P. 338 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33701-33800   *Has distinct variable group(s)

Type	Label	Description
Statement
21.3.10.21  Localization of rings
Syntax	cerl 33701	Syntax for ring localization equivalence class operation.
class ~RL
Syntax	crloc 33702	Syntax for ring localization operation.
class RLocal
Definition	df-erl 33703*	Define the operation giving the equivalence relation used in the localization of a ring 𝑟 by a set 𝑠. Two pairs 𝑎 = ⟨𝑥, 𝑦⟩ and 𝑏 = ⟨𝑧, 𝑤⟩ are equivalent if there exists 𝑡 ∈ 𝑠 such that 𝑡 · (𝑥 · 𝑤 − 𝑧 · 𝑦) = 0. This corresponds to the usual comparison of fractions 𝑥 / 𝑦 and 𝑧 / 𝑤. (Contributed by Thierry Arnoux, 28-Apr-2025.)
⊢ ~RL = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋(.r‘𝑟) / 𝑥⦌⦋((Base‘𝑟) × 𝑠) / 𝑤⦌{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ∃𝑡 ∈ 𝑠 (𝑡𝑥(((1st ‘𝑎)𝑥(2nd ‘𝑏))(-g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))) = (0g‘𝑟))})
Definition	df-rloc 33704*	Define the operation giving the localization of a ring 𝑟 by a given set 𝑠. The localized ring 𝑟 RLocal 𝑠 is the set of equivalence classes of pairs of elements in 𝑟 over the relation 𝑟 ~RL 𝑠 with addition and multiplication defined naturally. (Contributed by Thierry Arnoux, 27-Apr-2025.)
⊢ RLocal = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋(.r‘𝑟) / 𝑥⦌⦋((Base‘𝑟) × 𝑠) / 𝑤⦌((({⟨(Base‘ndx), 𝑤⟩, ⟨(+g‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ ⟨(((1st ‘𝑎)𝑥(2nd ‘𝑏))(+g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎))), ((2nd ‘𝑎)𝑥(2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ ⟨((1st ‘𝑎)𝑥(1st ‘𝑏)), ((2nd ‘𝑎)𝑥(2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑟)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑟)), 𝑎 ∈ 𝑤 ↦ ⟨(𝑘( ·𝑠 ‘𝑟)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑟) ×t ((TopSet‘𝑟) ↾t 𝑠))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ((1st ‘𝑎)𝑥(2nd ‘𝑏))(le‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ (((1st ‘𝑎)𝑥(2nd ‘𝑏))(dist‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎))))⟩}) /s (𝑟 ~RL 𝑠)))
Theorem	reldmrloc 33705	Ring localization is a proper operator, so it can be used with ovprc1 7456. (Contributed by Thierry Arnoux, 10-May-2025.)
⊢ Rel dom RLocal
Theorem	erlval 33706*	Value of the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    &   ⊢ 𝑊 = (𝐵 × 𝑆)    &   ⊢ ∼ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )}    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑅 ~RL 𝑆) = ∼ )
Theorem	rlocval 33707*	Expand the value of the ring localization operation. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ≤ = (le‘𝑅)    &   ⊢ 𝐹 = (Scalar‘𝑅)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝐶 = ( ·𝑠 ‘𝑅)    &   ⊢ 𝑊 = (𝐵 × 𝑆)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ 𝐽 = (TopSet‘𝑅)    &   ⊢ 𝐷 = (dist‘𝑅)    &   ⊢ ⊕ = (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)    &   ⊢ ⊗ = (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)    &   ⊢ × = (𝑘 ∈ 𝐾, 𝑎 ∈ 𝑊 ↦ ⟨(𝑘𝐶(1st ‘𝑎)), (2nd ‘𝑎)⟩)    &   ⊢ ≲ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1st ‘𝑎) · (2nd ‘𝑏)) ≤ ((1st ‘𝑏) · (2nd ‘𝑎)))}    &   ⊢ 𝐸 = (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1st ‘𝑎) · (2nd ‘𝑏))𝐷((1st ‘𝑏) · (2nd ‘𝑎))))    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑅 RLocal 𝑆) = ((({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), ⊗ ⟩} ∪ {⟨(Scalar‘ndx), 𝐹⟩, ⟨( ·𝑠 ‘ndx), × ⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), (𝐽 ×t (𝐽 ↾t 𝑆))⟩, ⟨(le‘ndx), ≲ ⟩, ⟨(dist‘ndx), 𝐸⟩}) /s ∼ ))
