P. 412 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 41101-41200   *Has distinct variable group(s)

Type	Label	Description
Statement
20.31.1.14  RP ADDTO: The reflexive and transitive properties of relations
Theorem	undmrnresiss 41101*	Two ways of saying the identity relation restricted to the union of the domain and range of a relation is a subset of a relation. Generalization of reflexg 41102. (Contributed by RP, 26-Sep-2020.)
⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥 ∧ 𝑦𝐵𝑦)))
Theorem	reflexg 41102*	Two ways of saying a relation is reflexive over its domain and range. (Contributed by RP, 4-Aug-2020.)
⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)))
Theorem	cnvssco 41103*	A condition weaker than reflexivity. (Contributed by RP, 3-Aug-2020.)
⊢ (◡𝐴 ⊆ ◡(𝐵 ∘ 𝐶) ↔ ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)))
Theorem	refimssco 41104	Reflexive relations are subsets of their self-composition. (Contributed by RP, 4-Aug-2020.)
⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 → ◡𝐴 ⊆ ◡(𝐴 ∘ 𝐴))
20.31.1.15  RP ADDTO: Basic properties of closures
Theorem	cleq2lem 41105	Equality implies bijection. (Contributed by RP, 24-Jul-2020.)
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ 𝜑) ↔ (𝑅 ⊆ 𝐵 ∧ 𝜓)))
Theorem	cbvcllem 41106*	Change of bound variable in class of supersets of a with a property. (Contributed by RP, 24-Jul-2020.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜑)} = {𝑦 ∣ (𝑋 ⊆ 𝑦 ∧ 𝜓)}
Theorem	clublem 41107*	If a superset 𝑌 of 𝑋 possesses the property parameterized in 𝑥 in 𝜓, then 𝑌 is a superset of the closure of that property for the set 𝑋. (Contributed by RP, 23-Jul-2020.)
⊢ (𝜑 → 𝑌 ∈ V)    &   ⊢ (𝑥 = 𝑌 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑌)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ 𝑌)
Theorem	clss2lem 41108*	The closure of a property is a superset of the closure of a less restrictive property. (Contributed by RP, 24-Jul-2020.)
⊢ (𝜑 → (𝜒 → 𝜓))    ⇒   ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)})
Theorem	dfid7 41109*	Definition of identity relation as the trivial closure. (Contributed by RP, 26-Jul-2020.)
⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)})
Theorem	mptrcllem 41110*	Show two versions of a closure with reflexive properties are equal. (Contributed by RP, 19-Oct-2020.)
⊢ (𝑥 ∈ 𝑉 → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∈ V)    &   ⊢ (𝑥 ∈ 𝑉 → ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} ∈ V)    &   ⊢ (𝑥 ∈ 𝑉 → 𝜒)    &   ⊢ (𝑥 ∈ 𝑉 → 𝜃)    &   ⊢ (𝑥 ∈ 𝑉 → 𝜏)    &   ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ 𝜃))    &   ⊢ (𝑧 = ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → (𝜓 ↔ 𝜏))    ⇒   ⊢ (𝑥 ∈ 𝑉 ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) = (𝑥 ∈ 𝑉 ↦ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)})
Theorem	cotrintab 41111	The intersection of a class is a transitive relation if membership in the class implies the member is a transitive relation. (Contributed by RP, 28-Oct-2020.)
⊢ (𝜑 → (𝑥 ∘ 𝑥) ⊆ 𝑥)    ⇒   ⊢ (∩ {𝑥 ∣ 𝜑} ∘ ∩ {𝑥 ∣ 𝜑}) ⊆ ∩ {𝑥 ∣ 𝜑}
Theorem	rclexi 41112*	The reflexive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
⊢ 𝐴 ∈ 𝑉    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V
Theorem	rtrclexlem 41113	Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 1-Nov-2020.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V)
Theorem	rtrclex 41114*	The reflexive-transitive closure of a set exists. (Contributed by RP, 1-Nov-2020.)
⊢ (𝐴 ∈ V ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V)
Theorem	trclubgNEW 41115*	If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
Theorem	trclubNEW 41116*	If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → Rel 𝑅)    ⇒   ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅))
Theorem	trclexi 41117*	The transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
⊢ 𝐴 ∈ 𝑉    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V
Theorem	rtrclexi 41118*	The reflexive-transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
⊢ 𝐴 ∈ 𝑉    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V
Theorem	clrellem 41119*	When the property 𝜓 holds for a relation substituted for 𝑥, then the closure on that property is a relation if the base set is a relation. (Contributed by RP, 30-Jul-2020.)
