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Theorem List for Metamath Proof Explorer - 32101-32200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembj-eldiag2 32101 Characterization of the elements 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 32103 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 9636 and df-1r 9637 but intervals are defined for real numers, 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 9633, df-ltr 9635, df-0r 9636, df-1r 9637, df-ltr 9635. The idea is then to define the order relation directly on ℝ̅, skipping .

 
Syntaxcinftyexpi 32102 Syntax for the function inftyexpi parameterizing .
class inftyexpi
 
Definitiondf-bj-inftyexpi 32103 Definition of the auxiliary function inftyexpi parameterizing the circle at infinity in ℂ̅. We use coupling with to simplify the proof of bj-ccinftydisj 32109. 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 32104 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 32105 Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 32104 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 32106 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 32107 Syntax for the circle at infinity .
class
 
Definitiondf-bj-ccinfty 32108 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 32109 The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.)
(ℂ ∩ ℂ) = ∅
 
Theorembj-elccinfty 32110 A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
(𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) ∈ ℂ)
 
Syntaxcccbar 32111 Syntax for the set of extended complex numbers ℂ̅.
class ℂ̅
 
Definitiondf-bj-ccbar 32112 Definition of the set of extended complex numbers ℂ̅. (Contributed by BJ, 22-Jun-2019.)
ℂ̅ = (ℂ ∪ ℂ)
 
Theorembj-ccssccbar 32113 Complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
ℂ ⊆ ℂ̅
 
Theorembj-ccinftyssccbar 32114 Infinite extended complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊆ ℂ̅
 
Syntaxcpinfty 32115 Syntax for +∞.
class +∞
 
Definitiondf-bj-pinfty 32116 Definition of +∞. (Contributed by BJ, 27-Jun-2019.)
+∞ = (inftyexpi ‘0)
 
Theorembj-pinftyccb 32117 The class +∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
+∞ ∈ ℂ̅
 
Theorembj-pinftynrr 32118 The extended complex number +∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
¬ +∞ ∈ ℂ
 
Syntaxcminfty 32119 Syntax for -∞.
class -∞
 
Definitiondf-bj-minfty 32120 Definition of -∞. (Contributed by BJ, 27-Jun-2019.)
-∞ = (inftyexpi ‘π)
 
Theorembj-minftyccb 32121 The class -∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
-∞ ∈ ℂ̅
 
Theorembj-minftynrr 32122 The extended complex number -∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
¬ -∞ ∈ ℂ
 
Theorembj-pinftynminfty 32123 The extended complex numbers +∞ and -∞ are different. (Contributed by BJ, 27-Jun-2019.)
+∞ ≠ -∞
 
Syntaxcrrbar 32124 Syntax for the set of extended real numbers ℝ̅.
class ℝ̅
 
Definitiondf-bj-rrbar 32125 Definition of the set of extended real numbers ℝ̅. See df-xr 9832. (Contributed by BJ, 29-Jun-2019.)
ℝ̅ = (ℝ ∪ {-∞, +∞})
 
Syntaxcinfty 32126 Syntax for .
class
 
Definitiondf-bj-infty 32127 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 32128 Syntax for ℂ̂.
class ℂ̂
 
Definitiondf-bj-cchat 32129 Define the complex projective line, or Riemann sphere. (Contributed by BJ, 27-Jun-2019.)
ℂ̂ = (ℂ ∪ {∞})
 
Syntaxcrrhat 32130 Syntax for ℝ̂.
class ℝ̂
 
Definitiondf-bj-rrhat 32131 Define the real projective line. (Contributed by BJ, 27-Jun-2019.)
ℝ̂ = (ℝ ∪ {∞})
 
Theorembj-rrhatsscchat 32132 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 32133 Syntax for the addition of extended complex numbers.
class +ℂ̅
 
Definitiondf-bj-addc 32134 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 32135 Syntax for the opposite of extended complex numbers.
class -ℂ̅
 
Definitiondf-bj-oppc 32136 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 32137 Syntax for the function prcpal.
class prcpal
 
Definitiondf-bj-prcpal 32138 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 32139 Syntax for the argument of a nonzero extended complex number.
class Arg
 
Definitiondf-bj-arg 32140 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 32141 Syntax for the multiplication of extended complex numbers.
class ·ℂ̅
 
Definitiondf-bj-mulc 32142 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 32143 Syntax for the inverse of nonzero extended complex numbers.
class -1ℂ̅
 
Definitiondf-bj-invc 32144 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 17922 and subsequents. The first few statements of this subsection can be put very early after ccmn 17922. Proposal: in the main part, make separate subsections of commutative monoids and abelian groups.

Relabel cabl 17923 to "cabl" or, preferably, other labels containing "abl" to "abel", for consistency.

 
Theorembj-cmnssmnd 32145 Commutative monoids are monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
CMnd ⊆ Mnd
 
Theorembj-cmnssmndel 32146 Commutative monoids are monoids (elemental version). This is a more direct proof of cmnmnd 17937, which relies on iscmn 17929. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ CMnd → 𝐴 ∈ Mnd)
 
Theorembj-ablssgrp 32147 Abelian groups are groups. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
Abel ⊆ Grp
 
Theorembj-ablssgrpel 32148 Abelian groups are groups (elemental version). This is a shorter proof of ablgrp 17927. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ Abel → 𝐴 ∈ Grp)
 
Theorembj-ablsscmn 32149 Abelian groups are commutative monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
Abel ⊆ CMnd
 
Theorembj-ablsscmnel 32150 Abelian groups are commutative monoids (elemental version). This is a shorter proof of ablcmn 17928. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
(𝐴 ∈ Abel → 𝐴 ∈ CMnd)
 
Theorembj-modssabl 32151 (The additive groups of) modules are abelian groups. (The elemental version is lmodabl 18635; see also lmodgrp 18598 and lmodcmn 18636.) (Contributed by BJ, 9-Jun-2019.)
LMod ⊆ Abel
 
Theorembj-vecssmod 32152 Vector spaces are modules. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
LVec ⊆ LMod
 
Theorembj-vecssmodel 32153 Vector spaces are modules (elemental version). This is a shorter proof of lveclmod 18829. (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 15808 (although it mixes finite and infinite sums, which makes it harder to understand).

 
Syntaxcfinsum 32154 Syntax for the class "finite summation in monoids".
class FinSum
 
Definitiondf-bj-finsum 32155* 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 32156* 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 32157 Syntax for the class of real vector spaces.
class ℝ-Vec
 
Definitiondf-bj-rrvec 32158 Definition of the class of real vector spaces. (Contributed by BJ, 9-Jun-2019.)
ℝ-Vec = {𝑥 ∈ LVec ∣ (Scalar‘𝑥) = ℝfld}
 
Theorembj-rrvecssvec 32159 Real vector spaces are vector spaces. (Contributed by BJ, 9-Jun-2019.)
ℝ-Vec ⊆ LVec
 
Theorembj-rrvecssvecel 32160 Real vector spaces are vector spaces (elemental version). (Contributed by BJ, 9-Jun-2019.)
(𝐴 ∈ ℝ-Vec → 𝐴 ∈ LVec)
 
Theorembj-rrvecsscmn 32161 (The additive groups of) real vector spaces are commutative monoids. (Contributed by BJ, 9-Jun-2019.)
ℝ-Vec ⊆ CMnd
 
Theorembj-rrvecsscmnel 32162 (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 32163 A consequence of commutativity of multiplication. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) − (𝐵 · 𝐴)) = 0)
 
Theorembj-ldiv 32164 Left-division. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 · 𝐵) = 𝐶𝐴 = (𝐶 / 𝐵)))
 
Theorembj-rdiv 32165 Right-division. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → ((𝐴 · 𝐵) = 𝐶𝐵 = (𝐶 / 𝐴)))
 
Theorembj-mdiv 32166 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 32167 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 32168 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 is proved by bj-bary1 32171 (which computes them).

 
Theorembj-bary1lem 32169 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 32170 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 32171 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
 
Syntaxctau 32172 Extend class notation to include tau = 6.283185....
class τ
 
Definitiondf-tau 32173 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 32174 Lemma for tau-related theorems . (Contributed by Jim Kingdon, 16-Feb-2019.)
(𝐴 ∈ (ℝ+ ∩ (cos “ {1})) ↔ (𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1))
 
Theoremtaupilemrplb 32175* 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 32176 Lemma for taupi 32178. 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 32177 Lemma for taupi 32178. 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 32178 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)
 
20.16  Mathbox for ML
 
Theoremcsbdif 32179 Distribution of class substitution over difference of two classes. (Contributed by ML, 14-Jul-2020.)
𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
 
Theoremcsbpredg 32180 Move class substitution in and out of the predecessor class of a relationship. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑋) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑋))
 
Theoremcsbwrecsg 32181 Move class substitution in and out of the well-founded recursive function generator . (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥wrecs(𝑅, 𝐷, 𝐹) = wrecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹))
 
Theoremcsbrecsg 32182 Move class substitution in and out of recs. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥recs(𝐹) = recs(𝐴 / 𝑥𝐹))
 
Theoremcsbrdgg 32183 Move class substitution in and out of the recursive function generator. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥rec(𝐹, 𝐼) = rec(𝐴 / 𝑥𝐹, 𝐴 / 𝑥𝐼))
 
Theoremcsboprabg 32184* Move class substitution in and out of class abstractions of nested ordered pairs. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]𝜑})
 
Theoremcsbmpt22g 32185* Move class substitution in and out of maps-to notation for operations. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥(𝑦𝑌, 𝑧𝑍𝐷) = (𝑦𝐴 / 𝑥𝑌, 𝑧𝐴 / 𝑥𝑍𝐴 / 𝑥𝐷))
 
Theoremmpnanrd 32186 Eliminate the right side of a negated conjunction in an implication. (Contributed by ML, 17-Oct-2020.)
(𝜑𝜓)    &   (𝜑 → ¬ (𝜓𝜒))       (𝜑 → ¬ 𝜒)
 
Theoremcon1bii2 32187 A contraposition inference. (Contributed by ML, 18-Oct-2020.)
𝜑𝜓)       (𝜑 ↔ ¬ 𝜓)
 
Theoremcon2bii2 32188 A contraposition inference. (Contributed by ML, 18-Oct-2020.)
(𝜑 ↔ ¬ 𝜓)       𝜑𝜓)
 
Theoremvtoclefex 32189* Implicit substitution of a class for a setvar variable. (Contributed by ML, 17-Oct-2020.)
𝑥𝜑    &   (𝑥 = 𝐴𝜑)       (𝐴𝑉𝜑)
 
Theoremrnmptsn 32190* The range of a function mapping to singletons. (Contributed by ML, 15-Jul-2020.)
ran (𝑥𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
 
Theoremf1omptsnlem 32191* This is the core of the proof of f1omptsn 32192, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 15-Jul-2020.)
𝐹 = (𝑥𝐴 ↦ {𝑥})    &   𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       𝐹:𝐴1-1-onto𝑅
 
Theoremf1omptsn 32192* A function mapping to singletons is bijective onto a set of singletons. (Contributed by ML, 16-Jul-2020.)
𝐹 = (𝑥𝐴 ↦ {𝑥})    &   𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       𝐹:𝐴1-1-onto𝑅
 
Theoremmptsnunlem 32193* This is the core of the proof of mptsnun 32194, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.)
𝐹 = (𝑥𝐴 ↦ {𝑥})    &   𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       (𝐵𝐴𝐵 = (𝐹𝐵))
 
Theoremmptsnun 32194* 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 = (𝑥𝐴 ↦ {𝑥})    &   𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       (𝐵𝐴𝐵 = (𝐹𝐵))
 
Theoremdissneqlem 32195* This is the core of the proof of dissneq 32196, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.)
𝐶 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
 
Theoremdissneq 32196* Any topology that contains every single-point set is the discrete topology. (Contributed by ML, 16-Jul-2020.)
𝐶 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
 
Theoremexlimim 32197* Closed form of exlimimd 32198. (Contributed by ML, 17-Jul-2020.)
((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → 𝜓)
 
Theoremexlimimd 32198* Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.)
(𝜑 → ∃𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)
 
Theoremexlimimdd 32199 Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → ∃𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)
 
Theoremexellim 32200* Closed form of exellimddv 32201. See also exlimim 32197 for a more general theorem. (Contributed by ML, 17-Jul-2020.)
((∃𝑥 𝑥𝐴 ∧ ∀𝑥(𝑥𝐴𝜑)) → 𝜑)
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