Theorem	erlcl1 33708	Closure for the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑈 ∼ 𝑉)    ⇒   ⊢ (𝜑 → 𝑈 ∈ (𝐵 × 𝑆))
Theorem	erlcl2 33709	Closure for the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑈 ∼ 𝑉)    ⇒   ⊢ (𝜑 → 𝑉 ∈ (𝐵 × 𝑆))
Theorem	erldi 33710*	Main property of the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    &   ⊢ (𝜑 → 𝑈 ∼ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑈) · (2nd ‘𝑉)) − ((1st ‘𝑉) · (2nd ‘𝑈)))) = 0 )
Theorem	erlbrd 33711	Deduce the ring localization equivalence relation. If for some 𝑇 ∈ 𝑆 we have 𝑇 · (𝐸 · 𝐻 − 𝐹 · 𝐺) = 0, then pairs ⟨𝐸, 𝐺⟩ and ⟨𝐹, 𝐻⟩ are equivalent under the localization relation. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    &   ⊢ (𝜑 → 𝑈 = ⟨𝐸, 𝐺⟩)    &   ⊢ (𝜑 → 𝑉 = ⟨𝐹, 𝐻⟩)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (𝑇 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 )    ⇒   ⊢ (𝜑 → 𝑈 ∼ 𝑉)
Theorem	erlbr2d 33712	Deduce the ring localization equivalence relation. Pairs ⟨𝐸, 𝐺⟩ and ⟨𝑇 · 𝐸, 𝑇 · 𝐺⟩ for 𝑇 ∈ 𝑆 are equivalent under the localization relation. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑈 = ⟨𝐸, 𝐺⟩)    &   ⊢ (𝜑 → 𝑉 = ⟨𝐹, 𝐻⟩)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑇 · 𝐸))    &   ⊢ (𝜑 → 𝐻 = (𝑇 · 𝐺))    ⇒   ⊢ (𝜑 → 𝑈 ∼ 𝑉)
Theorem	erler 33713	The relation used to build the ring localization is an equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    &   ⊢ 𝑊 = (𝐵 × 𝑆)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    ⇒   ⊢ (𝜑 → ∼ Er 𝑊)
Theorem	erld2 33714*	Main property of the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑆)    &   ⊢ (𝜑 → [⟨𝑋, 𝑌⟩] ∼ = [⟨𝑍, 𝑊⟩] ∼ )    ⇒   ⊢ (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · (𝑋 · 𝑊)) = (𝑡 · (𝑍 · 𝑌)))
Theorem	elrlocbasi 33715*	Membership in the basis of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ (𝜑 → 𝑋 ∈ ((𝐵 × 𝑆) / ∼ ))    ⇒   ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 ∃𝑏 ∈ 𝑆 𝑋 = [⟨𝑎, 𝑏⟩] ∼ )
Theorem	rlocbas 33716	The base set of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    &   ⊢ 𝑊 = (𝐵 × 𝑆)    &   ⊢ 𝐿 = (𝑅 RLocal 𝑆)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐿))
Theorem	rlocaddval 33717	Value of the addition in the ring localization, given two representatives. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐿 = (𝑅 RLocal 𝑆)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑆)    &   ⊢ ⊕ = (+g‘𝐿)    ⇒   ⊢ (𝜑 → ([⟨𝐸, 𝐺⟩] ∼ ⊕ [⟨𝐹, 𝐻⟩] ∼ ) = [⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩] ∼ )
Theorem	rlocmulval 33718	Value of the addition in the ring localization, given two representatives. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐿 = (𝑅 RLocal 𝑆)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑆)    &   ⊢ ⊗ = (.r‘𝐿)    ⇒   ⊢ (𝜑 → ([⟨𝐸, 𝐺⟩] ∼ ⊗ [⟨𝐹, 𝐻⟩] ∼ ) = [⟨(𝐸 · 𝐹), (𝐺 · 𝐻)⟩] ∼ )
Theorem	rloccring 33719	The ring localization 𝐿 of a commutative ring 𝑅 by a multiplicatively closed set 𝑆 is itself a commutative ring. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐿 = (𝑅 RLocal 𝑆)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    ⇒   ⊢ (𝜑 → 𝐿 ∈ CRing)
Theorem	rloc0g 33720	The zero of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐿 = (𝑅 RLocal 𝑆)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   ⊢ 𝑂 = [⟨ 0 , 1 ⟩] ∼    ⇒   ⊢ (𝜑 → 𝑂 = (0g‘𝐿))
Theorem	rloc1r 33721	The multiplicative identity of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐿 = (𝑅 RLocal 𝑆)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   ⊢ 𝐼 = [⟨ 1 , 1 ⟩] ∼    ⇒   ⊢ (𝜑 → 𝐼 = (1r‘𝐿))
Theorem	rlocf1 33722*	The embedding 𝐹 of a ring 𝑅 into its localization 𝐿. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐿 = (𝑅 RLocal 𝑆)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [⟨𝑥, 1 ⟩] ∼ )    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   ⊢ (𝜑 → 𝑆 ⊆ (RLReg‘𝑅))    ⇒   ⊢ (𝜑 → (𝐹:𝐵–1-1→((𝐵 × 𝑆) / ∼ ) ∧ 𝐹 ∈ (𝑅 RingHom 𝐿)))
Theorem	rlocinvunit 33723	In the localization of a ring 𝑅 at 𝑆, inverses of elements of 𝑆 are units. (Contributed by Thierry Arnoux, 6-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ 𝐿 = (𝑅 RLocal 𝑆)    &   ⊢ 𝑊 = (Unit‘𝐿)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   ⊢ (𝜑 → 𝑄 ∈ 𝑆)    ⇒   ⊢ (𝜑 → [⟨ 1 , 𝑄⟩] ∼ ∈ 𝑊)
Theorem	rlocisunit 33724*	Characterize the units of the localization 𝐿 of a ring 𝑅 at 𝑆 as the elements with a "numerator" 𝑃 in the saturation 𝑇 of 𝑆. (Contributed by Thierry Arnoux, 6-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐿 = (𝑅 RLocal 𝑆)    &   ⊢ 𝑊 = (Unit‘𝐿)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   ⊢ ∼ = (𝑅 ~RL 𝑆)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑆)    &   ⊢ 𝑇 = {𝑟 ∈ 𝐵 ∣ ∃𝑠 ∈ 𝐵 (𝑟 · 𝑠) ∈ 𝑆}    ⇒   ⊢ (𝜑 → ([⟨𝑃, 𝑄⟩] ∼ ∈ 𝑊 ↔ 𝑃 ∈ 𝑇))
21.3.10.22  Integral Domains
Theorem	domnmuln0rd 33725	In a domain, factors of a nonzero product are nonzero. (Contributed by Thierry Arnoux, 8-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋 · 𝑌) ≠ 0 )    ⇒   ⊢ (𝜑 → (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ))
Theorem	domnprodn0 33726	In a domain, a finite product of nonzero terms is nonzero. (Contributed by Thierry Arnoux, 6-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → 𝐹 ∈ Word (𝐵 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝑀 Σg 𝐹) ≠ 0 )
Theorem	domnprodeq0 33727	A product over a domain is zero exactly when one of the factors is zero. Generalization of domneq0 20876 for any number of factors. See also domnprodn0 33726. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → ((𝑀 Σg 𝐹) = 0 ↔ 0 ∈ ran 𝐹))
Theorem	domnpropd 33728*	If two structures have the same components (properties), one is a domain iff the other one is. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ Domn ↔ 𝐿 ∈ Domn))
Theorem	idompropd 33729*	If two structures have the same components (properties), one is a integral domain iff the other one is. See also domnpropd 33728. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ IDomn ↔ 𝐿 ∈ IDomn))
Theorem	idomrcan 33730	Right-cancellation law for integral domains. (Contributed by Thierry Arnoux, 22-Mar-2025.) (Proof shortened by SN, 21-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	1rrg 33731	The multiplicative identity is a left-regular element. (Contributed by Thierry Arnoux, 6-May-2025.)
⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 1 ∈ 𝐸)
Theorem	rrgsubm 33732	The left regular elements of a ring form a submonoid of the multiplicative group. (Contributed by Thierry Arnoux, 10-May-2025.)
⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝐸 ∈ (SubMnd‘𝑀))
Theorem	subrdom 33733	A subring of a domain is a domain. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    ⇒   ⊢ (𝜑 → (𝑅 ↾s 𝑆) ∈ Domn)
Theorem	subridom 33734	A subring of an integral domain is an integral domain. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    ⇒   ⊢ (𝜑 → (𝑅 ↾s 𝑆) ∈ IDomn)
Theorem	subrfld 33735	A subring of a field is an integral domain. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ (𝜑 → 𝑅 ∈ Field)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    ⇒   ⊢ (𝜑 → (𝑅 ↾s 𝑆) ∈ IDomn)
Theorem	ricnzr1 33736	A ring isomorphism maps a nonzero ring to a nonzero ring. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ NzRing) → 𝑆 ∈ NzRing)
Theorem	ricdomn1 33737	A ring isomorphism maps a domain to a domain. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) → 𝑆 ∈ Domn)
Theorem	ricdomn 33738	A ring is a domain if and only if an isomorphic ring is a domain. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ (𝑅 ≃𝑟 𝑆 → (𝑅 ∈ Domn ↔ 𝑆 ∈ Domn))
21.3.10.23  Euclidean Domains
Syntax	ceuf 33739	Declare the syntax for the Euclidean function index extractor.
class EuclF
Definition	df-euf 33740	Define the Euclidean function. (Contributed by Thierry Arnoux, 22-Mar-2025.) Use its index-independent form eufid 33742 instead. (New usage is discouraged.)
⊢ EuclF = Slot ;21
Theorem	eufndx 33741	Index value of the Euclidean function slot. Use ndxarg 17294. (Contributed by Thierry Arnoux, 22-Mar-2025.) (New usage is discouraged.)
⊢ (EuclF‘ndx) = ;21
Theorem	eufid 33742	Utility theorem: index-independent form of df-euf 33740. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ EuclF = Slot (EuclF‘ndx)
Syntax	cedom 33743	Declare the syntax for the Euclidean Domain.
class EDomn
Definition	df-edom 33744*	Define Euclidean Domains. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ EDomn = {𝑑 ∈ IDomn ∣ [(EuclF‘𝑑) / 𝑒][(Base‘𝑑) / 𝑣](Fun 𝑒 ∧ (𝑒 “ (𝑣 ∖ {(0g‘𝑑)})) ⊆ (0[,)+∞) ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ (𝑣 ∖ {(0g‘𝑑)})∃𝑞 ∈ 𝑣 ∃𝑟 ∈ 𝑣 (𝑎 = ((𝑏(.r‘𝑑)𝑞)(+g‘𝑑)𝑟) ∧ (𝑟 = (0g‘𝑑) ∨ (𝑒‘𝑟) < (𝑒‘𝑏))))}
21.3.10.24  Division Rings
Theorem	rndrhmcl 33745	The image of a division ring by a ring homomorphism is a division ring. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 𝑅 = (𝑁 ↾s ran 𝐹)    &   ⊢ 0 = (0g‘𝑁)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁))    &   ⊢ (𝜑 → ran 𝐹 ≠ { 0 })    &   ⊢ (𝜑 → 𝑀 ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝑅 ∈ DivRing)
21.3.10.25  The field of rational numbers
Theorem	qfld 33746	The field of rational numbers is a field. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ 𝑄 ∈ Field
21.3.10.26  Subfields
Theorem	subsdrg 33747	A subring of a sub-division-ring is a sub-division-ring. See also subsubrg 20766. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ (SubDRing‘𝑅))    ⇒   ⊢ (𝜑 → (𝐵 ∈ (SubDRing‘𝑆) ↔ (𝐵 ∈ (SubDRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)))
Theorem	sdrgdvcl 33748	A sub-division-ring is closed under the ring division operation. (Contributed by Thierry Arnoux, 15-Jan-2025.)
⊢ / = (/r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐴 ∈ (SubDRing‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝑋 / 𝑌) ∈ 𝐴)
Theorem	sdrginvcl 33749	A sub-division-ring is closed under the ring inverse operation. (Contributed by Thierry Arnoux, 15-Jan-2025.)