⊢ (𝜑 → 𝑌 ∈ V)    &   ⊢ (𝜑 → Rel 𝑋)    &   ⊢ (𝑥 = ◡◡𝑌 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑌)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})
Theorem	clcnvlem 41120*	When 𝐴, an upper bound of the closure, exists and certain substitutions hold the converse of the closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
⊢ ((𝜑 ∧ 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))) → (𝜒 → 𝜓))    &   ⊢ ((𝜑 ∧ 𝑦 = ◡𝑥) → (𝜓 → 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝜃)    ⇒   ⊢ (𝜑 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ 𝜒)})
Theorem	cnvtrucl0 41121*	The converse of the trivial closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ⊤)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ⊤)})
Theorem	cnvrcl0 41122*	The converse of the reflexive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)})
Theorem	cnvtrcl0 41123*	The converse of the transitive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)})
Theorem	dmtrcl 41124*	The domain of the transitive closure is equal to the domain of its base relation. (Contributed by RP, 1-Nov-2020.)
⊢ (𝑋 ∈ 𝑉 → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = dom 𝑋)
Theorem	rntrcl 41125*	The range of the transitive closure is equal to the range of its base relation. (Contributed by RP, 1-Nov-2020.)
⊢ (𝑋 ∈ 𝑉 → ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ran 𝑋)
Theorem	dfrtrcl5 41126*	Definition of reflexive-transitive closure as a standard closure. (Contributed by RP, 1-Nov-2020.)
⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦))})
20.31.1.16  RP REPLACE: Definitions and basic properties of transitive closures
Theorem	trcleq2lemRP 41127	Equality implies bijection. (Contributed by RP, 5-May-2020.) (Proof modification is discouraged.)
⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴) ↔ (𝑅 ⊆ 𝐵 ∧ (𝐵 ∘ 𝐵) ⊆ 𝐵)))
20.31.1.17  Additions for square root; absolute value This is based on the observation that the real and imaginary parts of a complex number can be calculated from the number's absolute and real part and the sign of its imaginary part. Formalization of the formula in sqrtcval 41138 was motivated by a short Michael Penn video.
Theorem	sqrtcvallem1 41128	Two ways of saying a complex number does not lie on the positive real axis. Lemma for sqrtcval 41138. (Contributed by RP, 17-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (((ℑ‘𝐴) = 0 → (ℜ‘𝐴) ≤ 0) ↔ ¬ 𝐴 ∈ ℝ+))
Theorem	reabsifneg 41129	Alternate expression for the absolute value of a real number. Lemma for sqrtcval 41138. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(𝐴 < 0, -𝐴, 𝐴))
Theorem	reabsifnpos 41130	Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(𝐴 ≤ 0, -𝐴, 𝐴))
Theorem	reabsifpos 41131	Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(0 < 𝐴, 𝐴, -𝐴))
Theorem	reabsifnneg 41132	Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(0 ≤ 𝐴, 𝐴, -𝐴))
Theorem	reabssgn 41133	Alternate expression for the absolute value of a real number. (Contributed by RP, 22-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = ((sgn‘𝐴) · 𝐴))
Theorem	sqrtcvallem2 41134	Equivalent to saying that the square of the imaginary component of the square root of a complex number is a nonnegative real number. Lemma for sqrtcval 41138. See imsqrtval 41141. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) − (ℜ‘𝐴)) / 2))
Theorem	sqrtcvallem3 41135	Equivalent to saying that the absolute value of the imaginary component of the square root of a complex number is a real number. Lemma for sqrtcval 41138, sqrtcval2 41139, resqrtval 41140, and imsqrtval 41141. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2)) ∈ ℝ)
Theorem	sqrtcvallem4 41136	Equivalent to saying that the square of the real component of the square root of a complex number is a nonnegative real number. Lemma for sqrtcval 41138. See resqrtval 41140. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) + (ℜ‘𝐴)) / 2))
Theorem	sqrtcvallem5 41137	Equivalent to saying that the real component of the square root of a complex number is a real number. Lemma for resqrtval 41140 and imsqrtval 41141. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → (√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) ∈ ℝ)
Theorem	sqrtcval 41138	Explicit formula for the complex square root in terms of the square root of nonnegative reals. The right-hand side is decomposed into real and imaginary parts in the format expected by crrei 14831 and crimi 14832. This formula can be found in section 3.7.27 of Handbook of Mathematical Functions, ed. M. Abramowitz and I. A. Stegun (1965, Dover Press). (Contributed by RP, 18-May-2024.)
