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Type | Label | Description |
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Statement | ||
Theorem | qus1 21301 | 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 21302 | 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 21303* | 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 21304 | 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 21305 | 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 21306 | 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 21307 | 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 21308 | 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 21309 | Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 21310. Similar to qusmul2idl 21306. (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 21310 | 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 21311 | 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 21312* | 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)) | ||
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 21334 (and variants rngringbd 21335 and ring2idlqusb 21337). 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 20586. 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 21346. An example for such a construction is given in pzriprng1ALT 21524, for the case mentioned in the comment for df-subrg 20586. | ||
Theorem | rngqiprng1elbas 21313 | 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 21314 | Lemma 1 for rngqiprngghm 21326. (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 21315 | Lemma 2 for rngqiprngghm 21326. (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 21316 | Lemma 3 for rngqiprngghm 21326. (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 21317 | Lemma for rngqiprngimfo 21328. (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 21318 | Lemma 1 for rngqiprnglin 21329. (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 21319 | Lemma 2 for rngqiprnglin 21329. (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 21320 | Lemma 3 for rngqiprnglin 21329. (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 21321 | Lemma for rngqiprngimf1 21327. (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 21322 | 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 21323 | 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 21324* | 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 21290, (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 21325* | 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 21326* | 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 21327* | 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 21328* | 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 21329* | 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 21330* | 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 21331* | 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 21332 | 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 21324) is an isomorphism. In our proof, however we show that 𝐹 is linear regarding the multiplication (rngqiprnglin 21329) via rngqiprnglinlem1 21318 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 21333 | 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 21334 | 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 21335 | 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 21336* | 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 21337* | 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 21338* | Lemma 1 for rngqiprngfu 21344 (and lemma for rngqiprngu 21345). (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 21339 | Lemma 2 for rngqiprngfu 21344 (and lemma for rngqiprngu 21345). (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 21340 | Lemma 3 for rngqiprngfu 21344 (and lemma for rngqiprngu 21345). (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 21341 | Lemma 4 for rngqiprngfu 21344. (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 21342 | Lemma 5 for rngqiprngfu 21344. (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 21343 | 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 21344* | 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 21345 | 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 21346 | 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 ))) | ||
Syntax | clpidl 21347 | Ring left-principal-ideal function. |
class LPIdeal | ||
Syntax | clpir 21348 | Class of left principal ideal rings. |
class LPIR | ||
Definition | df-lpidl 21349* | 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 21350 | 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 21351* | 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 21352* | 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 21353 | 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 21354 | 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 21355 | 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 21356 | 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 21357 | 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 21358 | Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ (𝑅 ∈ LPIR → 𝑅 ∈ Ring) | ||
Theorem | drnglpir 21359 | Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ LPIR) | ||
Theorem | rspsn 21360* | 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 21361* | 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 21362* | 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)) | ||
Syntax | cpid 21363 | Class of principal ideal domains. |
class PID | ||
Definition | df-pid 21364 | 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) | ||
Syntax | cpsmet 21365 | Extend class notation with the class of all pseudometric spaces. |
class PsMet | ||
Syntax | cxmet 21366 | Extend class notation with the class of all extended metric spaces. |
class ∞Met | ||
Syntax | cmet 21367 | Extend class notation with the class of all metrics. |
class Met | ||
Syntax | cbl 21368 | Extend class notation with the metric space ball function. |
class ball | ||
Syntax | cfbas 21369 | Extend class definition to include the class of filter bases. |
class fBas | ||
Syntax | cfg 21370 | Extend class definition to include the filter generating function. |
class filGen | ||
Syntax | cmopn 21371 | Extend class notation with a function mapping each metric space to the family of its open sets. |
class MetOpen | ||
Syntax | cmetu 21372 | Extend class notation with the function mapping metrics to the uniform structure generated by that metric. |
class metUnif | ||
Definition | df-psmet 21373* | 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 21374* | Define the set of all extended metrics on a given base set. The definition is similar to df-met 21375, 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 21375* | Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 24346. 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 24368, metgt0 24384, metsym 24375, and mettri 24377. (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 21376* | Define the metric space ball function. See blval 24411 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 21377 | Define a function whose value is the family of open sets of a metric space. See elmopn 24467 for its main property. (Contributed by NM, 1-Sep-2006.) |
⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))) | ||
Definition | df-fbas 21378* | 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 21379* | 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 21380* | 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 21381 | Extend class notation with the field of complex numbers. |
class ℂfld | ||
Definition | df-cnfld 21382* |
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 21384, cnfldadd 21387, cnfldmul 21389, cnfldcj 21390, cnfldtset 21391, cnfldle 21392, cnfldds 21393, and cnfldbas 21385. 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 21383 | 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 21382. (Revised by GG, 31-Mar-2025.) |
⊢ ℂfld Struct 〈1, ;13〉 | ||
Theorem | cnfldex 21384 | 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 5370. (Revised by GG, 16-Mar-2025.) |
⊢ ℂfld ∈ V | ||
Theorem | cnfldbas 21385 | 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 21382. (Revised by GG, 31-Mar-2025.) |
⊢ ℂ = (Base‘ℂfld) | ||
Theorem | mpocnfldadd 21386* | The addition operation of the field of complex numbers. Version of cnfldadd 21387 using maps-to notation, which does not require ax-addf 11231. (Contributed by GG, 31-Mar-2025.) |
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (+g‘ℂfld) | ||
Theorem | cnfldadd 21387 | 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 21382. (Revised by GG, 27-Apr-2025.) |
⊢ + = (+g‘ℂfld) | ||
Theorem | mpocnfldmul 21388* | The multiplication operation of the field of complex numbers. Version of cnfldmul 21389 using maps-to notation, which does not require ax-mulf 11232. (Contributed by GG, 31-Mar-2025.) |
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (.r‘ℂfld) | ||
Theorem | cnfldmul 21389 | The multiplication 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 21382. (Revised by GG, 27-Apr-2025.) |
⊢ · = (.r‘ℂfld) | ||
Theorem | cnfldcj 21390 | The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21382. (Revised by GG, 31-Mar-2025.) |
⊢ ∗ = (*𝑟‘ℂfld) | ||
Theorem | cnfldtset 21391 | The topology component of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21382. (Revised by GG, 31-Mar-2025.) |
⊢ (MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld) | ||
Theorem | cnfldle 21392 | The ordering of the field of complex numbers. Note that this is not actually an ordering on ℂ, but we put it in the structure anyway because restricting to ℝ does not affect this component, so that (ℂfld ↾s ℝ) is an ordered field even though ℂfld itself is not. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21382. (Revised by GG, 31-Mar-2025.) |
⊢ ≤ = (le‘ℂfld) | ||
Theorem | cnfldds 21393 | The metric of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21382. (Revised by GG, 31-Mar-2025.) |
⊢ (abs ∘ − ) = (dist‘ℂfld) | ||
Theorem | cnfldunif 21394 | The uniform structure component of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21382. (Revised by GG, 31-Mar-2025.) |
⊢ (metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld) | ||
Theorem | cnfldfun 21395 | The field of complex numbers is a function. The proof is much shorter than the proof of cnfldfunALT 21396 by using cnfldstr 21383 and structn0fun 17184: in addition, it must be shown that ∅ ∉ ℂfld. (Contributed by AV, 18-Nov-2021.) Revise df-cnfld 21382. (Revised by GG, 31-Mar-2025.) |
⊢ Fun ℂfld | ||
Theorem | cnfldfunALT 21396 | The field of complex numbers is a function. Alternate proof of cnfldfun 21395 not requiring that the index set of the components is ordered, but using quadratically many inequalities for the indices. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 11-Nov-2024.) Revise df-cnfld 21382. (Revised by GG, 31-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Fun ℂfld | ||
Theorem | dfcnfldOLD 21397 | Obsolete version of df-cnfld 21382 as of 27-Apr-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ℂfld = (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) | ||
Theorem | cnfldstrOLD 21398 | Obsolete version of cnfldstr 21383 as of 27-Apr-2025. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ℂfld Struct 〈1, ;13〉 | ||
Theorem | cnfldexOLD 21399 | Obsolete version of cnfldex 21384 as of 27-Apr-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ℂfld ∈ V | ||
Theorem | cnfldbasOLD 21400 | Obsolete version of cnfldbas 21385 as of 27-Apr-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ℂ = (Base‘ℂfld) |
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