P. 438 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43701-43800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	algmulr 43701	The multiplicative operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ ( × ∈ 𝑉 → × = (.r‘𝐴))
Theorem	algsca 43702	The set of scalars of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ (𝑆 ∈ 𝑉 → 𝑆 = (Scalar‘𝐴))
Theorem	algvsca 43703	The scalar product operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ ( · ∈ 𝑉 → · = ( ·𝑠 ‘𝐴))
Theorem	mendval 43704*	Value of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
⊢ 𝐵 = (𝑀 LMHom 𝑀)    &   ⊢ + = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦))    &   ⊢ × = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ · = (𝑥 ∈ (Base‘𝑆), 𝑦 ∈ 𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))    ⇒   ⊢ (𝑀 ∈ 𝑋 → (MEndo‘𝑀) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩}))
Theorem	mendbas 43705	Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
⊢ 𝐴 = (MEndo‘𝑀)    ⇒   ⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴)
Theorem	mendplusgfval 43706*	Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ (+g‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦))
Theorem	mendplusg 43707	A specific addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ + = (+g‘𝑀)    &   ⊢ ✚ = (+g‘𝐴)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ✚ 𝑌) = (𝑋 ∘f + 𝑌))
Theorem	mendmulrfval 43708*	Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ (.r‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))
Theorem	mendmulr 43709	A specific multiplication in the module endormoprhism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ · = (.r‘𝐴)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = (𝑋 ∘ 𝑌))
Theorem	mendsca 43710	The module endomorphism algebra has the same scalars as the underlying module. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    ⇒   ⊢ 𝑆 = (Scalar‘𝐴)
Theorem	mendvscafval 43711*	Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐸 = (Base‘𝑀)    ⇒   ⊢ ( ·𝑠 ‘𝐴) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))
Theorem	mendvsca 43712	A specific scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐸 = (Base‘𝑀)    &   ⊢ ∙ = ( ·𝑠 ‘𝐴)    ⇒   ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∙ 𝑌) = ((𝐸 × {𝑋}) ∘f · 𝑌))
Theorem	mendring 43713	The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐴 = (MEndo‘𝑀)    ⇒   ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Ring)
Theorem	mendlmod 43714	The module endomorphism algebra is a left module. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)
Theorem	mendassa 43715	The module endomorphism algebra is an algebra. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ 𝐴 = (MEndo‘𝑀)    &   ⊢ 𝑆 = (Scalar‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ AssAlg)
21.33.47  Cyclic groups and order
Theorem	idomodle 43716*	Limit on the number of 𝑁-th roots of unity in an integral domain. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → (♯‘{𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑁}) ≤ 𝑁)
Theorem	fiuneneq 43717	Two finite sets of equal size have a union of the same size iff they were equal. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → ((𝐴 ∪ 𝐵) ≈ 𝐴 ↔ 𝐴 = 𝐵))
Theorem	idomsubgmo 43718*	The units of an integral domain have at most one subgroup of any single finite cardinality. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Revised by NM, 17-Jun-2017.)
⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))    ⇒   ⊢ ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → ∃*𝑦 ∈ (SubGrp‘𝐺)(♯‘𝑦) = 𝑁)
Theorem	proot1mul 43719	Any primitive 𝑁-th root of unity is a multiple of any other. (Contributed by Stefan O'Rear, 2-Nov-2015.)
⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    ⇒   ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑋 ∈ (◡𝑂 “ {𝑁}) ∧ 𝑌 ∈ (◡𝑂 “ {𝑁}))) → 𝑋 ∈ (𝐾‘{𝑌}))
Theorem	proot1hash 43720	If an integral domain has a primitive 𝑁-th root of unity, it has exactly (ϕ‘𝑁) of them. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ (◡𝑂 “ {𝑁})) → (♯‘(◡𝑂 “ {𝑁})) = (ϕ‘𝑁))
Theorem	proot1ex 43721	The complex field has primitive 𝑁-th roots of unity for all 𝑁. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝐺 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ (𝑁 ∈ ℕ → (-1↑𝑐(2 / 𝑁)) ∈ (◡𝑂 “ {𝑁}))
21.33.48  Cyclotomic polynomials
Syntax	ccytp 43722	Syntax for the sequence of cyclotomic polynomials.