⊢ 𝐼 = (invr‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝐴 ∈ (SubDRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐴)
Theorem	primefldchr 33750	The characteristic of a prime field is the same as the characteristic of the main field. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝑃 = (𝑅 ↾s ∩ (SubDRing‘𝑅))    ⇒   ⊢ (𝑅 ∈ DivRing → (chr‘𝑃) = (chr‘𝑅))
21.3.10.27  Field of fractions
Syntax	cfrac 33751	Syntax for the field of fractions of a given integral domain.
class Frac
Definition	df-frac 33752	Define the field of fractions of a given integral domain. (Contributed by Thierry Arnoux, 26-Apr-2025.)
⊢ Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟)))
Theorem	fracval 33753	Value of the field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.)
⊢ ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅))
Theorem	fracbas 33754	The base of the field of fractions. (Contributed by Thierry Arnoux, 10-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 𝐹 = ( Frac ‘𝑅)    &   ⊢ ∼ = (𝑅 ~RL 𝐸)    ⇒   ⊢ ((𝐵 × 𝐸) / ∼ ) = (Base‘𝐹)
Theorem	fracerl 33755	Rewrite the ring localization equivalence relation in the case of a field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∼ = (𝑅 ~RL (RLReg‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (RLReg‘𝑅))    &   ⊢ (𝜑 → 𝐻 ∈ (RLReg‘𝑅))    ⇒   ⊢ (𝜑 → (⟨𝐸, 𝐹⟩ ∼ ⟨𝐺, 𝐻⟩ ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹)))
Theorem	fracf1 33756*	The embedding of a commutative ring 𝑅 into its field of fractions. (Contributed by Thierry Arnoux, 10-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ ∼ = (𝑅 ~RL 𝐸)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [⟨𝑥, 1 ⟩] ∼ )    ⇒   ⊢ (𝜑 → (𝐹:𝐵–1-1→((𝐵 × 𝐸) / ∼ ) ∧ 𝐹 ∈ (𝑅 RingHom ( Frac ‘𝑅))))
Theorem	fracfld 33757	The field of fractions of an integral domain is a field. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → ( Frac ‘𝑅) ∈ Field)
Theorem	idomsubr 33758*	Every integral domain is isomorphic with a subring of some field. (Proposed by Gerard Lang, 10-May-2025.) (Contributed by Thierry Arnoux, 10-May-2025.)
⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ Field ∃𝑠 ∈ (SubRing‘𝑓)𝑅 ≃𝑟 (𝑓 ↾s 𝑠))
21.3.10.28  Field extensions generated by a set
Syntax	cfldgen 33759	Syntax for a function generating sub-fields.
class fldGen
Definition	df-fldgen 33760*	Define a function generating the smallest sub-division-ring of a given ring containing a given set. If the base structure is a division ring, then this is also a division ring (see fldgensdrg 33763). If the base structure is a field, this is a subfield (see fldgenfld 33769 and fldsdrgfld 20970). In general this will be used in the context of fields, hence the name fldGen. (Contributed by Saveliy Skresanov and Thierry Arnoux, 9-Jan-2025.)
⊢ fldGen = (𝑓 ∈ V, 𝑠 ∈ V ↦ ∩ {𝑎 ∈ (SubDRing‘𝑓) ∣ 𝑠 ⊆ 𝑎})
Theorem	fldgenval 33761*	Value of the field generating function: (𝐹 fldGen 𝑆) is the smallest sub-division-ring of 𝐹 containing 𝑆. (Contributed by Thierry Arnoux, 11-Jan-2025.)
⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 fldGen 𝑆) = ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
Theorem	fldgenssid 33762	The field generated by a set of elements contains those elements. See lspssid 21175. (Contributed by Thierry Arnoux, 15-Jan-2025.)
⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝑆 ⊆ (𝐹 fldGen 𝑆))
Theorem	fldgensdrg 33763	A generated subfield is a sub-division-ring. (Contributed by Thierry Arnoux, 11-Jan-2025.)
⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 fldGen 𝑆) ∈ (SubDRing‘𝐹))
Theorem	fldgenssv 33764	A generated subfield is a subset of the field's base. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 fldGen 𝑆) ⊆ 𝐵)
Theorem	fldgenss 33765	Generated subfields preserve subset ordering. ( see lspss 21174 and spanss 31837) (Contributed by Thierry Arnoux, 15-Jan-2025.)
⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹 fldGen 𝑇) ⊆ (𝐹 fldGen 𝑆))
Theorem	fldgenidfld 33766	The subfield generated by a subfield is the subfield itself. (Contributed by Thierry Arnoux, 15-Jan-2025.)
⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝐹))    ⇒   ⊢ (𝜑 → (𝐹 fldGen 𝑆) = 𝑆)
Theorem	fldgenssp 33767	The field generated by a set of elements in a division ring is contained in any sub-division-ring which contains those elements. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝐹))    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹 fldGen 𝑇) ⊆ 𝑆)
Theorem	fldgenid 33768	The subfield of a field 𝐹 generated by the whole base set of 𝐹 is 𝐹 itself. (Contributed by Thierry Arnoux, 11-Jan-2025.)
⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    ⇒   ⊢ (𝜑 → (𝐹 fldGen 𝐵) = 𝐵)
Theorem	fldgenfld 33769	A generated subfield is a field. (Contributed by Thierry Arnoux, 11-Jan-2025.)
⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ Field)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ↾s (𝐹 fldGen 𝑆)) ∈ Field)
Theorem	primefldgen1 33770	The prime field of a division ring is the subfield generated by the multiplicative identity element. In general, we should write "prime division ring", but since most later usages are in the case where the ambient ring is commutative, we keep the term "prime field". (Contributed by Thierry Arnoux, 11-Jan-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    ⇒   ⊢ (𝜑 → ∩ (SubDRing‘𝑅) = (𝑅 fldGen { 1 }))
Theorem	1fldgenq 33771	The field of rational numbers ℚ is generated by 1 in ℂfld, that is, ℚ is the prime field of ℂfld. (Contributed by Thierry Arnoux, 15-Jan-2025.)
⊢ (ℂfld fldGen {1}) = ℚ
21.3.10.29  Ring homomorphisms - misc additions
Theorem	rhmdvd 33772	A ring homomorphism preserves ratios. (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ 𝑈 = (Unit‘𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ / = (/r‘𝑆)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → ((𝐹‘𝐴) / (𝐹‘𝐵)) = ((𝐹‘(𝐴 · 𝐶)) / (𝐹‘(𝐵 · 𝐶))))
Theorem	kerunit 33773	If a unit element lies in the kernel of a ring homomorphism, then 0 = 1, i.e. the target ring is the zero ring. (Contributed by Thierry Arnoux, 24-Oct-2017.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 1 = (1r‘𝑆)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑈 ∩ (◡𝐹 “ { 0 })) ≠ ∅) → 1 = 0 )
21.3.10.30  Scalar restriction operation
Syntax	cresv 33774	Extend class notation with the scalar restriction operation.
class ↾v
Definition	df-resv 33775*	Define an operator to restrict the scalar field component of an extended structure. (Contributed by Thierry Arnoux, 5-Sep-2018.)
⊢ ↾v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))
Theorem	reldmresv 33776	The scalar restriction is a proper operator, so it can be used with ovprc1 7456. (Contributed by Thierry Arnoux, 6-Sep-2018.)
⊢ Rel dom ↾v
Theorem	resvval 33777	Value of structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
⊢ 𝑅 = (𝑊 ↾v 𝐴)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)))
Theorem	resvid2 33778	General behavior of trivial restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
⊢ 𝑅 = (𝑊 ↾v 𝐴)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊)
Theorem	resvval2 33779	Value of nontrivial structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
⊢ 𝑅 = (𝑊 ↾v 𝐴)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩))
Theorem	resvsca 33780	Base set of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
⊢ 𝑅 = (𝑊 ↾v 𝐴)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐹 ↾s 𝐴) = (Scalar‘𝑅))
Theorem	resvlem 33781	Other elements of a scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
⊢ 𝑅 = (𝑊 ↾v 𝐴)    &   ⊢ 𝐶 = (𝐸‘𝑊)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅))
Theorem	resvbas 33782	Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
⊢ 𝐻 = (𝐺 ↾v 𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐵 = (Base‘𝐻))
Theorem	resvplusg 33783	+g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
⊢ 𝐻 = (𝐺 ↾v 𝐴)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → + = (+g‘𝐻))
Theorem	resvvsca 33784	·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Proof shortened by AV, 31-Oct-2024.)
⊢ 𝐻 = (𝐺 ↾v 𝐴)    &   ⊢ · = ( ·𝑠 ‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘𝐻))
Theorem	resvmulr 33785	.r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
⊢ 𝐻 = (𝐺 ↾v 𝐴)    &   ⊢ · = (.r‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → · = (.r‘𝐻))
Theorem	resv0g 33786	0g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
⊢ 𝐻 = (𝐺 ↾v 𝐴)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 0 = (0g‘𝐻))
Theorem	resv1r 33787	1r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
⊢ 𝐻 = (𝐺 ↾v 𝐴)    &   ⊢ 1 = (1r‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 1 = (1r‘𝐻))
Theorem	resvcmn 33788	Scalar restriction preserves commutative monoids. (Contributed by Thierry Arnoux, 6-Sep-2018.)
⊢ 𝐻 = (𝐺 ↾v 𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd))
21.3.10.31  The commutative ring of gaussian integers
Theorem	gzcrng 33789	The gaussian integers form a commutative ring. (Contributed by Thierry Arnoux, 18-Mar-2018.)
⊢ (ℂfld ↾s ℤ[i]) ∈ CRing
21.3.10.32  The archimedean ordered field of real numbers
Theorem	cnfldfld 33790	The complex numbers form a field. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ ℂfld ∈ Field
Theorem	reofld 33791	The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
⊢ ℝfld ∈ oField
Theorem	nn0omnd 33792	The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.)
⊢ (ℂfld ↾s ℕ0) ∈ oMnd
Theorem	gsumind 33793	The group sum of an indicator function of the set 𝐴 gives the size of 𝐴. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑂)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → (ℂfld Σg ((𝟭‘𝑂)‘𝐴)) = (♯‘𝐴))
Theorem	rearchi 33794	The field of the real numbers is Archimedean. See also arch 12529. (Contributed by Thierry Arnoux, 9-Apr-2018.)
⊢ ℝfld ∈ Archi
Theorem	nn0archi 33795	The monoid of the nonnegative integers is Archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
⊢ (ℂfld ↾s ℕ0) ∈ Archi
Theorem	xrge0slmod 33796	The extended nonnegative real numbers form a semiring left module. One could also have used subringAlg to get the same structure. (Contributed by Thierry Arnoux, 6-Sep-2018.)
⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))    &   ⊢ 𝑊 = (𝐺 ↾v (0[,)+∞))    ⇒   ⊢ 𝑊 ∈ SLMod
21.3.10.33  The quotient map and quotient modules
Theorem	qusker 33797*	The kernel of a quotient map. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝑉 = (Base‘𝑀)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺))    &   ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))    &   ⊢ 0 = (0g‘𝑁)    ⇒   ⊢ (𝐺 ∈ (NrmSGrp‘𝑀) → (◡𝐹 “ { 0 }) = 𝐺)
Theorem	eqgvscpbl 33798	The left coset equivalence relation is compatible with the scalar multiplication operation. (Contributed by Thierry Arnoux, 18-May-2023.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ∼ = (𝑀 ~QG 𝐺)    &   ⊢ 𝑆 = (Base‘(Scalar‘𝑀))    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑋 ∼ 𝑌 → (𝐾 · 𝑋) ∼ (𝐾 · 𝑌)))
Theorem	qusvscpbl 33799*	The quotient map distributes over the scalar multiplication. (Contributed by Thierry Arnoux, 18-May-2023.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ∼ = (𝑀 ~QG 𝐺)    &   ⊢ 𝑆 = (Base‘(Scalar‘𝑀))    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑆)    &   ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))    &   ⊢ ∙ = ( ·𝑠 ‘𝑁)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) → (𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉))))
Theorem	qusvsval 33800	Value of the scalar multiplication operation on the quotient structure. (Contributed by Thierry Arnoux, 18-May-2023.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ∼ = (𝑀 ~QG 𝐺)    &   ⊢ 𝑆 = (Base‘(Scalar‘𝑀))    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑆)    &   ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))    &   ⊢ ∙ = ( ·𝑠 ‘𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐾 ∙ [𝑋](𝑀 ~QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺))