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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | drngringd 20301 | A division ring is a ring. (Contributed by SN, 16-May-2024.) |
⊢ (𝜑 → 𝑅 ∈ DivRing) ⇒ ⊢ (𝜑 → 𝑅 ∈ Ring) | ||
Theorem | drnggrpd 20302 | A division ring is a group (deduction form). (Contributed by SN, 16-May-2024.) |
⊢ (𝜑 → 𝑅 ∈ DivRing) ⇒ ⊢ (𝜑 → 𝑅 ∈ Grp) | ||
Theorem | drnggrp 20303 | A division ring is a group (closed form). (Contributed by NM, 8-Sep-2011.) |
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp) | ||
Theorem | isfld 20304 | A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.) |
⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing)) | ||
Theorem | fldcrngd 20305 | A field is a commutative ring. (Contributed by SN, 23-Nov-2024.) |
⊢ (𝜑 → 𝑅 ∈ Field) ⇒ ⊢ (𝜑 → 𝑅 ∈ CRing) | ||
Theorem | isdrng2 20306 | A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ⇒ ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp)) | ||
Theorem | drngprop 20307 | If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.) |
⊢ (Base‘𝐾) = (Base‘𝐿) & ⊢ (+g‘𝐾) = (+g‘𝐿) & ⊢ (.r‘𝐾) = (.r‘𝐿) ⇒ ⊢ (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing) | ||
Theorem | drngmgp 20308 | A division ring contains a multiplicative group. (Contributed by NM, 8-Sep-2011.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ⇒ ⊢ (𝑅 ∈ DivRing → 𝐺 ∈ Grp) | ||
Theorem | drngmcl 20309 | The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ (𝐵 ∖ { 0 }) ∧ 𝑌 ∈ (𝐵 ∖ { 0 })) → (𝑋 · 𝑌) ∈ (𝐵 ∖ { 0 })) | ||
Theorem | drngid 20310 | A division ring's unity is the identity element of its multiplicative group. (Contributed by NM, 7-Sep-2011.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ⇒ ⊢ (𝑅 ∈ DivRing → 1 = (0g‘𝐺)) | ||
Theorem | drngunz 20311 | A division ring's unity is different from its zero. (Contributed by NM, 8-Sep-2011.) |
⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → 1 ≠ 0 ) | ||
Theorem | drngnzr 20312 | All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing) | ||
Theorem | drngid2 20313 | Properties showing that an element 𝐼 is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ 1 = 𝐼)) | ||
Theorem | drnginvrcl 20314 | Closure of the multiplicative inverse in a division ring. (reccl 11866 analog). (Contributed by NM, 19-Apr-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐵) | ||
Theorem | drnginvrn0 20315 | The multiplicative inverse in a division ring is nonzero. (recne0 11872 analog). (Contributed by NM, 19-Apr-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ≠ 0 ) | ||
Theorem | drnginvrcld 20316 | Closure of the multiplicative inverse in a division ring. (reccld 11970 analog). (Contributed by SN, 14-Aug-2024.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝐼‘𝑋) ∈ 𝐵) | ||
Theorem | drnginvrl 20317 | Property of the multiplicative inverse in a division ring. (recid2 11874 analog). (Contributed by NM, 19-Apr-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ((𝐼‘𝑋) · 𝑋) = 1 ) | ||
Theorem | drnginvrr 20318 | Property of the multiplicative inverse in a division ring. (recid 11873 analog). (Contributed by NM, 19-Apr-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝑋 · (𝐼‘𝑋)) = 1 ) | ||
Theorem | drnginvrld 20319 | Property of the multiplicative inverse in a division ring. (recid2d 11973 analog). (Contributed by SN, 14-Aug-2024.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 0 ) ⇒ ⊢ (𝜑 → ((𝐼‘𝑋) · 𝑋) = 1 ) | ||
Theorem | drnginvrrd 20320 | Property of the multiplicative inverse in a division ring. (recidd 11972 analog). (Contributed by SN, 14-Aug-2024.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝑋 · (𝐼‘𝑋)) = 1 ) | ||
Theorem | drngmul0or 20321 | A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 ))) | ||
Theorem | drngmulne0 20322 | A product is nonzero iff both its factors are nonzero. (Contributed by NM, 18-Oct-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) ≠ 0 ↔ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ))) | ||
Theorem | drngmuleq0 20323 | An element is zero iff its product with a nonzero element is zero. (Contributed by NM, 8-Oct-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ≠ 0 ) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑋 = 0 )) | ||
Theorem | opprdrng 20324 | The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.) |
⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing ↔ 𝑂 ∈ DivRing) | ||
Theorem | isdrngd 20325* | Properties that characterize a division ring among rings: it should be nonzero, have no nonzero zero-divisors, and every nonzero element 𝑥 should have a left-inverse 𝐼(𝑥). See isdrngrd 20326 for the characterization using right-inverses. (Contributed by NM, 2-Aug-2013.) Remove hypothesis. (Revised by SN, 19-Feb-2025.) |
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → 0 = (0g‘𝑅)) & ⊢ (𝜑 → 1 = (1r‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 ) & ⊢ (𝜑 → 1 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝐼 · 𝑥) = 1 ) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
Theorem | isdrngrd 20326* | Properties that characterize a division ring among rings: it should be nonzero, have no nonzero zero-divisors, and every nonzero element 𝑥 should have a right-inverse 𝐼(𝑥). See isdrngd 20325 for the characterization using left-inverses. (Contributed by NM, 10-Aug-2013.) Remove hypothesis. (Revised by SN, 19-Feb-2025.) |
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → 0 = (0g‘𝑅)) & ⊢ (𝜑 → 1 = (1r‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 ) & ⊢ (𝜑 → 1 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝑥 · 𝐼) = 1 ) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
Theorem | isdrngdOLD 20327* | Obsolete version of isdrngd 20325 as of 19-Feb-2025. (Contributed by NM, 2-Aug-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → 0 = (0g‘𝑅)) & ⊢ (𝜑 → 1 = (1r‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 ) & ⊢ (𝜑 → 1 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝐼 · 𝑥) = 1 ) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
Theorem | isdrngrdOLD 20328* | Obsolete version of isdrngrd 20326 as of 19-Feb-2025. (Contributed by NM, 10-Aug-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → 0 = (0g‘𝑅)) & ⊢ (𝜑 → 1 = (1r‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 ) & ⊢ (𝜑 → 1 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝑥 · 𝐼) = 1 ) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
Theorem | drngpropd 20329* | If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing)) | ||
Theorem | fldpropd 20330* | If two structures have the same group components (properties), one is a field iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ Field ↔ 𝐿 ∈ Field)) | ||
Theorem | rng1nnzr 20331 | The (smallest) structure representing a zero ring is not a nonzero ring. (Contributed by AV, 29-Apr-2019.) |
⊢ 𝑀 = {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} ⇒ ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∉ NzRing) | ||
Theorem | ring1zr 20332 | The only (unital) ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 7-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ ∗ = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) | ||
Theorem | rngen1zr 20333 | The only (unital) ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ ∗ = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 ≈ 1o ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) | ||
Theorem | ringen1zr 20334 | The only unital ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 15-Feb-2010.) (Revised by AV, 25-Jan-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ ∗ = (.r‘𝑅) & ⊢ 𝑍 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1o ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) | ||
Theorem | rng1nfld 20335 | The zero ring is not a field. (Contributed by AV, 29-Apr-2019.) |
⊢ 𝑀 = {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} ⇒ ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∉ Field) | ||
Syntax | csubrg 20336 | Extend class notation with all subrings of a ring. |
class SubRing | ||
Syntax | crgspn 20337 | Extend class notation with span of a set of elements over a ring. |
class RingSpan | ||
Definition | df-subrg 20338* |
Define a subring of a ring as a set of elements that is a ring in its
own right and contains the multiplicative identity.
The additional constraint is necessary because the multiplicative identity of a ring, unlike the additive identity of a ring/group or the multiplicative identity of a field, cannot be identified by a local property. Thus, it is possible for a subset of a ring to be a ring while not containing the true identity if it contains a false identity. For instance, the subset (ℤ × {0}) of (ℤ × ℤ) (where multiplication is componentwise) contains the false identity 〈1, 0〉 which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)}) | ||
Definition | df-rgspn 20339* | The ring-span of a set of elements in a ring is the smallest subring which contains all of them. (Contributed by Stefan O'Rear, 7-Dec-2014.) |
⊢ RingSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠 ⊆ 𝑡})) | ||
Theorem | issubrg 20340 | The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Proof shortened by AV, 12-Oct-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴))) | ||
Theorem | subrgss 20341 | A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ 𝐵) | ||
Theorem | subrgid 20342 | Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) | ||
Theorem | subrgring 20343 | A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring) | ||
Theorem | subrgcrng 20344 | A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ CRing) | ||
Theorem | subrgrcl 20345 | Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.) |
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | ||
Theorem | subrgsubg 20346 | A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.) |
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) | ||
Theorem | subrg0 20347 | A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g‘𝑆)) | ||
Theorem | subrg1cl 20348 | A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 ∈ 𝐴) | ||
Theorem | subrgbas 20349 | Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) | ||
Theorem | subrg1 20350 | A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 = (1r‘𝑆)) | ||
Theorem | subrgacl 20351 | A subring is closed under addition. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ + = (+g‘𝑅) ⇒ ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 + 𝑌) ∈ 𝐴) | ||
Theorem | subrgmcl 20352 | A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴) | ||
Theorem | subrgsubm 20353 | A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubMnd‘𝑀)) | ||
Theorem | subrgdvds 20354 | If an element divides another in a subring, then it also divides the other in the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 𝐸 = (∥r‘𝑆) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ ) | ||
Theorem | subrguss 20355 | A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑉 = (Unit‘𝑆) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈) | ||
Theorem | subrginv 20356 | A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ 𝑈 = (Unit‘𝑆) & ⊢ 𝐽 = (invr‘𝑆) ⇒ ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) = (𝐽‘𝑋)) | ||
Theorem | subrgdv 20357 | A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ / = (/r‘𝑅) & ⊢ 𝑈 = (Unit‘𝑆) & ⊢ 𝐸 = (/r‘𝑆) ⇒ ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋𝐸𝑌)) | ||
Theorem | subrgunit 20358 | An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑉 = (Unit‘𝑆) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴))) | ||
Theorem | subrgugrp 20359 | The units of a subring form a subgroup of the unit group of the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑉 = (Unit‘𝑆) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺)) | ||
Theorem | issubrg2 20360* | Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴))) | ||
Theorem | opprsubrg 20361 | Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.) |
⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (SubRing‘𝑅) = (SubRing‘𝑂) | ||
Theorem | subrgnzr 20362 | A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ NzRing) | ||
Theorem | subrgint 20363 | The intersection of a nonempty collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.) |
⊢ ((𝑆 ⊆ (SubRing‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubRing‘𝑅)) | ||
Theorem | subrgin 20364 | The intersection of two subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.) |
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑅)) → (𝐴 ∩ 𝐵) ∈ (SubRing‘𝑅)) | ||
Theorem | subrgmre 20365 | The subrings of a ring are a Moore system. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (Moore‘𝐵)) | ||
Theorem | issubdrg 20366* | Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑆 ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴)) | ||
Theorem | subsubrg 20367 | A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐵 ∈ (SubRing‘𝑆) ↔ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴))) | ||
Theorem | subsubrg2 20368 | The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → (SubRing‘𝑆) = ((SubRing‘𝑅) ∩ 𝒫 𝐴)) | ||
Theorem | issubrg3 20369 | A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.) |
⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝑆 ∈ (SubRing‘𝑅) ↔ (𝑆 ∈ (SubGrp‘𝑅) ∧ 𝑆 ∈ (SubMnd‘𝑀)))) | ||
Theorem | resrhm 20370 | Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.) |
⊢ 𝑈 = (𝑆 ↾s 𝑋) ⇒ ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 RingHom 𝑇)) | ||
Theorem | resrhm2b 20371 | Restriction of the codomain of a (ring) homomorphism. resghm2b 19095 analog. (Contributed by SN, 7-Feb-2025.) |
⊢ 𝑈 = (𝑇 ↾s 𝑋) ⇒ ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ 𝐹 ∈ (𝑆 RingHom 𝑈))) | ||
Theorem | rhmeql 20372 | The equalizer of two ring homomorphisms is a subring. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubRing‘𝑆)) | ||
Theorem | rhmima 20373 | The homomorphic image of a subring is a subring. (Contributed by Stefan O'Rear, 10-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubRing‘𝑁)) | ||
Theorem | rnrhmsubrg 20374 | The range of a ring homomorphism is a subring. (Contributed by SN, 18-Nov-2023.) |
⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → ran 𝐹 ∈ (SubRing‘𝑁)) | ||
Theorem | cntzsubr 20375 | Centralizers in a ring are subrings. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubRing‘𝑅)) | ||
Theorem | pwsdiagrhm 20376* | Diagonal homomorphism into a structure power (Rings). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 RingHom 𝑌)) | ||
Theorem | subrgpropd 20377* | If two structures have the same group components (properties), they have the same set of subrings. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (SubRing‘𝐾) = (SubRing‘𝐿)) | ||
Theorem | rhmpropd 20378* | Ring homomorphism depends only on the ring attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐽)) & ⊢ (𝜑 → 𝐶 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐶 = (Base‘𝑀)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐽)𝑦) = (𝑥(.r‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝑀)𝑦)) ⇒ ⊢ (𝜑 → (𝐽 RingHom 𝐾) = (𝐿 RingHom 𝑀)) | ||
Syntax | csdrg 20379 | Syntax for subfields (sub-division-rings). |
class SubDRing | ||
Definition | df-sdrg 20380* | Define the function associating with a ring the set of its sub-division-rings. A sub-division-ring of a ring is a subset of its base set which is a division ring when equipped with the induced structure (sum, multiplication, zero, and unity). If a ring is commutative (resp., a field), then its sub-division-rings are commutative (resp., are fields) (fldsdrgfld 20391), so we do not make a specific definition for subfields. (Contributed by Stefan O'Rear, 3-Oct-2015.) TODO: extend this definition to a function with domain V or at least Ring and not only DivRing. |
⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing}) | ||
Theorem | issdrg 20381 | Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.) |
⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)) | ||
Theorem | sdrgrcl 20382 | Reverse closure for a sub-division-ring predicate. (Contributed by SN, 19-Feb-2025.) |
⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing) | ||
Theorem | sdrgdrng 20383 | A sub-division-ring is a division ring. (Contributed by SN, 19-Feb-2025.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑆 ∈ DivRing) | ||
Theorem | sdrgsubrg 20384 | A sub-division-ring is a subring. (Contributed by SN, 19-Feb-2025.) |
⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 ∈ (SubRing‘𝑅)) | ||
Theorem | sdrgid 20385 | Every division ring is a division subring of itself. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → 𝐵 ∈ (SubDRing‘𝑅)) | ||
Theorem | sdrgss 20386 | A division subring is a subset of the base set. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑆 ∈ (SubDRing‘𝑅) → 𝑆 ⊆ 𝐵) | ||
Theorem | sdrgbas 20387 | Base set of a sub-division-ring structure. (Contributed by SN, 19-Feb-2025.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 = (Base‘𝑆)) | ||
Theorem | issdrg2 20388* | Property of a division subring (closure version). (Contributed by Mario Carneiro, 3-Oct-2015.) |
⊢ 𝐼 = (invr‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ ∀𝑥 ∈ (𝑆 ∖ { 0 })(𝐼‘𝑥) ∈ 𝑆)) | ||
Theorem | sdrgunit 20389 | A unit of a sub-division-ring is a nonzero element of the subring. (Contributed by SN, 19-Feb-2025.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑈 = (Unit‘𝑆) ⇒ ⊢ (𝐴 ∈ (SubDRing‘𝑅) → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ 0 ))) | ||
Theorem | imadrhmcl 20390 | The image of a (nontrivial) division ring homomorphism is a division ring. (Contributed by SN, 17-Feb-2025.) |
⊢ 𝑅 = (𝑁 ↾s (𝐹 “ 𝑆)) & ⊢ 0 = (0g‘𝑁) & ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁)) & ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝑀)) & ⊢ (𝜑 → ran 𝐹 ≠ { 0 }) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
Theorem | fldsdrgfld 20391 | A sub-division-ring of a field is itself a field, so it is a subfield. We can therefore use SubDRing to express subfields. (Contributed by Thierry Arnoux, 11-Jan-2025.) |
⊢ ((𝐹 ∈ Field ∧ 𝐴 ∈ (SubDRing‘𝐹)) → (𝐹 ↾s 𝐴) ∈ Field) | ||
Theorem | acsfn1p 20392* | Construction of a closure rule from a one-parameter partial operation. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎 ∩ 𝑌)𝐸 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
Theorem | subrgacs 20393 | Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵)) | ||
Theorem | sdrgacs 20394 | Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) ∈ (ACS‘𝐵)) | ||
Theorem | cntzsdrg 20395 | Centralizers in division rings/fields are subfields. (Contributed by Mario Carneiro, 3-Oct-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubDRing‘𝑅)) | ||
Theorem | subdrgint 20396* | The intersection of a nonempty collection of sub division rings is a sub division ring. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
⊢ 𝐿 = (𝑅 ↾s ∩ 𝑆) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝑆 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑅 ↾s 𝑠) ∈ DivRing) ⇒ ⊢ (𝜑 → 𝐿 ∈ DivRing) | ||
Theorem | sdrgint 20397 | The intersection of a nonempty collection of sub division rings is a sub division ring. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ⊆ (SubDRing‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubDRing‘𝑅)) | ||
Theorem | primefld 20398 | The smallest sub division ring of a division ring, here named 𝑃, is a field, called the Prime Field of 𝑅. (Suggested by GL, 4-Aug-2023.) (Contributed by Thierry Arnoux, 21-Aug-2023.) |
⊢ 𝑃 = (𝑅 ↾s ∩ (SubDRing‘𝑅)) ⇒ ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Field) | ||
Theorem | primefld0cl 20399 | The prime field contains the zero element of the division ring. (Contributed by Thierry Arnoux, 22-Aug-2023.) |
⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → 0 ∈ ∩ (SubDRing‘𝑅)) | ||
Theorem | primefld1cl 20400 | The prime field contains the unity element of the division ring. (Contributed by Thierry Arnoux, 22-Aug-2023.) |
⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → 1 ∈ ∩ (SubDRing‘𝑅)) |
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