class CytP
Definition	df-cytp 43723*	The Nth cyclotomic polynomial is the polynomial which has as its zeros precisely the primitive Nth roots of unity. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ CytP = (𝑛 ∈ ℕ ↦ ((mulGrp‘(Poly1‘ℂfld)) Σg (𝑟 ∈ (◡(od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛}) ↦ ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟)))))
Theorem	mon1psubm 43724	Monic polynomials are a multiplicative submonoid. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑀 = (Monic1p‘𝑅)    &   ⊢ 𝑈 = (mulGrp‘𝑃)    ⇒   ⊢ (𝑅 ∈ NzRing → 𝑀 ∈ (SubMnd‘𝑈))
Theorem	deg1mhm 43725	Homomorphic property of the polynomial degree. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝑌 = ((mulGrp‘𝑃) ↾s (𝐵 ∖ { 0 }))    &   ⊢ 𝑁 = (ℂfld ↾s ℕ0)    ⇒   ⊢ (𝑅 ∈ Domn → (𝐷 ↾ (𝐵 ∖ { 0 })) ∈ (𝑌 MndHom 𝑁))
Theorem	cytpfn 43726	Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ CytP Fn ℕ
Theorem	cytpval 43727*	Substitutions for the Nth cyclotomic polynomial. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝑇 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))    &   ⊢ 𝑂 = (od‘𝑇)    &   ⊢ 𝑃 = (Poly1‘ℂfld)    &   ⊢ 𝑋 = (var1‘ℂfld)    &   ⊢ 𝑄 = (mulGrp‘𝑃)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ (𝑁 ∈ ℕ → (CytP‘𝑁) = (𝑄 Σg (𝑟 ∈ (◡𝑂 “ {𝑁}) ↦ (𝑋 − (𝐴‘𝑟)))))
21.33.49  Miscellaneous topology
Theorem	fgraphopab 43728*	Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)})
Theorem	fgraphxp 43729*	Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)})
Theorem	hausgraph 43730	The graph of a continuous function into a Hausdorff space is closed. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (Clsd‘(𝐽 ×t 𝐾)))
Syntax	ctopsep 43731	The class of separable topologies.
class TopSep
Syntax	ctoplnd 43732	The class of Lindelöf topologies.
class TopLnd
Definition	df-topsep 43733*	A topology is separable iff it has a countable dense subset. (Contributed by Stefan O'Rear, 8-Jan-2015.)
⊢ TopSep = {𝑗 ∈ Top ∣ ∃𝑥 ∈ 𝒫 ∪ 𝑗(𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = ∪ 𝑗)}
Definition	df-toplnd 43734*	A topology is Lindelöf iff every open cover has a countable subcover. (Contributed by Stefan O'Rear, 8-Jan-2015.)
⊢ TopLnd = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ 𝒫 𝑥(𝑧 ≼ ω ∧ ∪ 𝑥 = ∪ 𝑧))}
21.34  Mathbox for Noam Pasman
Theorem	r1sssucd 43735	Deductive form of r1sssuc 9731. (Contributed by Noam Pasman, 19-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ On)    ⇒   ⊢ (𝜑 → (𝑅1‘𝐴) ⊆ (𝑅1‘suc 𝐴))
21.35  Mathbox for Jon Pennant
Theorem	iocunico 43736	Split an open interval into two pieces at point B, Co-author TA. (Contributed by Jon Pennant, 8-Jun-2019.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶))
Theorem	iocinico 43737	The intersection of two sets that meet at a point is that point. (Contributed by Jon Pennant, 12-Jun-2019.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) = {𝐵})
Theorem	iocmbl 43738	An open-below, closed-above real interval is measurable. (Contributed by Jon Pennant, 12-Jun-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ∈ dom vol)
Theorem	cnioobibld 43739*	A bounded, continuous function on an open bounded interval is integrable. The function must be bounded. For a counterexample, consider 𝐹 = (𝑥 ∈ (0(,)1) ↦ (1 / 𝑥)). See cniccibl 25876 for closed bounded intervals. (Contributed by Jon Pennant, 31-May-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)
Theorem	arearect 43740	The area of a rectangle whose sides are parallel to the coordinate axes in (ℝ × ℝ) is its width multiplied by its height. (Contributed by Jon Pennant, 19-Mar-2019.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 𝐷 ∈ ℝ    &   ⊢ 𝐴 ≤ 𝐵    &   ⊢ 𝐶 ≤ 𝐷    &   ⊢ 𝑆 = ((𝐴[,]𝐵) × (𝐶[,]𝐷))    ⇒   ⊢ (area‘𝑆) = ((𝐵 − 𝐴) · (𝐷 − 𝐶))