⊢ (𝐴 ∈ ℂ → (√‘𝐴) = ((√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) + (i · (if((ℑ‘𝐴) < 0, -1, 1) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2))))))
Theorem	sqrtcval2 41139	Explicit formula for the complex square root in terms of the square root of nonnegative reals. The right side is slightly more compact than sqrtcval 41138. (Contributed by RP, 18-May-2024.)
⊢ (𝐴 ∈ ℂ → (√‘𝐴) = ((√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) + (if((ℑ‘𝐴) < 0, -i, i) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2)))))
Theorem	resqrtval 41140	Real part of the complex square root. (Contributed by RP, 18-May-2024.)
⊢ (𝐴 ∈ ℂ → (ℜ‘(√‘𝐴)) = (√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)))
Theorem	imsqrtval 41141	Imaginary part of the complex square root. (Contributed by RP, 18-May-2024.)
⊢ (𝐴 ∈ ℂ → (ℑ‘(√‘𝐴)) = (if((ℑ‘𝐴) < 0, -1, 1) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2))))
Theorem	resqrtvalex 41142	Example for resqrtval 41140. (Contributed by RP, 21-May-2024.)
⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4
Theorem	imsqrtvalex 41143	Example for imsqrtval 41141. (Contributed by RP, 21-May-2024.)
⊢ (ℑ‘(√‘(;15 + (i · 8)))) = 1
20.31.2  Additional statements on relations and subclasses
Theorem	al3im 41144	Version of ax-4 1813 for a nested implication. (Contributed by RP, 13-Apr-2020.)
⊢ (∀𝑥(𝜑 → (𝜓 → (𝜒 → 𝜃))) → (∀𝑥𝜑 → (∀𝑥𝜓 → (∀𝑥𝜒 → ∀𝑥𝜃))))
Theorem	intima0 41145*	Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.)
⊢ ∩ 𝑎 ∈ 𝐴 (𝑎 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}
Theorem	elimaint 41146*	Element of image of intersection. (Contributed by RP, 13-Apr-2020.)
⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
Theorem	cnviun 41147*	Converse of indexed union. (Contributed by RP, 20-Jun-2020.)
⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵
Theorem	imaiun1 41148*	The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶)
Theorem	coiun1 41149*	Composition with an indexed union. Proof analgous to that of coiun 6149. (Contributed by RP, 20-Jun-2020.)
⊢ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
Theorem	elintima 41150*	Element of intersection of images. (Contributed by RP, 13-Apr-2020.)
⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
Theorem	intimass 41151*	The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)
⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}
Theorem	intimass2 41152*	The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)
⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ 𝑥 ∈ 𝐴 (𝑥 “ 𝐵)
Theorem	intimag 41153*	Requirement for the image under the intersection of relations to equal the intersection of the images of those relations. (Contributed by RP, 13-Apr-2020.)
⊢ (∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) → (∩ 𝐴 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)})
Theorem	intimasn 41154*	Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)
⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ {𝐵})})
Theorem	intimasn2 41155*	Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)
⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ 𝑥 ∈ 𝐴 (𝑥 “ {𝐵}))
Theorem	ss2iundf 41156*	Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑦𝑌    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ Ⅎ𝑦𝐶    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ Ⅎ𝑦𝐺    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐺)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷)
Theorem	ss2iundv 41157*	Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐺)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷)
Theorem	cbviuneq12df 41158*	Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝑋    &   ⊢ Ⅎ𝑦𝑌    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ Ⅎ𝑦𝐶    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑦𝐺    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝑋 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐹)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐷 = 𝐹)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷)
Theorem	cbviuneq12dv 41159*	Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝑋 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐹)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐷 = 𝐹)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷)
Theorem	conrel1d 41160	Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → ◡𝐴 = ∅)    ⇒   ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅)
Theorem	conrel2d 41161	Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → ◡𝐴 = ∅)    ⇒   ⊢ (𝜑 → (𝐵 ∘ 𝐴) = ∅)
