P. 213 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21201-21300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rlmplusg 21201	Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (+g‘𝑅) = (+g‘(ringLMod‘𝑅))
Theorem	rlm0 21202	Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ (0g‘𝑅) = (0g‘(ringLMod‘𝑅))
Theorem	rlmsub 21203	Subtraction in the ring module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
⊢ (-g‘𝑅) = (-g‘(ringLMod‘𝑅))
Theorem	rlmmulr 21204	Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (.r‘𝑅) = (.r‘(ringLMod‘𝑅))
Theorem	rlmsca 21205	Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
⊢ (𝑅 ∈ 𝑋 → 𝑅 = (Scalar‘(ringLMod‘𝑅)))
Theorem	rlmsca2 21206	Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅))
Theorem	rlmvsca 21207	Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅))
Theorem	rlmtopn 21208	Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅))
Theorem	rlmds 21209	Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (dist‘𝑅) = (dist‘(ringLMod‘𝑅))
Theorem	rlmlmod 21210	The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)
Theorem	rlmlvec 21211	The ring module over a division ring is a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑅 ∈ DivRing → (ringLMod‘𝑅) ∈ LVec)
Theorem	rlmlsm 21212	Subgroup sum of the ring module. (Contributed by Thierry Arnoux, 9-Apr-2024.)
⊢ (𝑅 ∈ 𝑉 → (LSSum‘𝑅) = (LSSum‘(ringLMod‘𝑅)))
Theorem	rlmvneg 21213	Vector negation in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 5-Jun-2015.)
⊢ (invg‘𝑅) = (invg‘(ringLMod‘𝑅))
Theorem	rlmscaf 21214	Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (+𝑓‘(mulGrp‘𝑅)) = ( ·sf ‘(ringLMod‘𝑅))
Theorem	ixpsnbasval 21215*	The value of an infinite Cartesian product of the base of a left module over a ring with a singleton. (Contributed by AV, 3-Dec-2018.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → X𝑥 ∈ {𝑋} (Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ (Base‘𝑅))})
10.7.2  Left ideals and spans Remark: Usually, (left) ideals are defined as a subset of a (unital or non-unital) ring that is a subgroup of the additive group of the ring that "absorbs multiplication from the left by elements of the ring", see Wikipedia https://en.wikipedia.org/wiki/Ideal_(ring_theory) (19.02.2025), or the definition 4 in [BourbakiAlg1] p. 103 and the definition in [Lang] p.86, although a ring is to be considered unital (and commutative!) here, see definition 1 in [BourbakiAlg1] p. 96 resp. the definition in [Lang] p. 83, or definition in [Roman] p. 20. In contrast, the definition of (LIdeal‘𝑅), does not require the subset to be a subgroup of the additive group, as can be seen by islidl 21225. If 𝑅 is a unital ring, however, it can be proven that each ideal in (LIdeal‘𝑅) is a subgroup of the additive group of the ring, see lidlsubg 21233. This is not possible for arbitrary non-unital rings, because the proof uses the existence of the ring unity.
Syntax	clidl 21216	Ring left-ideal function.
class LIdeal
Syntax	crsp 21217	Ring span function.
class RSpan
Definition	df-lidl 21218	Define the class of left ideals of a given ring. An ideal is a submodule of the ring viewed as a module over itself. For the usual textbook definition of a (left) ideal of a ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring, see dflidl2rng 21228 and dflidl2 21237. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ LIdeal = (LSubSp ∘ ringLMod)
Definition	df-rsp 21219	Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ RSpan = (LSpan ∘ ringLMod)
Theorem	lidlval 21220	Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ (LIdeal‘𝑊) = (LSubSp‘(ringLMod‘𝑊))
Theorem	rspval 21221	Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ (RSpan‘𝑊) = (LSpan‘(ringLMod‘𝑊))
Theorem	lidlss 21222	An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐼 = (LIdeal‘𝑊)    ⇒   ⊢ (𝑈 ∈ 𝐼 → 𝑈 ⊆ 𝐵)
Theorem	lidlssbas 21223	The base set of the restriction of the ring to a (left) ideal is a subset of the base set of the ring. (Contributed by AV, 17-Feb-2020.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    ⇒   ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅))
Theorem	lidlbas 21224	A (left) ideal of a ring is the base set of the restriction of the ring to this ideal. (Contributed by AV, 17-Feb-2020.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    ⇒   ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) = 𝑈)
Theorem	islidl 21225*	Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝐼 ∈ 𝑈 ↔ (𝐼 ⊆ 𝐵 ∧ 𝐼 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐼 ((𝑥 · 𝑎) + 𝑏) ∈ 𝐼))
Theorem	rnglidlmcl 21226	A (left) ideal containing the zero element is closed under left-multiplication by elements of the full non-unital ring. If the ring is not a unital ring, and the ideal does not contain the zero element of the ring, then the closure cannot be proven as in lidlmcl 21235. (Contributed by AV, 18-Feb-2025.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Rng ∧ 𝐼 ∈ 𝑈 ∧ 0 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → (𝑋 · 𝑌) ∈ 𝐼)
Theorem	rngridlmcl 21227	A right ideal (which is a left ideal over the opposite ring) containing the zero element is closed under right-multiplication by elements of the full non-unital ring. (Contributed by AV, 19-Feb-2025.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))    ⇒   ⊢ (((𝑅 ∈ Rng ∧ 𝐼 ∈ 𝑈 ∧ 0 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → (𝑌 · 𝑋) ∈ 𝐼)
Theorem	dflidl2rng 21228*	Alternate (the usual textbook) definition of a (left) ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥 · 𝑦) ∈ 𝐼))
Theorem	isridlrng 21229*	A right ideal is a left ideal of the opposite non-unital ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼))
Theorem	lidl0cl 21230	An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 0 ∈ 𝐼)
Theorem	lidlacl 21231	An ideal is closed under addition. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋 + 𝑌) ∈ 𝐼)
Theorem	lidlnegcl 21232	An ideal contains negatives. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → (𝑁‘𝑋) ∈ 𝐼)
Theorem	lidlsubg 21233	An ideal is a subgroup of the additive group. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 𝐼 ∈ (SubGrp‘𝑅))
Theorem	lidlsubcl 21234	An ideal is closed under subtraction. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ − = (-g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋 − 𝑌) ∈ 𝐼)
Theorem	lidlmcl 21235	An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Proof shortened by AV, 31-Mar-2025.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → (𝑋 · 𝑌) ∈ 𝐼)
Theorem	lidl1el 21236	An ideal contains 1 iff it is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ( 1 ∈ 𝐼 ↔ 𝐼 = 𝐵))
Theorem	dflidl2 21237*	Alternate (the usual textbook) definition of a (left) ideal of a ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥 · 𝑦) ∈ 𝐼)))
Theorem	lidl0ALT 21238	Alternate proof for lidl0 21240 not using rnglidl0 21239: Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → { 0 } ∈ 𝑈)
Theorem	rnglidl0 21239	Every non-unital ring contains a zero ideal. (Contributed by AV, 19-Feb-2025.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng → { 0 } ∈ 𝑈)
Theorem	lidl0 21240	Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Proof shortened by AV, 18-Apr-2025.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → { 0 } ∈ 𝑈)
Theorem	lidl1ALT 21241	Alternate proof for lidl1 21243 not using rnglidl1 21242: Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐵 ∈ 𝑈)
Theorem	rnglidl1 21242	The base set of every non-unital ring is an ideal. For unital rings, such ideals are called "unit ideals", see lidl1 21243. (Contributed by AV, 19-Feb-2025.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng → 𝐵 ∈ 𝑈)
Theorem	lidl1 21243	Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Proof shortened by AV, 18-Apr-2025.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐵 ∈ 𝑈)
Theorem	lidlacs 21244	The ideal system is an algebraic closure system on the base set. (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐼 = (LIdeal‘𝑊)    ⇒   ⊢ (𝑊 ∈ Ring → 𝐼 ∈ (ACS‘𝐵))
Theorem	rspcl 21245	The span of a set of ring elements is an ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) ∈ 𝑈)
Theorem	rspssid 21246	The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ⊆ 𝐵) → 𝐺 ⊆ (𝐾‘𝐺))
Theorem	rsp1 21247	The span of the identity element is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐾‘{ 1 }) = 𝐵)
Theorem	rsp0 21248	The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐾‘{ 0 }) = { 0 })
Theorem	rspssp 21249	The ideal span of a set of elements in a ring is contained in any subring which contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ⊆ 𝐼) → (𝐾‘𝐺) ⊆ 𝐼)
Theorem	elrspsn 21250*	Membership in a principal ideal. Analogous to ellspsn 21001. (Contributed by Thierry Arnoux, 15-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝐼 ∈ (𝐾‘{𝑋}) ↔ ∃𝑥 ∈ 𝐵 𝐼 = (𝑥 · 𝑋)))
Theorem	mrcrsp 21251	Moore closure generalizes ideal span. (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝐹 = (mrCls‘𝑈)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐾 = 𝐹)
Theorem	lidlnz 21252*	A nonzero ideal contains a nonzero element. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐼 ≠ { 0 }) → ∃𝑥 ∈ 𝐼 𝑥 ≠ 0 )
Theorem	drngnidl 21253	A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → 𝑈 = {{ 0 }, 𝐵})
Theorem	lidlrsppropd 21254*	The left ideals and ring span of a ring depend only on the ring components. Here 𝑊 is expected to be either 𝐵 (when closure is available) or V (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿)))
Theorem	rnglidlmmgm 21255	The multiplicative group of a (left) ideal of a non-unital ring is a magma. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0 ∈ 𝑈 is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Mgm)
Theorem	rnglidlmsgrp 21256	The multiplicative group of a (left) ideal of a non-unital ring is a semigroup. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0 ∈ 𝑈 is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Smgrp)
Theorem	rnglidlrng 21257	A (left) ideal of a non-unital ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 𝑈 ∈ (SubGrp‘𝑅) is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Rng)
Theorem	lidlnsg 21258	An ideal is a normal subgroup. (Contributed by Thierry Arnoux, 14-Jan-2024.)
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
10.7.3  Two-sided ideals and quotient rings
Syntax	c2idl 21259	Ring two-sided ideal function.
class 2Ideal
Definition	df-2idl 21260	Define the class of two-sided ideals of a ring. A two-sided ideal is a left ideal which is also a right ideal (or a left ideal over the opposite ring). (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))))
Theorem	2idlval 21261	Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝐽 = (LIdeal‘𝑂)    &   ⊢ 𝑇 = (2Ideal‘𝑅)    ⇒   ⊢ 𝑇 = (𝐼 ∩ 𝐽)
Theorem	isridl 21262*	A right ideal is a left ideal of the opposite ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.)
⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼)))
Theorem	2idlelb 21263	Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl 21282. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝐽 = (LIdeal‘𝑂)    &   ⊢ 𝑇 = (2Ideal‘𝑅)    ⇒   ⊢ (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽))
Theorem	2idllidld 21264	A two-sided ideal is a left ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    ⇒   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
Theorem	2idlridld 21265	A two-sided ideal is a right ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑂))
Theorem	df2idl2rng 21266*	Alternate (the usual textbook) definition of a two-sided ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
⊢ 𝑈 = (2Ideal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 ((𝑥 · 𝑦) ∈ 𝐼 ∧ (𝑦 · 𝑥) ∈ 𝐼)))
Theorem	df2idl2 21267*	Alternate (the usual textbook) definition of a two-sided ideal of a ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
⊢ 𝑈 = (2Ideal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 ((𝑥 · 𝑦) ∈ 𝐼 ∧ (𝑦 · 𝑥) ∈ 𝐼))))
Theorem	ridl0 21268	Every ring contains a zero right ideal. (Contributed by AV, 13-Feb-2025.)
⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → { 0 } ∈ 𝑈)
Theorem	ridl1 21269	Every ring contains a unit right ideal. (Contributed by AV, 13-Feb-2025.)
⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐵 ∈ 𝑈)
Theorem	2idl0 21270	Every ring contains a zero two-sided ideal. (Contributed by AV, 13-Feb-2025.)
⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → { 0 } ∈ 𝐼)
Theorem	2idl1 21271	Every ring contains a unit two-sided ideal. (Contributed by AV, 13-Feb-2025.)
⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐵 ∈ 𝐼)
Theorem	2idlss 21272	A two-sided ideal is a subset of the base set. Formerly part of proof for 2idlcpbl 21282. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.) (Proof shortened by AV, 13-Mar-2025.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐼 = (2Ideal‘𝑊)    ⇒   ⊢ (𝑈 ∈ 𝐼 → 𝑈 ⊆ 𝐵)
Theorem	2idlbas 21273	The base set of a two-sided ideal as structure. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ 𝐵 = (Base‘𝐽)    ⇒   ⊢ (𝜑 → 𝐵 = 𝐼)
Theorem	2idlelbas 21274	The base set of a two-sided ideal as structure is a left and right ideal. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ 𝐵 = (Base‘𝐽)    ⇒   ⊢ (𝜑 → (𝐵 ∈ (LIdeal‘𝑅) ∧ 𝐵 ∈ (LIdeal‘(oppr‘𝑅))))
Theorem	rng2idlsubrng 21275	A two-sided ideal of a non-unital ring which is a non-unital ring is a subring of the ring. (Contributed by AV, 20-Feb-2025.) (Revised by AV, 11-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (SubRng‘𝑅))
Theorem	rng2idlnsg 21276	A two-sided ideal of a non-unital ring which is a non-unital ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
Theorem	rng2idl0 21277	The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a non-unital ring. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   ⊢ (𝜑 → (0g‘𝑅) ∈ 𝐼)
Theorem	rng2idlsubgsubrng 21278	A two-sided ideal of a non-unital ring which is a subgroup of the ring is a subring of the ring. (Contributed by AV, 11-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   ⊢ (𝜑 → 𝐼 ∈ (SubRng‘𝑅))
Theorem	rng2idlsubgnsg 21279	A two-sided ideal of a non-unital ring which is a subgroup of the ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
Theorem	rng2idlsubg0 21280	The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   ⊢ (𝜑 → (0g‘𝑅) ∈ 𝐼)
Theorem	2idlcpblrng 21281	The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) Generalization for non-unital rings and two-sided ideals which are subgroups of the additive group of the non-unital ring. (Revised by AV, 23-Feb-2025.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝐸 = (𝑅 ~QG 𝑆)    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
Theorem	2idlcpbl 21282	The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) (Proof shortened by AV, 31-Mar-2025.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝐸 = (𝑅 ~QG 𝑆)    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
Theorem	qus2idrng 21283	The quotient of a non-unital ring modulo a two-sided ideal, which is a subgroup of the additive group of the non-unital ring, is a non-unital ring (qusring 21285 analog). (Contributed by AV, 23-Feb-2025.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑈 ∈ Rng)
Theorem	qus1 21284	The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [ 1 ](𝑅 ~QG 𝑆) = (1r‘𝑈)))
Theorem	qusring 21285	If 𝑆 is a two-sided ideal in 𝑅, then 𝑈 = 𝑅 / 𝑆 is a ring, called the quotient ring of 𝑅 by 𝑆. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ Ring)
Theorem	qusrhm 21286*	If 𝑆 is a two-sided ideal in 𝑅, then the "natural map" from elements to their cosets is a ring homomorphism from 𝑅 to 𝑅 / 𝑆. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝑅 ~QG 𝑆))    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈))
Theorem	rhmpreimaidl 21287	The preimage of an ideal by a ring homomorphism is an ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.)
⊢ 𝐼 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝐼)
Theorem	kerlidl 21288	The kernel of a ring homomorphism is an ideal. (Contributed by Thierry Arnoux, 1-Jul-2024.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (◡𝐹 “ { 0 }) ∈ 𝐼)
Theorem	qusmul2idl 21289	Value of the ring operation in a quotient ring by a two-sided ideal. (Contributed by Thierry Arnoux, 1-Sep-2024.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑄)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ([𝑋](𝑅 ~QG 𝐼) × [𝑌](𝑅 ~QG 𝐼)) = [(𝑋 · 𝑌)](𝑅 ~QG 𝐼))
Theorem	crngridl 21290	In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐼 = (LIdeal‘𝑂))
Theorem	crng2idl 21291	In a commutative ring, a two-sided ideal is the same as a left ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐼 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐼 = (2Ideal‘𝑅))
Theorem	qusmulrng 21292	Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 21293. Similar to qusmul2idl 21289. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 28-Feb-2025.)