Theorem	areaquad 43741*	The area of a quadrilateral with two sides which are parallel to the y-axis in (ℝ × ℝ) is its width multiplied by the average height of its higher edge minus the average height of its lower edge. Co-author TA. (Contributed by Jon Pennant, 31-May-2019.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 𝐷 ∈ ℝ    &   ⊢ 𝐸 ∈ ℝ    &   ⊢ 𝐹 ∈ ℝ    &   ⊢ 𝐴 < 𝐵    &   ⊢ 𝐶 ≤ 𝐸    &   ⊢ 𝐷 ≤ 𝐹    &   ⊢ 𝑈 = (𝐶 + (((𝑥 − 𝐴) / (𝐵 − 𝐴)) · (𝐷 − 𝐶)))    &   ⊢ 𝑉 = (𝐸 + (((𝑥 − 𝐴) / (𝐵 − 𝐴)) · (𝐹 − 𝐸)))    &   ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝑈[,]𝑉))}    ⇒   ⊢ (area‘𝑆) = ((((𝐹 + 𝐸) / 2) − ((𝐷 + 𝐶) / 2)) · (𝐵 − 𝐴))
21.36  Mathbox for Richard Penner
21.36.1  Set Theory and Ordinal Numbers
Theorem	uniel 43742*	Two ways to say a union is an element of a class. (Contributed by RP, 27-Jan-2025.)
⊢ (∪ 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
Theorem	unielss 43743*	Two ways to say the union of a class is an element of a subclass. (Contributed by RP, 29-Jan-2025.)
⊢ (𝐴 ⊆ 𝐵 → (∪ 𝐵 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥))
Theorem	unielid 43744*	Two ways to say the union of a class is an element of that class. (Contributed by RP, 27-Jan-2025.)
⊢ (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
Theorem	ssunib 43745*	Two ways to say a class is a subclass of a union. (Contributed by RP, 27-Jan-2025.)
⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)
Theorem	rp-intrabeq 43746*	Equality theorem for supremum of sets of ordinals. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 = 𝐵 → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥})
Theorem	rp-unirabeq 43747*	Equality theorem for infimum of non-empty classes of ordinals. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 = 𝐵 → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦})
Theorem	onmaxnelsup 43748*	Two ways to say the maximum element of a class of ordinals is also the supremum of that class. (Contributed by RP, 27-Jan-2025.)
⊢ (𝐴 ⊆ On → (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))
Theorem	onsupneqmaxlim0 43749	If the supremum of a class of ordinals is not in that class, then the supremum is a limit ordinal or empty. (Contributed by RP, 27-Jan-2025.)
⊢ (𝐴 ⊆ On → (𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))
Theorem	onsupcl2 43750	The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 ∈ 𝒫 On → ∪ 𝐴 ∈ On)
Theorem	onuniintrab 43751*	The union of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. Closed form of uniordint 7773. (Contributed by RP, 28-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
Theorem	onintunirab 43752*	The intersection of a non-empty class of ordinals is the union of every ordinal less-than-or-equal to every element of that class. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})
Theorem	onsupnmax 43753	If the union of a class of ordinals is not the maximum element of that class, then the union is a limit ordinal or empty. But this isn't a biconditional since 𝐴 could be a non-empty set where a limit ordinal or the empty set happens to be the largest element. (Contributed by RP, 27-Jan-2025.)
⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))
Theorem	onsupuni 43754	The supremum of a set of ordinals is the union of that set. Lemma 2.10 of [Schloeder] p. 5. (Contributed by RP, 19-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴)
Theorem	onsupuni2 43755	The supremum of a set of ordinals is the union of that set. (Contributed by RP, 22-Jan-2025.)
