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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | subrgpropd 19501* | 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 19502* | 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 19503 | Syntax for subfields (sub-division-rings). |
class SubDRing | ||
Definition | df-sdrg 19504* | A sub-division-ring is a subset of a division ring's set which is a division ring under the induced operation. If the overring is commutative this is a field; no special consideration is made of the fields in the center of a skew field. (Contributed by Stefan O'Rear, 3-Oct-2015.) |
⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing}) | ||
Theorem | issdrg 19505 | 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 | sdrgid 19506 | 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 19507 | 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 | issdrg2 19508* | 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 | acsfn1p 19509* | 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 19510 | Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵)) | ||
Theorem | sdrgacs 19511 | Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) ∈ (ACS‘𝐵)) | ||
Theorem | cntzsdrg 19512 | 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 19513* | 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 19514 | 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 19515 | 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 19516 | The prime field contains the neutral element of the division ring. (Contributed by Thierry Arnoux, 22-Aug-2023.) |
⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → 0 ∈ ∩ (SubDRing‘𝑅)) | ||
Theorem | primefld1cl 19517 | The prime field contains the multiplicative neutral element of the division ring. (Contributed by Thierry Arnoux, 22-Aug-2023.) |
⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → 1 ∈ ∩ (SubDRing‘𝑅)) | ||
Syntax | cabv 19518 | The set of absolute values on a ring. |
class AbsVal | ||
Definition | df-abv 19519* | Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 14585 to arbitrary rings. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ AbsVal = (𝑟 ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞) ↑m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝑟)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))}) | ||
Theorem | abvfval 19520* | Value of the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝐴 = {𝑓 ∈ ((0[,)+∞) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 (((𝑓‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))}) | ||
Theorem | isabv 19521* | Elementhood in the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐹 ∈ 𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))))) | ||
Theorem | isabvd 19522* | Properties that determine an absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 4-Dec-2014.) |
⊢ (𝜑 → 𝐴 = (AbsVal‘𝑅)) & ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → 0 = (0g‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹:𝐵⟶ℝ) & ⊢ (𝜑 → (𝐹‘ 0 ) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) → 0 < (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))) ⇒ ⊢ (𝜑 → 𝐹 ∈ 𝐴) | ||
Theorem | abvrcl 19523 | Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) | ||
Theorem | abvfge0 19524 | An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐴 → 𝐹:𝐵⟶(0[,)+∞)) | ||
Theorem | abvf 19525 | An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐴 → 𝐹:𝐵⟶ℝ) | ||
Theorem | abvcl 19526 | An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ ℝ) | ||
Theorem | abvge0 19527 | The absolute value of a number is greater than or equal to zero. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → 0 ≤ (𝐹‘𝑋)) | ||
Theorem | abveq0 19528 | The value of an absolute value is zero iff the argument is zero. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋) = 0 ↔ 𝑋 = 0 )) | ||
Theorem | abvne0 19529 | The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐹‘𝑋) ≠ 0) | ||
Theorem | abvgt0 19530 | The absolute value of a nonzero number is strictly positive. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 0 < (𝐹‘𝑋)) | ||
Theorem | abvmul 19531 | An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) · (𝐹‘𝑌))) | ||
Theorem | abvtri 19532 | An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹‘𝑋) + (𝐹‘𝑌))) | ||
Theorem | abv0 19533 | The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐴 → (𝐹‘ 0 ) = 0) | ||
Theorem | abv1z 19534 | The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) = 1) | ||
Theorem | abv1 19535 | The absolute value of one is one in a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) → (𝐹‘ 1 ) = 1) | ||
Theorem | abvneg 19536 | The absolute value of a negative is the same as that of the positive. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁‘𝑋)) = (𝐹‘𝑋)) | ||
Theorem | abvsubtri 19537 | An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ − = (-g‘𝑅) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 − 𝑌)) ≤ ((𝐹‘𝑋) + (𝐹‘𝑌))) | ||
Theorem | abvrec 19538 | The absolute value distributes under reciprocal. (Contributed by Mario Carneiro, 10-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝐹‘(𝐼‘𝑋)) = (1 / (𝐹‘𝑋))) | ||
Theorem | abvdiv 19539 | The absolute value distributes under division. (Contributed by Mario Carneiro, 10-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ / = (/r‘𝑅) ⇒ ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 0 )) → (𝐹‘(𝑋 / 𝑌)) = ((𝐹‘𝑋) / (𝐹‘𝑌))) | ||
Theorem | abvdom 19540 | Any ring with an absolute value is a domain, which is to say that it contains no zero divisors. (Contributed by Mario Carneiro, 10-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 0 )) → (𝑋 · 𝑌) ≠ 0 ) | ||
Theorem | abvres 19541 | The restriction of an absolute value to a subring is an absolute value. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝑆 = (𝑅 ↾s 𝐶) & ⊢ 𝐵 = (AbsVal‘𝑆) ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → (𝐹 ↾ 𝐶) ∈ 𝐵) | ||
Theorem | abvtrivd 19542* | The trivial absolute value. (Contributed by Mario Carneiro, 6-May-2015.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, 1)) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 ) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧 ≠ 0 )) → (𝑦 · 𝑧) ≠ 0 ) ⇒ ⊢ (𝜑 → 𝐹 ∈ 𝐴) | ||
Theorem | abvtriv 19543* | The trivial absolute value. (This theorem is true as long as 𝑅 is a domain, but it is not true for rings with zero divisors, which violate the multiplication axiom; abvdom 19540 is the converse of this remark.) (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, 1)) ⇒ ⊢ (𝑅 ∈ DivRing → 𝐹 ∈ 𝐴) | ||
Theorem | abvpropd 19544* | If two structures have the same ring components, they have the same collection of absolute values. (Contributed by Mario Carneiro, 4-Oct-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 → (AbsVal‘𝐾) = (AbsVal‘𝐿)) | ||
Syntax | cstf 19545 | Extend class notation with the functionalization of the *-ring involution. |
class *rf | ||
Syntax | csr 19546 | Extend class notation with class of all *-rings. |
class *-Ring | ||
Definition | df-staf 19547* | Define the functionalization of the involution in a star ring. This is not strictly necessary but by having *𝑟 as an actual function we can state the principal properties of an involution much more cleanly. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥))) | ||
Definition | df-srng 19548* | Define class of all star rings. A star ring is a ring with an involution (conjugation) function. Involution (unlike say the ring zero) is not unique and therefore must be added as a new component to the ring. For example, two possible involutions for complex numbers are the identity function and complex conjugation. Definition of involution in [Holland95] p. 204. (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.) |
⊢ *-Ring = {𝑓 ∣ [(*rf‘𝑓) / 𝑖](𝑖 ∈ (𝑓 RingHom (oppr‘𝑓)) ∧ 𝑖 = ◡𝑖)} | ||
Theorem | staffval 19549* | The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∗ = (*𝑟‘𝑅) & ⊢ ∙ = (*rf‘𝑅) ⇒ ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) | ||
Theorem | stafval 19550 | The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∗ = (*𝑟‘𝑅) & ⊢ ∙ = (*rf‘𝑅) ⇒ ⊢ (𝐴 ∈ 𝐵 → ( ∙ ‘𝐴) = ( ∗ ‘𝐴)) | ||
Theorem | staffn 19551 | The functionalization is equal to the original function, if it is a function on the right base set. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∗ = (*𝑟‘𝑅) & ⊢ ∙ = (*rf‘𝑅) ⇒ ⊢ ( ∗ Fn 𝐵 → ∙ = ∗ ) | ||
Theorem | issrng 19552 | The predicate "is a star ring." (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ ∗ = (*rf‘𝑅) ⇒ ⊢ (𝑅 ∈ *-Ring ↔ ( ∗ ∈ (𝑅 RingHom 𝑂) ∧ ∗ = ◡ ∗ )) | ||
Theorem | srngrhm 19553 | The involution function in a star ring is an antiautomorphism. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ ∗ = (*rf‘𝑅) ⇒ ⊢ (𝑅 ∈ *-Ring → ∗ ∈ (𝑅 RingHom 𝑂)) | ||
Theorem | srngring 19554 | A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ (𝑅 ∈ *-Ring → 𝑅 ∈ Ring) | ||
Theorem | srngcnv 19555 | The involution function in a star ring is its own inverse function. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ ∗ = (*rf‘𝑅) ⇒ ⊢ (𝑅 ∈ *-Ring → ∗ = ◡ ∗ ) | ||
Theorem | srngf1o 19556 | The involution function in a star ring is a bijection. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ ∗ = (*rf‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ *-Ring → ∗ :𝐵–1-1-onto→𝐵) | ||
Theorem | srngcl 19557 | The involution function in a star ring is closed in the ring. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ ∗ = (*𝑟‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ( ∗ ‘𝑋) ∈ 𝐵) | ||
Theorem | srngnvl 19558 | The involution function in a star ring is an involution. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ ∗ = (*𝑟‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ( ∗ ‘( ∗ ‘𝑋)) = 𝑋) | ||
Theorem | srngadd 19559 | The involution function in a star ring distributes over addition. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ ∗ = (*𝑟‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) ⇒ ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ∗ ‘(𝑋 + 𝑌)) = (( ∗ ‘𝑋) + ( ∗ ‘𝑌))) | ||
Theorem | srngmul 19560 | The involution function in a star ring distributes over multiplication, with a change in the order of the factors. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ ∗ = (*𝑟‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ∗ ‘(𝑋 · 𝑌)) = (( ∗ ‘𝑌) · ( ∗ ‘𝑋))) | ||
Theorem | srng1 19561 | The conjugate of the ring identity is the identity. (This is sometimes taken as an axiom, and indeed the proof here follows because we defined *𝑟 to be a ring homomorphism, which preserves 1; nevertheless, it is redundant, as can be seen from the proof of issrngd 19563.) (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ ∗ = (*𝑟‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ *-Ring → ( ∗ ‘ 1 ) = 1 ) | ||
Theorem | srng0 19562 | The conjugate of the ring zero is zero. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ ∗ = (*𝑟‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ *-Ring → ( ∗ ‘ 0 ) = 0 ) | ||
Theorem | issrngd 19563* | Properties that determine a star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2015.) |
⊢ (𝜑 → 𝐾 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → ∗ = (*𝑟‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ( ∗ ‘𝑥) ∈ 𝐾) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → ( ∗ ‘(𝑥 + 𝑦)) = (( ∗ ‘𝑥) + ( ∗ ‘𝑦))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → ( ∗ ‘(𝑥 · 𝑦)) = (( ∗ ‘𝑦) · ( ∗ ‘𝑥))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ( ∗ ‘( ∗ ‘𝑥)) = 𝑥) ⇒ ⊢ (𝜑 → 𝑅 ∈ *-Ring) | ||
Theorem | idsrngd 19564* | A commutative ring is a star ring when the conjugate operation is the identity. (Contributed by Thierry Arnoux, 19-Apr-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∗ = (*𝑟‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥) ⇒ ⊢ (𝜑 → 𝑅 ∈ *-Ring) | ||
Syntax | clmod 19565 | Extend class notation with class of all left modules. |
class LMod | ||
Syntax | cscaf 19566 | The functionalization of the scalar multiplication operation. |
class ·sf | ||
Definition | df-lmod 19567* | Define the class of all left modules, which are generalizations of left vector spaces. A left module over a ring is an (Abelian) group (vectors) together with a ring (scalars) and a left scalar product connecting them. (Contributed by NM, 4-Nov-2013.) |
⊢ LMod = {𝑔 ∈ Grp ∣ [(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][( ·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))} | ||
Definition | df-scaf 19568* | Define the functionalization of the ·𝑠 operator. This restricts the value of ·𝑠 to the stated domain, which is necessary when working with restricted structures, whose operations may be defined on a larger set than the true base. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ ·sf = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠 ‘𝑔)𝑦))) | ||
Theorem | islmod 19569* | The predicate "is a left module". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ⨣ = (+g‘𝐹) & ⊢ × = (.r‘𝐹) & ⊢ 1 = (1r‘𝐹) ⇒ ⊢ (𝑊 ∈ LMod ↔ (𝑊 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤)))) | ||
Theorem | lmodlema 19570 | Lemma for properties of a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ⨣ = (+g‘𝐹) & ⊢ × = (.r‘𝐹) & ⊢ 1 = (1r‘𝐹) ⇒ ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝑅 · 𝑌) ∈ 𝑉 ∧ (𝑅 · (𝑌 + 𝑋)) = ((𝑅 · 𝑌) + (𝑅 · 𝑋)) ∧ ((𝑄 ⨣ 𝑅) · 𝑌) = ((𝑄 · 𝑌) + (𝑅 · 𝑌))) ∧ (((𝑄 × 𝑅) · 𝑌) = (𝑄 · (𝑅 · 𝑌)) ∧ ( 1 · 𝑌) = 𝑌))) | ||
Theorem | islmodd 19571* | Properties that determine a left module. See note in isgrpd2 18063 regarding the 𝜑 on hypotheses that name structure components. (Contributed by Mario Carneiro, 22-Jun-2014.) |
⊢ (𝜑 → 𝑉 = (Base‘𝑊)) & ⊢ (𝜑 → + = (+g‘𝑊)) & ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊)) & ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐹)) & ⊢ (𝜑 → ⨣ = (+g‘𝐹)) & ⊢ (𝜑 → × = (.r‘𝐹)) & ⊢ (𝜑 → 1 = (1r‘𝐹)) & ⊢ (𝜑 → 𝐹 ∈ Ring) & ⊢ (𝜑 → 𝑊 ∈ Grp) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑉) → (𝑥 · 𝑦) ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 ⨣ 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 × 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 1 · 𝑥) = 𝑥) ⇒ ⊢ (𝜑 → 𝑊 ∈ LMod) | ||
Theorem | lmodgrp 19572 | A left module is a group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.) |
⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | ||
Theorem | lmodring 19573 | The scalar component of a left module is a ring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) | ||
Theorem | lmodfgrp 19574 | The scalar component of a left module is an additive group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp) | ||
Theorem | lmodbn0 19575 | The base set of a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → 𝐵 ≠ ∅) | ||
Theorem | lmodacl 19576 | Closure of ring addition for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ + = (+g‘𝐹) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) | ||
Theorem | lmodmcl 19577 | Closure of ring multiplication for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = (.r‘𝐹) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 · 𝑌) ∈ 𝐾) | ||
Theorem | lmodsn0 19578 | The set of scalars in a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ LMod → 𝐵 ≠ ∅) | ||
Theorem | lmodvacl 19579 | Closure of vector addition for a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) | ||
Theorem | lmodass 19580 | Left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) | ||
Theorem | lmodlcan 19581 | Left cancellation law for vector sum. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌)) | ||
Theorem | lmodvscl 19582 | Closure of scalar product for a left module. (hvmulcl 28718 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · 𝑋) ∈ 𝑉) | ||
Theorem | scaffval 19583* | The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ∙ = ( ·sf ‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) | ||
Theorem | scafval 19584 | The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ∙ = ( ·sf ‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∙ 𝑌) = (𝑋 · 𝑌)) | ||
Theorem | scafeq 19585 | If the scalar multiplication operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ∙ = ( ·sf ‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ( · Fn (𝐾 × 𝐵) → ∙ = · ) | ||
Theorem | scaffn 19586 | The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ∙ = ( ·sf ‘𝑊) ⇒ ⊢ ∙ Fn (𝐾 × 𝐵) | ||
Theorem | lmodscaf 19587 | The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ∙ = ( ·sf ‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → ∙ :(𝐾 × 𝐵)⟶𝐵) | ||
Theorem | lmodvsdi 19588 | Distributive law for scalar product (left-distributivity). (ax-hvdistr1 28713 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))) | ||
Theorem | lmodvsdir 19589 | Distributive law for scalar product (right-distributivity). (ax-hvdistr1 28713 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ⨣ = (+g‘𝐹) ⇒ ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) | ||
Theorem | lmodvsass 19590 | Associative law for scalar product. (ax-hvmulass 28712 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ × = (.r‘𝐹) ⇒ ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) | ||
Theorem | lmod0cl 19591 | The ring zero in a left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 0 = (0g‘𝐹) ⇒ ⊢ (𝑊 ∈ LMod → 0 ∈ 𝐾) | ||
Theorem | lmod1cl 19592 | The ring unit in a left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 1 = (1r‘𝐹) ⇒ ⊢ (𝑊 ∈ LMod → 1 ∈ 𝐾) | ||
Theorem | lmodvs1 19593 | Scalar product with ring unit. (ax-hvmulid 28711 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 1 = (1r‘𝐹) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ( 1 · 𝑋) = 𝑋) | ||
Theorem | lmod0vcl 19594 | The zero vector is a vector. (ax-hv0cl 28708 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → 0 ∈ 𝑉) | ||
Theorem | lmod0vlid 19595 | Left identity law for the zero vector. (hvaddid2 28728 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋) | ||
Theorem | lmod0vrid 19596 | Right identity law for the zero vector. (ax-hvaddid 28709 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋 + 0 ) = 𝑋) | ||
Theorem | lmod0vid 19597 | Identity equivalent to the value of the zero vector. Provides a convenient way to compute the value. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋)) | ||
Theorem | lmod0vs 19598 | Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 28715 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑂 = (0g‘𝐹) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 ) | ||
Theorem | lmodvs0 19599 | Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 28729 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 ) | ||
Theorem | lmodvsmmulgdi 19600 | Distributive law for a group multiple of a scalar multiplication. (Contributed by AV, 2-Sep-2019.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ↑ = (.g‘𝑊) & ⊢ 𝐸 = (.g‘𝐹) ⇒ ⊢ ((𝑊 ∈ LMod ∧ (𝐶 ∈ 𝐾 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉)) → (𝑁 ↑ (𝐶 · 𝑋)) = ((𝑁𝐸𝐶) · 𝑋)) |
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