhat 33252, and the point at infinity is denoted by , defined in df-bj-infty 33248.

Both and also have "directional compactifications", denoted respectively by ℝ̅, defined in df-bj-rrbar 33246 (already defined as *, see df-xr 10116) and ℂ̅, defined in df-bj-ccbar 33233.

Since ℂ̅ does not seem to be standard, we describe it in some detail. It is obtained by adding to a "point at infinity at the end of each ray with origin at 0". Although ℂ̅ is not an important object in itself, the motivation for introducing it is to provide a common superset to both ℝ̅ and and to define algebraic operations (addition, opposite, multiplication, inverse) as widely as reasonably possible.

Mathematically, ℂ̅ is the quotient of ((ℂ × ℝ≥0) ∖ {⟨0, 0⟩}) by the diagonal multiplicative action of >0 (think of the closed "northern hemisphere" in ^3 identified with (ℂ × ℝ), that each open ray from 0 included in the closed northern half-space intersects exactly once).

Since in set.mm, we want to have a genuine inclusion ℂ ⊆ ℂ̅, we instead define ℂ̅ as the (disjoint) union of with a circle at infinity denoted by . To have a genuine inclusion ℝ̅ ⊆ ℂ̅, we define +∞ and -∞ as certain points in .

Thanks to this framework, one has the genuine inclusions ℝ ⊆ ℝ̅ and ℝ ⊆ ℝ̂ and similarly ℂ ⊆ ℂ̅ and ℂ ⊆ ℂ̂. Furthermore, one has ℝ ⊆ ℂ as well as ℝ̅ ⊆ ℂ̅ and ℝ̂ ⊆ ℂ̂.

Furthermore, we define the main algebraic operations on (ℂ̅ ∪ ℂ̂), which is not very mathematical, but "overloads" the operations, so that one can use the same notation in all cases.

20.14.6.1  Diagonal in a Cartesian square

Complements on the idendity relation and definition of the diagonal in the Cartesian square of a set.

Theorembj-elid 33215 Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
(𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))

Theorembj-elid2 33216 Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
(𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))

Theorembj-elid3 33217 Characterization of the elements of I. (Contributed by BJ, 29-Mar-2020.)
(⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntaxcdiag2 33218 Syntax for the diagonal of the Cartesian square of a set.
class Diag

Definitiondf-bj-diag 33219 Define the diagonal of the Cartesian square of a set. (Contributed by BJ, 22-Jun-2019.)
Diag = (𝑥 ∈ V ↦ ( I ∩ (𝑥 × 𝑥)))

Theorembj-diagval 33220 Value of the diagonal. (Contributed by BJ, 22-Jun-2019.)
(𝐴𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))

Theorembj-eldiag 33221 Characterization of the elements of the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
(𝐴𝑉 → (𝐵 ∈ (Diag‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵))))

Theorembj-eldiag2 33222 Characterization of the elements of the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
(𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Diag‘𝐴) ↔ (𝐵𝐴𝐵 = 𝐶)))

20.14.6.2  Extended numbers and projective lines as sets

TODO(?): replace df-bj-inftyexpi 33224 with a function inftyexpi2pi defined on (0[,)1) since we plan to put this section as early as possible, before the definition of π. It would be best to use df-0r 9920 and df-1r 9921 but intervals are defined for real numbers, and not these temporary reals.

It looks like to define the sets, the addition and the opposite, one only needs some basic results about addition, opposite and ordering, which could use df-plr 9917, df-ltr 9919, df-0r 9920, df-1r 9921, df-ltr 9919. The idea is then to define the order relation directly on ℝ̅, skipping .

Syntaxcinftyexpi 33223 Syntax for the function inftyexpi parameterizing .
class inftyexpi

Definitiondf-bj-inftyexpi 33224 Definition of the auxiliary function inftyexpi parameterizing the circle at infinity in ℂ̅. We use coupling with to simplify the proof of bj-ccinftydisj 33230. It could seem more natural to define inftyexpi on all of using prcpal but we want to use only basic functions in the definition of ℂ̅. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)

Theorembj-inftyexpiinv 33225 Utility theorem for the inverse of inftyexpi. (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 ‘(inftyexpi ‘𝐴)) = 𝐴)

Theorembj-inftyexpiinj 33226 Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 33225 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 ↔ (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵)))

Theorembj-inftyexpidisj 33227 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.)
¬ (inftyexpi ‘𝐴) ∈ ℂ

Syntaxcccinfty 33228 Syntax for the circle at infinity .
class

Definitiondf-bj-ccinfty 33229 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 inftyexpi

Theorembj-ccinftydisj 33230 The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.)
(ℂ ∩ ℂ) = ∅

Theorembj-elccinfty 33231 A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
(𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) ∈ ℂ)

Syntaxcccbar 33232 Syntax for the set of extended complex numbers ℂ̅.
class ℂ̅

Definitiondf-bj-ccbar 33233 Definition of the set of extended complex numbers ℂ̅. (Contributed by BJ, 22-Jun-2019.)
ℂ̅ = (ℂ ∪ ℂ)

Theorembj-ccssccbar 33234 Complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
ℂ ⊆ ℂ̅

Theorembj-ccinftyssccbar 33235 Infinite extended complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊆ ℂ̅

Syntaxcpinfty 33236 Syntax for +∞.
class +∞

Definitiondf-bj-pinfty 33237 Definition of +∞. (Contributed by BJ, 27-Jun-2019.)
+∞ = (inftyexpi ‘0)

Theorembj-pinftyccb 33238 The class +∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
+∞ ∈ ℂ̅

Theorembj-pinftynrr 33239 The extended complex number +∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
¬ +∞ ∈ ℂ

Syntaxcminfty 33240 Syntax for -∞.
class -∞

Definitiondf-bj-minfty 33241 Definition of -∞. (Contributed by BJ, 27-Jun-2019.)
-∞ = (inftyexpi ‘π)

Theorembj-minftyccb 33242 The class -∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
-∞ ∈ ℂ̅

Theorembj-minftynrr 33243 The extended complex number -∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
¬ -∞ ∈ ℂ

Theorembj-pinftynminfty 33244 The extended complex numbers +∞ and -∞ are different. (Contributed by BJ, 27-Jun-2019.)
+∞ ≠ -∞

Syntaxcrrbar 33245 Syntax for the set of extended real numbers ℝ̅.
class ℝ̅

Definitiondf-bj-rrbar 33246 Definition of the set of extended real numbers ℝ̅. See df-xr 10116. (Contributed by BJ, 29-Jun-2019.)
ℝ̅ = (ℝ ∪ {-∞, +∞})

Syntaxcinfty 33247 Syntax for .
class

Definitiondf-bj-infty 33248 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 33249 Syntax for ℂ̂.
class ℂ̂

Definitiondf-bj-cchat 33250 Define the complex projective line, or Riemann sphere. (Contributed by BJ, 27-Jun-2019.)
ℂ̂ = (ℂ ∪ {∞})

Syntaxcrrhat 33251 Syntax for ℝ̂.
class ℝ̂

Definitiondf-bj-rrhat 33252 Define the real projective line. (Contributed by BJ, 27-Jun-2019.)
ℝ̂ = (ℝ ∪ {∞})

Theorembj-rrhatsscchat 33253 The real projective line is included in the complex projective line. (Contributed by BJ, 27-Jun-2019.)
ℝ̂ ⊆ ℂ̂

20.14.6.3  Addition and opposite

We define the operations on the extended real and complex numbers and on the real and complex projective lines simultaneously, thus "overloading" the operations.

Syntaxcaddcc 33254 Syntax for the addition of extended complex numbers.
class +ℂ̅

Definitiondf-bj-addc 33255 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). (Contributed by BJ, 22-Jun-2019.)
+ℂ̅ = (𝑥 ∈ (((ℂ × ℂ̅) ∪ (ℂ̅ × ℂ)) ∪ ((ℂ̂ × ℂ̂) ∪ (Diag‘ℂ))) ↦ if(((1st𝑥) = ∞ ∨ (2nd𝑥) = ∞), ∞, if((1st𝑥) ∈ ℂ, if((2nd𝑥) ∈ ℂ, ((1st𝑥) + (2nd𝑥)), (2nd𝑥)), (1st𝑥))))

Syntaxcoppcc 33256 Syntax for the opposite of extended complex numbers.
class -ℂ̅

Definitiondf-bj-oppc 33257 Define the negation (operation givin the opposite) the set of extended complex numbers and the complex projective line (Riemann sphere). One could use the prcpal function in the infinite case, but we want to use only basic functions at this point. (Contributed by BJ, 22-Jun-2019.)
-ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = ∞, ∞, if(𝑥 ∈ ℂ, -𝑥, (inftyexpi ‘if(0 < (1st𝑥), ((1st𝑥) − π), ((1st𝑥) + π))))))

20.14.6.4  Argument, multiplication and inverse

Since one needs arguments in order to define multiplication in ℂ̅, it seems harder to put this at the very beginning of the introduction of complex numbers.

Syntaxcprcpal 33258 Syntax for the function prcpal.
class prcpal

Definitiondf-bj-prcpal 33259 Define the function prcpal. (Contributed by BJ, 22-Jun-2019.)
prcpal = (𝑥 ∈ ℝ ↦ ((𝑥 mod (2 · π)) − if((𝑥 mod (2 · π)) ≤ π, 0, (2 · π))))

Syntaxcarg 33260 Syntax for the argument of a nonzero extended complex number.
class Arg

Definitiondf-bj-arg 33261 Define the argument of a nonzero extended complex number. By convention, it has values in (-π, π]. Another convention chooses [0, 2π) but the present one simplifies formulas giving the argument as an arctangent. (Contributed by BJ, 22-Jun-2019.)
Arg = (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), (1st𝑥)))

Syntaxcmulc 33262 Syntax for the multiplication of extended complex numbers.
class ·ℂ̅

Definitiondf-bj-mulc 33263 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. (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)), (inftyexpi ‘(prcpal‘((Arg‘(1st𝑥)) + (Arg‘(2nd𝑥)))))))))

Syntaxcinvc 33264 Syntax for the inverse of nonzero extended complex numbers.
class -1ℂ̅

Definitiondf-bj-invc 33265 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. (Contributed by BJ, 22-Jun-2019.)
-1ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = 0, ∞, if(𝑥 ∈ ℂ, (1 / 𝑥), 0)))

20.14.7  Monoids

See ccmn 18239 and subsequents. The first few statements of this subsection can be put very early after ccmn 18239. Proposal: in the main part, make separate subsections of commutative monoids and abelian groups.

Relabel cabl 18240 to "cabl" or, preferably, other labels containing "abl" to "abel", for consistency.

Theorembj-cmnssmnd 33266 Commutative monoids are monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
CMnd ⊆ Mnd

Theorembj-cmnssmndel 33267 Commutative monoids are monoids (elemental version). This is a more direct proof of cmnmnd 18254, which relies on iscmn 18246. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ CMnd → 𝐴 ∈ Mnd)

Theorembj-ablssgrp 33268 Abelian groups are groups. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
Abel ⊆ Grp

Theorembj-ablssgrpel 33269 Abelian groups are groups (elemental version). This is a shorter proof of ablgrp 18244. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ Abel → 𝐴 ∈ Grp)

Theorembj-ablsscmn 33270 Abelian groups are commutative monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
Abel ⊆ CMnd

Theorembj-ablsscmnel 33271 Abelian groups are commutative monoids (elemental version). This is a shorter proof of ablcmn 18245. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ Abel → 𝐴 ∈ CMnd)

Theorembj-modssabl 33272 (The additive groups of) modules are abelian groups. (The elemental version is lmodabl 18958; see also lmodgrp 18918 and lmodcmn 18959.) (Contributed by BJ, 9-Jun-2019.)
LMod ⊆ Abel

Theorembj-vecssmod 33273 Vector spaces are modules. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
LVec ⊆ LMod

Theorembj-vecssmodel 33274 Vector spaces are modules (elemental version). This is a shorter proof of lveclmod 19154. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ LVec → 𝐴 ∈ LMod)

20.14.7.1  Finite sums in monoids

UPDATE: a similar summation is already defined as df-gsum 16150 (although it mixes finite and infinite sums, which makes it harder to understand).

Syntaxcfinsum 33275 Syntax for the class "finite summation in monoids".
class FinSum

Definitiondf-bj-finsum 33276* 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))))

Theorembj-finsumval0 33277* Value of a finite sum. (Contributed by BJ, 9-Jun-2019.) (Proof shortened by AV, 5-May-2021.)
(𝜑𝐴 ∈ CMnd)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐵:𝐼⟶(Base‘𝐴))       (𝜑 → (𝐴 FinSum 𝐵) = (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(#‘𝐼)))))

20.14.8  Affine, Euclidean, and Cartesian geometry

A few basic theorems to start affine, Euclidean, and Cartesian geometry.

20.14.8.1  Convex hull in real vector spaces

A few basic definitions and theorems about convex hulls in real vector spaces. TODO: prove inclusion in the class of subcomplex vector spaces.

Syntaxcrrvec 33278 Syntax for the class of real vector spaces.
class ℝ-Vec

Definitiondf-bj-rrvec 33279 Definition of the class of real vector spaces. (Contributed by BJ, 9-Jun-2019.)
ℝ-Vec = {𝑥 ∈ LVec ∣ (Scalar‘𝑥) = ℝfld}

Theorembj-rrvecssvec 33280 Real vector spaces are vector spaces. (Contributed by BJ, 9-Jun-2019.)
ℝ-Vec ⊆ LVec

Theorembj-rrvecssvecel 33281 Real vector spaces are vector spaces (elemental version). (Contributed by BJ, 9-Jun-2019.)
(𝐴 ∈ ℝ-Vec → 𝐴 ∈ LVec)

Theorembj-rrvecsscmn 33282 (The additive groups of) real vector spaces are commutative monoids. (Contributed by BJ, 9-Jun-2019.)
ℝ-Vec ⊆ CMnd

Theorembj-rrvecsscmnel 33283 (The additive groups of) real vector spaces are commutative monoids (elemental version). (Contributed by BJ, 9-Jun-2019.)
(𝐴 ∈ ℝ-Vec → 𝐴 ∈ CMnd)

20.14.8.2  Complex numbers (supplements)

Some lemmas to ease algebraic manipulations.

Theorembj-subcom 33284 A consequence of commutativity of multiplication. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) − (𝐵 · 𝐴)) = 0)

Theorembj-ldiv 33285 Left-division. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 · 𝐵) = 𝐶𝐴 = (𝐶 / 𝐵)))

Theorembj-rdiv 33286 Right-division. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → ((𝐴 · 𝐵) = 𝐶𝐵 = (𝐶 / 𝐴)))

Theorembj-mdiv 33287 A division law. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 = (𝐶 / 𝐵) ↔ 𝐵 = (𝐶 / 𝐴)))

Theorembj-lineq 33288 Solution of a (scalar) linear equation. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑌 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (((𝐴 · 𝑋) + 𝐵) = 𝑌𝑋 = ((𝑌𝐵) / 𝐴)))

Theorembj-lineqi 33289 Solution of a (scalar) linear equation. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑌 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑 → ((𝐴 · 𝑋) + 𝐵) = 𝑌)       (𝜑𝑋 = ((𝑌𝐵) / 𝐴))

20.14.8.3  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 33292 (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.

Theorembj-bary1lem 33290 A lemma for barycentric coordinates in one dimension. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑𝑋 = ((((𝐵𝑋) / (𝐵𝐴)) · 𝐴) + (((𝑋𝐴) / (𝐵𝐴)) · 𝐵)))

Theorembj-bary1lem1 33291 Existence and uniqueness (and actual computation) of barycentric coordinates in dimension 1 (complex line). (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑆 ∈ ℂ)    &   (𝜑𝑇 ∈ ℂ)       (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) → 𝑇 = ((𝑋𝐴) / (𝐵𝐴))))

Theorembj-bary1 33292 Barycentric coordinates in one dimension. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑆 ∈ ℂ)    &   (𝜑𝑇 ∈ ℂ)       (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) ↔ (𝑆 = ((𝐵𝑋) / (𝐵𝐴)) ∧ 𝑇 = ((𝑋𝐴) / (𝐵𝐴)))))

20.15  Mathbox for Jim Kingdon

20.15.0.1  Circle constant

Syntaxctau 33293 Extend class notation to include tau = 6.283185....
class τ

Definitiondf-tau 33294 Define tau = 6.283185..., which is the smallest positive real number whose cosine is one. Various notations have been used or proposed for this number including τ, a three-legged variant of π, or . Note the difference between this constant τ and the variable 𝜏 which is a variable representing a propositional logic formula. Only the latter is italic, and the colors are different. (Contributed by Jim Kingdon, 9-Apr-2018.) (Revised by AV, 1-Oct-2020.)
τ = inf((ℝ+ ∩ (cos “ {1})), ℝ, < )

Theoremtaupilem3 33295 Lemma for tau-related theorems . (Contributed by Jim Kingdon, 16-Feb-2019.)
(𝐴 ∈ (ℝ+ ∩ (cos “ {1})) ↔ (𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1))

Theoremtaupilemrplb 33296* A set of positive reals has (in the reals) a lower bound. (Contributed by Jim Kingdon, 19-Feb-2019.)
𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ+𝐴)𝑥𝑦

Theoremtaupilem1 33297 Lemma for taupi 33299. 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)

Theoremtaupilem2 33298 Lemma for taupi 33299. 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)

Theoremtaupi 33299 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)

20.15.0.2  Number theory

Theoremdfgcd3 33300* Alternate definition of the gcd operator. (Contributed by Jim Kingdon, 31-Dec-2021.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (𝑑 ∈ ℕ0𝑧 ∈ ℤ (𝑧𝑑 ↔ (𝑧𝑀𝑧𝑁))))

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