⊢ (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = ∪ 𝐴)
Theorem	onsupintrab 43756*	The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. Definition 2.9 of [Schloeder] p. 5. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
Theorem	onsupintrab2 43757*	The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
Theorem	onsupcl3 43758*	The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} ∈ On)
Theorem	onsupex3 43759*	The supremum of a set of ordinals exists. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} ∈ V)
Theorem	onuniintrab2 43760*	The union of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 ∈ 𝒫 On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
Theorem	oninfint 43761	The infimum of a non-empty class of ordinals is the intersection of that class. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → inf(𝐴, On, E ) = ∩ 𝐴)
Theorem	oninfunirab 43762*	The infimum of a non-empty class of ordinals is the union of every ordinal less-than-or-equal to every element of that class. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → inf(𝐴, On, E ) = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})
Theorem	oninfcl2 43763*	The infimum of a non-empty class of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∈ On)
Theorem	onsupmaxb 43764	The union of a class of ordinals is an element is an element of that class if and only if there is a maximum element of that class under the epsilon relation, which is to say that the domain of the restricted epsilon relation is not the whole class. (Contributed by RP, 25-Jan-2025.)
⊢ (𝐴 ⊆ On → (dom ( E ∩ (𝐴 × 𝐴)) = 𝐴 ↔ ¬ ∪ 𝐴 ∈ 𝐴))
Theorem	onexgt 43765*	For any ordinal, there is always a larger ordinal. (Contributed by RP, 1-Feb-2025.)
⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)
Theorem	onexomgt 43766*	For any ordinal, there is always a larger product of omega. (Contributed by RP, 1-Feb-2025.)
⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))
Theorem	omlimcl2 43767	The product of a limit ordinal with any nonzero ordinal is a limit ordinal. (Contributed by RP, 8-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐵 ·o 𝐴))
Theorem	onexlimgt 43768*	For any ordinal, there is always a larger limit ordinal. (Contributed by RP, 1-Feb-2025.)
⊢ (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))
Theorem	onexoegt 43769*	For any ordinal, there is always a larger power of omega. (Contributed by RP, 1-Feb-2025.)
⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ (ω ↑o 𝑥))
Theorem	oninfex2 43770*	The infimum of a non-empty class of ordinals exists. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∈ V)
Theorem	onsupeqmax 43771*	Condition when the supremum of a set of ordinals is the maximum element of that set. (Contributed by RP, 24-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∪ 𝐴 ∈ 𝐴))
Theorem	onsupeqnmax 43772*	Condition when the supremum of a class of ordinals is not the maximum element of that class. (Contributed by RP, 27-Jan-2025.)
⊢ (𝐴 ⊆ On → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ (∪ 𝐴 = ∪ ∪ 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴)))
Theorem	onsuplub 43773*	The supremum of a set of ordinals is the least upper bound. (Contributed by RP, 27-Jan-2025.)
⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∈ On) → (𝐵 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝐵 ∈ 𝑧))
Theorem	onsupnub 43774*	An upper bound of a set of ordinals is not less than the supremum. (Contributed by RP, 27-Jan-2025.)
⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)) → ∪ 𝐴 ⊆ 𝐵)
Theorem	onfisupcl 43775	Sufficient condition when the supremum of a set of ordinals is the maximum element of that set. See ordunifi 9223. (Contributed by RP, 27-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴))
Theorem	onelord 43776	Every element of a ordinal is an ordinal. Lemma 1.3 of [Schloeder] p. 1. Based on onelon 6360 and eloni 6345. (Contributed by RP, 15-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
Theorem	onepsuc 43777	Every ordinal is less than its successor, relationship version. Lemma 1.7 of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
⊢ (𝐴 ∈ On → 𝐴 E suc 𝐴)
Theorem	epsoon 43778	The ordinals are strictly and completely (linearly) ordered. Theorem 1.9 of [Schloeder] p. 1. Based on epweon 7747 and weso 5631. (Contributed by RP, 15-Jan-2025.)
⊢ E Or On
Theorem	epirron 43779	The strict order on the ordinals is irreflexive. Theorem 1.9(i) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
⊢ (𝐴 ∈ On → ¬ 𝐴 E 𝐴)
Theorem	oneptr 43780	The strict order on the ordinals is transitive. Theorem 1.9(ii) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵 ∧ 𝐵 E 𝐶) → 𝐴 E 𝐶))
Theorem	oneltr 43781	The elementhood relation on the ordinals is transitive. Theorem 1.9(ii) of [Schloeder] p. 1. See ontr1 6382. (Contributed by RP, 15-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
Theorem	oneptri 43782	The strict, complete (linear) order on the ordinals is complete. Theorem 1.9(iii) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ∨ 𝐴 = 𝐵))
Theorem	ordeldif 43783	Membership in the difference of ordinals. (Contributed by RP, 15-Jan-2025.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐶 ∈ (𝐴 ∖ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶)))
Theorem	ordeldifsucon 43784	Membership in the difference of ordinal and successor ordinal. (Contributed by RP, 16-Jan-2025.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐶 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)))
Theorem	ordeldif1o 43785	Membership in the difference of ordinal and ordinal one. (Contributed by RP, 16-Jan-2025.)
⊢ (Ord 𝐴 → (𝐵 ∈ (𝐴 ∖ 1o) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ≠ ∅)))
Theorem	ordne0gt0 43786	Ordinal zero is less than every nonzero ordinal. Theorem 1.10 of [Schloeder] p. 2. Closely related to ord0eln0 6391. (Contributed by RP, 16-Jan-2025.)
⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∅ ∈ 𝐴)
Theorem	ondif1i 43787	Ordinal zero is less than every nonzero ordinal, class difference version. Theorem 1.10 of [Schloeder] p. 2. See ondif1 8458. (Contributed by RP, 16-Jan-2025.)
⊢ (𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴)
Theorem	onsucelab 43788*	The successor of every ordinal is an element of the class of successor ordinals. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})
Theorem	dflim6 43789*	A limit ordinal is a nonzero ordinal which is not a successor ordinal. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
Theorem	limnsuc 43790*	A limit ordinal is not an element of the class of successor ordinals. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)
⊢ (Lim 𝐴 → ¬ 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})
Theorem	onsucss 43791	If one ordinal is less than another, then the successor of the first is less than or equal to the second. Lemma 1.13 of [Schloeder] p. 2. See ordsucss 7787. (Contributed by RP, 16-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
Theorem	ordnexbtwnsuc 43792*	For any distinct pair of ordinals, if there is no ordinal between the lesser and the greater, the greater is the successor of the lesser. Lemma 1.16 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → 𝐵 = suc 𝐴))
Theorem	orddif0suc 43793	For any distinct pair of ordinals, if the set difference between the greater and the successor of the lesser is empty, the greater is the successor of the lesser. Lemma 1.16 of [Schloeder] p. 2. (Contributed by RP, 17-Jan-2025.)
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐵 ∖ suc 𝐴) = ∅ → 𝐵 = suc 𝐴))
Theorem	onsucf1lem 43794*	For ordinals, the successor operation is injective, so there is at most one ordinal that a given ordinal could be the successor of. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
⊢ (𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
Theorem	onsucf1olem 43795*	The successor operation is bijective between the ordinals and the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃!𝑏 ∈ On 𝐴 = suc 𝑏)
Theorem	onsucrn 43796*	The successor operation is surjective onto its range, the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥)    ⇒   ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}
Theorem	onsucf1o 43797*	The successor operation is a bijective function between the ordinals and the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥)    ⇒   ⊢ 𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}
Theorem	dflim7 43798*	A limit ordinal is a nonzero ordinal that contains all the successors of its elements. Lemma 1.18 of [Schloeder] p. 2. Closely related to dflim4 7817. (Contributed by RP, 17-Jan-2025.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∀𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ∧ 𝐴 ≠ ∅))
Theorem	onov0suclim 43799	Compactly express rules for binary operations on ordinals. (Contributed by RP, 18-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐴 ⊗ ∅) = 𝐷)    &   ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊗ suc 𝐶) = 𝐸)    &   ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐴 ⊗ 𝐵) = 𝐹)    ⇒   ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹)))
Theorem	oa0suclim 43800*	Closed form expression of the value of ordinal addition for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.3 of [Schloeder] p. 4. See oa0 8473, oasuc 8481, and oalim 8489. (Contributed by RP, 18-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 +o 𝐵) = 𝐴) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 +o 𝐵) = suc (𝐴 +o 𝐶)) ∧ (Lim 𝐵 → (𝐴 +o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 +o 𝑐))))