20.31.2.1  Transitive relations (not to be confused with transitive classes).
Theorem	trrelind 41162	The intersection of transitive relations is a transitive relation. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)    &   ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)    &   ⊢ (𝜑 → 𝑇 = (𝑅 ∩ 𝑆))    ⇒   ⊢ (𝜑 → (𝑇 ∘ 𝑇) ⊆ 𝑇)
Theorem	xpintrreld 41163	The intersection of a transitive relation with a Cartesian product is a transitive relation. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)    &   ⊢ (𝜑 → 𝑆 = (𝑅 ∩ (𝐴 × 𝐵)))    ⇒   ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
Theorem	restrreld 41164	The restriction of a transitive relation is a transitive relation. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)    &   ⊢ (𝜑 → 𝑆 = (𝑅 ↾ 𝐴))    ⇒   ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
Theorem	trrelsuperreldg 41165	Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 25-Dec-2019.)
⊢ (𝜑 → Rel 𝑅)    &   ⊢ (𝜑 → 𝑆 = (dom 𝑅 × ran 𝑅))    ⇒   ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆))
Theorem	trficl 41166*	The class of all transitive relations has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.)
⊢ 𝐴 = {𝑧 ∣ (𝑧 ∘ 𝑧) ⊆ 𝑧}    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴
Theorem	cnvtrrel 41167	The converse of a transitive relation is a transitive relation. (Contributed by RP, 25-Dec-2019.)
⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆)
Theorem	trrelsuperrel2dg 41168	Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 20-Jul-2020.)
⊢ (𝜑 → 𝑆 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)))    ⇒   ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆))
20.31.2.2  Reflexive closures
Syntax	crcl 41169	Extend class notation with reflexive closure.
class r*
Definition	df-rcl 41170*	Reflexive closure of a relation. This is the smallest superset which has the reflexive property. (Contributed by RP, 5-Jun-2020.)
⊢ r* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
Theorem	dfrcl2 41171	Reflexive closure of a relation as union with restricted identity relation. (Contributed by RP, 6-Jun-2020.)
⊢ r* = (𝑥 ∈ V ↦ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
Theorem	dfrcl3 41172	Reflexive closure of a relation as union of powers of the relation. (Contributed by RP, 6-Jun-2020.)
⊢ r* = (𝑥 ∈ V ↦ ((𝑥↑𝑟0) ∪ (𝑥↑𝑟1)))
Theorem	dfrcl4 41173*	Reflexive closure of a relation as indexed union of powers of the relation. (Contributed by RP, 8-Jun-2020.)
⊢ r* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {0, 1} (𝑟↑𝑟𝑛))
20.31.2.3  Finite relationship composition. In order for theorems on the transitive closure of a relation to be grouped together before the concept of continuity, we really need an analogue of ↑𝑟 that works on finite ordinals or finite sets instead of natural numbers.
Theorem	relexp2 41174	A set operated on by the relation exponent to the second power is equal to the composition of the set with itself. (Contributed by RP, 1-Jun-2020.)
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟2) = (𝑅 ∘ 𝑅))
Theorem	relexpnul 41175	If the domain and range of powers of a relation are disjoint then the relation raised to the sum of those exponents is empty. (Contributed by RP, 1-Jun-2020.)
