![]() |
Metamath
Proof Explorer Theorem List (p. 193 of 437) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-28364) |
![]() (28365-29889) |
![]() (29890-43671) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | subrgunit 19201 | 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 19202 | 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 19203* | 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 19204 | Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.) |
⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (SubRing‘𝑅) = (SubRing‘𝑂) | ||
Theorem | subrgint 19205 | 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 19206 | 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 19207 | 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 19208* | 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 19209 | 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 19210 | 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 19211 | 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 19212 | 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 | rhmeql 19213 | 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 19214 | 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 | cntzsubr 19215 | 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 19216* | 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 19217* | 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 19218* | 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 | cabv 19219 | The set of absolute values on a ring. |
class AbsVal | ||
Definition | df-abv 19220* | Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 14389 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 19221* | 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 19222* | 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 19223* | 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 19224 | Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.) |
⊢ 𝐴 = (AbsVal‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) | ||
Theorem | abvfge0 19225 | 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 19226 | 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 19227 | 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 19228 | 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 19229 | 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 19230 | 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 19231 | 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 19232 | 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 19233 | 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 19234 | 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 19235 | 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 19236 | 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 19237 | 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 19238 | 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 19239 | 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 19240 | 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 19241 | 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 19242 | 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 19243* | 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 19244* | 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 19241 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 19245* | 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 19246 | Extend class notation with the functionalization of the *-ring involution. |
class *rf | ||
Syntax | csr 19247 | Extend class notation with class of all *-rings. |
class *-Ring | ||
Definition | df-staf 19248* | 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 19249* | 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 19250* | 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 19251 | 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 19252 | 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 19253 | 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 19254 | 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 19255 | A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ (𝑅 ∈ *-Ring → 𝑅 ∈ Ring) | ||
Theorem | srngcnv 19256 | 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 19257 | 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 19258 | 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 19259 | 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 19260 | 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 19261 | 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 19262 | 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 19264.) (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ ∗ = (*𝑟‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ *-Ring → ( ∗ ‘ 1 ) = 1 ) | ||
Theorem | srng0 19263 | 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 19264* | 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 19265* | 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 19266 | Extend class notation with class of all left modules. |
class LMod | ||
Syntax | cscaf 19267 | The functionalization of the scalar multiplication operation. |
class ·sf | ||
Definition | df-lmod 19268* | 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 19269* | 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 19270* | 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 19271 | 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 19272* | Properties that determine a left module. See note in isgrpd2 17840 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 19273 | 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 19274 | 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 19275 | 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 19276 | 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 19277 | 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 19278 | 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 19279 | 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 19280 | 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 19281 | 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 19282 | 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 19283 | Closure of scalar product for a left module. (hvmulcl 28459 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 19284* | 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)) | ||
Theorem | scafval 19285 | 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 19286 | 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 19287 | 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 19288 | 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 19289 | Distributive law for scalar product (left-distributivity). (ax-hvdistr1 28454 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 19290 | Distributive law for scalar product (right-distributivity). (ax-hvdistr1 28454 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 19291 | Associative law for scalar product. (ax-hvmulass 28453 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 19292 | 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 19293 | 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 19294 | Scalar product with ring unit. (ax-hvmulid 28452 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 19295 | The zero vector is a vector. (ax-hv0cl 28449 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 19296 | Left identity law for the zero vector. (hvaddid2 28469 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 19297 | Right identity law for the zero vector. (ax-hvaddid 28450 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 19298 | 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 19299 | Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 28456 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 19300 | Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 28470 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 ) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |