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	2idlss 21201	A two-sided ideal is a subset of the base set. Formerly part of proof for 2idlcpbl 21211. (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 21202	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 21203	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 21204	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 21205	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 21206	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 21207	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 21208	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 21209	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 21210	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 21211	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 21212	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 21214 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 21213	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 21214	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 21215*	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 21216	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 21217	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 21218	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 21219	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 21220	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 21221	Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 21222. Similar to qusmul2idl 21218. (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 21222	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 21223	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 21224*	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 21246 (and variants rngringbd 21247 and ring2idlqusb 21249). 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 20487. 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 21258. An example for such a construction is given in pzriprng1ALT 21435, for the case mentioned in the comment for df-subrg 20487.
Theorem	rngqiprng1elbas 21225	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 21226	Lemma 1 for rngqiprngghm 21238. (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 21227	Lemma 2 for rngqiprngghm 21238. (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 21228	Lemma 3 for rngqiprngghm 21238. (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 21229	Lemma for rngqiprngimfo 21240. (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)
Theorem	rngqiprnglinlem1 21230	Lemma 1 for rngqiprnglin 21241. (Contributed by AV, 28-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) · ( 1 · 𝐶)) = ( 1 · (𝐴 · 𝐶)))
Theorem	rngqiprnglinlem2 21231	Lemma 2 for rngqiprnglin 21241. (Contributed by AV, 28-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → [(𝐴 · 𝐶)] ∼ = ([𝐴] ∼ (.r‘𝑄)[𝐶] ∼ ))
Theorem	rngqiprnglinlem3 21232	Lemma 3 for rngqiprnglin 21241. (Contributed by AV, 28-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ([𝐴] ∼ (.r‘𝑄)[𝐶] ∼ ) ∈ (Base‘𝑄))
Theorem	rngqiprngimf1lem 21233	Lemma for rngqiprngimf1 21239. (Contributed by AV, 7-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → (([𝐴] ∼ = (0g‘𝑄) ∧ ( 1 · 𝐴) = (0g‘𝐽)) → 𝐴 = (0g‘𝑅)))
Theorem	rngqipbas 21234	The base set of the product of the quotient with a two-sided ideal and the two-sided ideal is the cartesian product of the base set of the quotient and the base set of the two-sided ideal. (Contributed by AV, 21-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    ⇒   ⊢ (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
Theorem	rngqiprng 21235	The product of the quotient with a two-sided ideal and the two-sided ideal is a non-unital ring. (Contributed by AV, 23-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    ⇒   ⊢ (𝜑 → 𝑃 ∈ Rng)
Theorem	rngqiprngimf 21236*	𝐹 is a function from (the base set of) a non-unital ring to the product of the (base set 𝐶 of the) quotient with a two-sided ideal and the (base set 𝐼 of the) two-sided ideal (because of 2idlbas 21202, (Base‘𝐽) = 𝐼!) (Contributed by AV, 21-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → 𝐹:𝐵⟶(𝐶 × 𝐼))
Theorem	rngqiprngimfv 21237*	The value of the function 𝐹 at an element of (the base set of) a non-unital ring. (Contributed by AV, 24-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩)
Theorem	rngqiprngghm 21238*	𝐹 is a homomorphism of the additive groups of non-unital rings. (Contributed by AV, 24-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑃))
Theorem	rngqiprngimf1 21239*	𝐹 is a one-to-one function from (the base set of) a non-unital ring to the product of the (base set of the) quotient with a two-sided ideal and the (base set of the) two-sided ideal. (Contributed by AV, 7-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → 𝐹:𝐵–1-1→(𝐶 × 𝐼))
Theorem	rngqiprngimfo 21240*	𝐹 is a function from (the base set of) a non-unital ring onto the product of the (base set of the) quotient with a two-sided ideal and the (base set of the) two-sided ideal. (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)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → 𝐹:𝐵–onto→(𝐶 × 𝐼))
Theorem	rngqiprnglin 21241*	𝐹 is linear with respect to the multiplication. (Contributed by AV, 28-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐹‘(𝑎 · 𝑏)) = ((𝐹‘𝑎)(.r‘𝑃)(𝐹‘𝑏)))
Theorem	rngqiprngho 21242*	𝐹 is a homomorphism of non-unital rings. (Contributed by AV, 21-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHom 𝑃))
Theorem	rngqiprngim 21243*	𝐹 is an isomorphism of non-unital rings. (Contributed by AV, 21-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngIso 𝑃))
Theorem	rng2idl1cntr 21244	The unity of a two-sided ideal of a non-unital ring is central, i.e., an element of the center of the multiplicative semigroup of the non-unital ring. This is part of the proof given in MathOverflow, which seems to be sufficient to show that 𝐹 given below (see rngqiprngimf 21236) is an isomorphism. In our proof, however we show that 𝐹 is linear regarding the multiplication (rngqiprnglin 21241) via rngqiprnglinlem1 21230 instead. (Contributed by AV, 13-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (𝜑 → 1 ∈ (Cntr‘𝑀))
Theorem	rngringbdlem1 21245	In a unital ring, the quotient of the ring and a two-sided ideal is unital. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ ((𝜑 ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
Theorem	rngringbdlem2 21246	A non-unital ring is unital if and only if there is a (two-sided) ideal of the ring which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 14-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ ((𝜑 ∧ 𝑄 ∈ Ring) → 𝑅 ∈ Ring)
Theorem	rngringbd 21247	A non-unital ring is unital if and only if there is a (two-sided) ideal of the ring which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ (𝜑 → (𝑅 ∈ Ring ↔ 𝑄 ∈ Ring))
Theorem	ring2idlqus 21248*	For every unital ring there is a (two-sided) ideal of the ring (in fact the base set of the ring itself) which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 13-Feb-2025.)
⊢ (𝑅 ∈ Ring → ∃𝑖 ∈ (2Ideal‘𝑅)((𝑅 ↾s 𝑖) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝑖)) ∈ Ring))
Theorem	ring2idlqusb 21249*	A non-unital ring is unital if and only if there is a (two-sided) ideal of the ring which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 20-Feb-2025.)
⊢ (𝑅 ∈ Rng → (𝑅 ∈ Ring ↔ ∃𝑖 ∈ (2Ideal‘𝑅)((𝑅 ↾s 𝑖) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝑖)) ∈ Ring)))
Theorem	rngqiprngfulem1 21250*	Lemma 1 for rngqiprngfu 21256 (and lemma for rngqiprngu 21257). (Contributed by AV, 16-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (1r‘𝑄) = [𝑥] ∼ )
Theorem	rngqiprngfulem2 21251	Lemma 2 for rngqiprngfu 21256 (and lemma for rngqiprngu 21257). (Contributed by AV, 16-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))    ⇒   ⊢ (𝜑 → 𝐸 ∈ 𝐵)
Theorem	rngqiprngfulem3 21252	Lemma 3 for rngqiprngfu 21256 (and lemma for rngqiprngu 21257). (Contributed by AV, 16-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    ⇒   ⊢ (𝜑 → 𝑈 ∈ 𝐵)
Theorem	rngqiprngfulem4 21253	Lemma 4 for rngqiprngfu 21256. (Contributed by AV, 16-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    ⇒   ⊢ (𝜑 → [𝑈] ∼ = [𝐸] ∼ )
Theorem	rngqiprngfulem5 21254	Lemma 5 for rngqiprngfu 21256. (Contributed by AV, 16-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    ⇒   ⊢ (𝜑 → ( 1 · 𝑈) = 1 )
Theorem	rngqipring1 21255	The ring unity of the product of the quotient with a two-sided ideal and the two-sided ideal, which both are rings. (Contributed by AV, 16-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    ⇒   ⊢ (𝜑 → (1r‘𝑃) = ⟨[𝐸] ∼ , 1 ⟩)
Theorem	rngqiprngfu 21256*	The function value of 𝐹 at the constructed expected ring unity of 𝑅 is the ring unity of the product of the quotient with the two-sided ideal and the two-sided ideal. (Contributed by AV, 16-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → (𝐹‘𝑈) = ⟨[𝐸] ∼ , 1 ⟩)
Theorem	rngqiprngu 21257	If a non-unital ring has a (two-sided) ideal which is unital, and the quotient of the ring and the ideal is also unital, then the ring is also unital with a ring unity which can be constructed from the ring unity of the ideal and a representative of the ring unity of the quotient. (Contributed by AV, 17-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    ⇒   ⊢ (𝜑 → (1r‘𝑅) = 𝑈)
Theorem	ring2idlqus1 21258	If a non-unital ring has a (two-sided) ideal which is unital, and the quotient of the ring and the ideal is also unital, then the ring is also unital with a ring unity which can be constructed from the ring unity of the ideal and a representative of the ring unity of the quotient. (Contributed by AV, 17-Mar-2025.)
⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘(𝑅 ↾s 𝐼))    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Rng ∧ 𝐼 ∈ (2Ideal‘𝑅)) ∧ ((𝑅 ↾s 𝐼) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝐼)) ∈ Ring) ∧ 𝑈 ∈ (1r‘(𝑅 /s (𝑅 ~QG 𝐼)))) → (𝑅 ∈ Ring ∧ (1r‘𝑅) = ((𝑈 − ( 1 · 𝑈)) + 1 )))
10.7.4  Principal ideal rings. Divisibility in the integers
Syntax	clpidl 21259	Ring left-principal-ideal function.
class LPIdeal
Syntax	clpir 21260	Class of left principal ideal rings.
class LPIR
Definition	df-lpidl 21261*	Define the class of left principal ideals of a ring, which are ideals with a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ LPIdeal = (𝑤 ∈ Ring ↦ ∪ 𝑔 ∈ (Base‘𝑤){((RSpan‘𝑤)‘{𝑔})})
Definition	df-lpir 21262	Define the class of left principal ideal rings, rings where every left ideal has a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}
Theorem	lpival 21263*	Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑃 = ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})})
Theorem	islpidl 21264*	Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑃 ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔})))
Theorem	lpi0 21265	The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → { 0 } ∈ 𝑃)
Theorem	lpi1 21266	The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐵 ∈ 𝑃)
Theorem	islpir 21267	Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃))
Theorem	lpiss 21268	Principal ideals are a subclass of ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑃 ⊆ 𝑈)
Theorem	islpir2 21269	Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 ⊆ 𝑃))
Theorem	lpirring 21270	Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝑅 ∈ LPIR → 𝑅 ∈ Ring)
Theorem	drnglpir 21271	Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ LPIR)
Theorem	rspsn 21272*	Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥})
Theorem	lidldvgen 21273*	An element generates an ideal iff it is contained in the ideal and all elements are right-divided by it. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → (𝐼 = (𝐾‘{𝐺}) ↔ (𝐺 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 𝐺 ∥ 𝑥)))
Theorem	lpigen 21274*	An ideal is principal iff it contains an element which right-divides all elements. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝐼 ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 𝑥 ∥ 𝑦))
10.7.5  Principal ideal domains
Syntax	cpid 21275	Class of principal ideal domains.
class PID
Definition	df-pid 21276	A principal ideal domain is an integral domain satisfying the left principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ PID = (IDomn ∩ LPIR)
10.8  The complex numbers as an algebraic extensible structure
10.8.1  Definition and basic properties
Syntax	cpsmet 21277	Extend class notation with the class of all pseudometric spaces.
class PsMet
Syntax	cxmet 21278	Extend class notation with the class of all extended metric spaces.
class ∞Met
Syntax	cmet 21279	Extend class notation with the class of all metrics.
class Met
Syntax	cbl 21280	Extend class notation with the metric space ball function.
class ball
Syntax	cfbas 21281	Extend class definition to include the class of filter bases.
class fBas
Syntax	cfg 21282	Extend class definition to include the filter generating function.
class filGen
Syntax	cmopn 21283	Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen
Syntax	cmetu 21284	Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif
Definition	df-psmet 21285*	Define the set of all pseudometrics on a given base set. In a pseudo metric, two distinct points may have a distance zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
Definition	df-xmet 21286*	Define the set of all extended metrics on a given base set. The definition is similar to df-met 21287, but we also allow the metric to take on the value +∞. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
Definition	df-met 21287*	Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 24237. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. The 4 properties in Gleason's definition are shown by met0 24259, metgt0 24275, metsym 24266, and mettri 24268. (Contributed by NM, 25-Aug-2006.)
⊢ Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
Definition	df-bl 21288*	Define the metric space ball function. See blval 24302 for its value. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)
⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
Definition	df-mopn 21289	Define a function whose value is the family of open sets of a metric space. See elmopn 24358 for its main property. (Contributed by NM, 1-Sep-2006.)
⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
Definition	df-fbas 21290*	Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
⊢ fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧)) ≠ ∅)})
Definition	df-fg 21291*	Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
⊢ filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})
Definition	df-metu 21292*	Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
⊢ metUnif = (𝑑 ∈ ∪ ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎)))))
Syntax	ccnfld 21293	Extend class notation with the field of complex numbers.
class ℂfld
Definition	df-cnfld 21294*	The field of complex numbers. Other number fields and rings can be constructed by applying the ↾s restriction operator, for instance (ℂfld ↾ 𝔸) is the field of algebraic numbers. The contract of this set is defined entirely by cnfldex 21296, cnfldadd 21299, cnfldmul 21301, cnfldcj 21302, cnfldtset 21303, cnfldle 21304, cnfldds 21305, and cnfldbas 21297. We may add additional members to this in the future. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) Use maps-to notation for addition and multiplication. (Revised by GG, 31-Mar-2025.) (New usage is discouraged.)
⊢ ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
Theorem	cnfldstr 21295	The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21294. (Revised by GG, 31-Mar-2025.)
⊢ ℂfld Struct ⟨1, ;13⟩
Theorem	cnfldex 21296	The field of complex numbers is a set. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Avoid complex number axioms and ax-pow 5305. (Revised by GG, 16-Mar-2025.)
⊢ ℂfld ∈ V
Theorem	cnfldbas 21297	The base set of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21294. (Revised by GG, 31-Mar-2025.)
⊢ ℂ = (Base‘ℂfld)
Theorem	mpocnfldadd 21298*	The addition operation of the field of complex numbers. Version of cnfldadd 21299 using maps-to notation, which does not require ax-addf 11092. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (+g‘ℂfld)
Theorem	cnfldadd 21299	The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21294. (Revised by GG, 27-Apr-2025.)
⊢ + = (+g‘ℂfld)
Theorem	mpocnfldmul 21300*	The multiplication operation of the field of complex numbers. Version of cnfldmul 21301 using maps-to notation, which does not require ax-mulf 11093. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (.r‘ℂfld)