⊢ ∼ = (𝑅 ~QG 𝑆)    &   ⊢ 𝐻 = (𝑅 /s ∼ )    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝐻)    ⇒   ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ (2Ideal‘𝑅) ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )
Theorem	quscrng 21293	The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) (Proof shortened by AV, 3-Apr-2025.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ CRing)
Theorem	qusmulcrng 21294	Value of the ring operation in a quotient ring of a commutative ring. (Contributed by Thierry Arnoux, 1-Sep-2024.) (Proof shortened by metakunt, 3-Jun-2025.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑄)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ([𝑋](𝑅 ~QG 𝐼) × [𝑌](𝑅 ~QG 𝐼)) = [(𝑋 · 𝑌)](𝑅 ~QG 𝐼))
Theorem	rhmqusnsg 21295*	The mapping 𝐽 induced by a ring homomorphism 𝐹 from a subring 𝑁 of the quotient group 𝑄 over 𝐹's kernel 𝐾 is a ring homomorphism. (Contributed by Thierry Arnoux, 13-May-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ (𝜑 → 𝐺 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ⊆ 𝐾)    &   ⊢ (𝜑 → 𝑁 ∈ (LIdeal‘𝐺))    ⇒   ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻))
10.7.3.1  Condition for a non-unital ring to be unital In MathOverflow, the following theorem is claimed: "Theorem 1. Let A be a rng (= nonunital associative ring). Let J be a (two-sided) ideal of A. Assume that both rngs J and A/J are unital. Then, the rng A is also unital.", see https://mathoverflow.net/questions/487676 (/unitality-of-rngs-is-inherited-by-extensions). This thread also contains some hints to literature: Clifford and Preston's book "The Algebraic Theory of Semigroups"(Chapter 5 on representation theory), and Okninski's book Semigroup Algebras, Corollary 8 in Chapter 4. In the following, this theorem is proven formally, see rngringbdlem2 21317 (and variants rngringbd 21318 and ring2idlqusb 21320). This theorem is not trivial, since it is possible for a subset of a ring, especially a subring of a non-unital ring or (left/two-sided) ideal, to be a unital ring with a different ring unity. See also the comment for df-subrg 20570. In the given case, however, the ring unity of the larger ring can be determined by the ring unity of the two-sided ideal and a representative of the ring unity of the corresponding quotient, see ring2idlqus1 21329. An example for such a construction is given in pzriprng1ALT 21507, for the case mentioned in the comment for df-subrg 20570.
Theorem	rngqiprng1elbas 21296	The ring unity of a two-sided ideal of a non-unital ring belongs to the base set of the ring. (Contributed by AV, 15-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    ⇒   ⊢ (𝜑 → 1 ∈ 𝐵)
Theorem	rngqiprngghmlem1 21297	Lemma 1 for rngqiprngghm 21309. (Contributed by AV, 25-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → ( 1 · 𝐴) ∈ (Base‘𝐽))
Theorem	rngqiprngghmlem2 21298	Lemma 2 for rngqiprngghm 21309. (Contributed by AV, 25-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴)(+g‘𝐽)( 1 · 𝐶)) ∈ (Base‘𝐽))
Theorem	rngqiprngghmlem3 21299	Lemma 3 for rngqiprngghm 21309. (Contributed by AV, 25-Feb-2025.) (Proof shortened by AV, 24-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ( 1 · (𝐴(+g‘𝑅)𝐶)) = (( 1 · 𝐴)(+g‘𝐽)( 1 · 𝐶)))
Theorem	rngqiprngimfolem 21300	Lemma for rngqiprngimfo 21311. (Contributed by AV, 5-Mar-2025.) (Proof shortened by AV, 24-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐼 ∧ 𝐶 ∈ 𝐵) → ( 1 · ((𝐶(-g‘𝑅)( 1 · 𝐶))(+g‘𝑅)𝐴)) = 𝐴)