⊢ (((𝑅 ∈ 𝑉 ∧ Rel 𝑅) ∧ (𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → ((dom (𝑅↑𝑟𝑁) ∩ ran (𝑅↑𝑟𝑀)) = ∅ ↔ (𝑅↑𝑟(𝑁 + 𝑀)) = ∅))
Theorem	eliunov2 41176*	Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the element is a member of that operator value. Generalized from dfrtrclrec2 14697. (Contributed by RP, 1-Jun-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))    ⇒   ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))
Theorem	eltrclrec 41177*	Membership in the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛))    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ ℕ 𝑋 ∈ (𝑅↑𝑟𝑛)))
Theorem	elrtrclrec 41178*	Membership in the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ ℕ0 𝑋 ∈ (𝑅↑𝑟𝑛)))
Theorem	briunov2 41179*	Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the two classes are related by that operator value. (Contributed by RP, 1-Jun-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))    ⇒   ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌))
Theorem	brmptiunrelexpd 41180*	If two elements are connected by an indexed union of relational powers, then they are connected via 𝑛 instances the relation, for some 𝑛. Generalization of dfrtrclrec2 14697. (Contributed by RP, 21-Jul-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))    &   ⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝑁 ⊆ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴(𝐶‘𝑅)𝐵 ↔ ∃𝑛 ∈ 𝑁 𝐴(𝑅↑𝑟𝑛)𝐵))
Theorem	fvmptiunrelexplb0d 41181*	If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))    &   ⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝑁 ∈ V)    &   ⊢ (𝜑 → 0 ∈ 𝑁)    ⇒   ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅))
Theorem	fvmptiunrelexplb0da 41182*	If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))    &   ⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝑁 ∈ V)    &   ⊢ (𝜑 → Rel 𝑅)    &   ⊢ (𝜑 → 0 ∈ 𝑁)    ⇒   ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝐶‘𝑅))
Theorem	fvmptiunrelexplb1d 41183*	If the indexed union ranges over the first power of the relation, then the relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))    &   ⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝑁 ∈ V)    &   ⊢ (𝜑 → 1 ∈ 𝑁)    ⇒   ⊢ (𝜑 → 𝑅 ⊆ (𝐶‘𝑅))
Theorem	brfvid 41184	If two elements are connected by a value of the identity relation, then they are connected via the argument. (Contributed by RP, 21-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵))
Theorem	brfvidRP 41185	If two elements are connected by a value of the identity relation, then they are connected via the argument. This is an example which uses brmptiunrelexpd 41180. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵))
Theorem	fvilbd 41186	A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → 𝑅 ⊆ ( I ‘𝑅))
Theorem	fvilbdRP 41187	A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → 𝑅 ⊆ ( I ‘𝑅))
Theorem	brfvrcld 41188	If two elements are connected by the reflexive closure of a relation, then they are connected via zero or one instances the relation. (Contributed by RP, 21-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅↑𝑟0)𝐵 ∨ 𝐴(𝑅↑𝑟1)𝐵)))
Theorem	brfvrcld2 41189	If two elements are connected by the reflexive closure of a relation, then they are equal or related by relation. (Contributed by RP, 21-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ∨ 𝐴𝑅𝐵)))
Theorem	fvrcllb0d 41190	A restriction of the identity relation is a subset of the reflexive closure of a set. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (r*‘𝑅))
Theorem	fvrcllb0da 41191	A restriction of the identity relation is a subset of the reflexive closure of a relation. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → Rel 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (r*‘𝑅))
Theorem	fvrcllb1d 41192	A set is a subset of its image under the reflexive closure. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → 𝑅 ⊆ (r*‘𝑅))
Theorem	brtrclrec 41193*	Two classes related by the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the two classes are related by that relationship power. (Contributed by RP, 2-Jun-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛))    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ 𝑋(𝑅↑𝑟𝑛)𝑌))
Theorem	brrtrclrec 41194*	Two classes related by the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the two classes are related by that relationship power. (Contributed by RP, 2-Jun-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅↑𝑟𝑛)𝑌))
Theorem	briunov2uz 41195*	Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the two classes are related by that operator value. The index set 𝑁 is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))    ⇒   ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 = (ℤ≥‘𝑀)) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌))
Theorem	eliunov2uz 41196*	Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the element is a member of that operator value. The index set 𝑁 is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))    ⇒   ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 = (ℤ≥‘𝑀)) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))
Theorem	ov2ssiunov2 41197*	Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 14696 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))    ⇒   ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑅 ↑ 𝑀) ⊆ (𝐶‘𝑅))
Theorem	relexp0eq 41198	The zeroth power of relationships is the same if and only if the union of their domain and ranges is the same. (Contributed by RP, 11-Jun-2020.)
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ (𝐴↑𝑟0) = (𝐵↑𝑟0)))
Theorem	iunrelexp0 41199*	Simplification of zeroth power of indexed union of powers of relations. (Contributed by RP, 19-Jun-2020.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (∪ 𝑥 ∈ 𝑍 (𝑅↑𝑟𝑥)↑𝑟0) = (𝑅↑𝑟0))
Theorem	relexpxpnnidm 41200	Any positive power of a Cartesian product of non-disjoint sets is itself. (Contributed by RP, 13-Jun-2020.)
⊢ (𝑁 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑𝑟𝑁) = (𝐴 × 𝐵)))