HomeTheorem List (p. 209 of 489)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | ringen1zr 20801 | 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 20802 | 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) | ||
| Theorem | issubdrg 20803* | 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 | drhmsubc 20804* | According to df-subc 17873, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17904 and subcss2 17907). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.) |
| ⊢ 𝐶 = (𝑈 ∩ DivRing) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈))) | ||
| Theorem | drngcat 20805* | The restriction of the category of (unital) rings to the set of division ring homomorphisms is a category, the "category of division rings". (Contributed by AV, 20-Feb-2020.) |
| ⊢ 𝐶 = (𝑈 ∩ DivRing) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat) | ||
| Theorem | fldcat 20806* | The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) |
| ⊢ 𝐶 = (𝑈 ∩ DivRing) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) & ⊢ 𝐷 = (𝑈 ∩ Field) & ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐹) ∈ Cat) | ||
| Theorem | fldc 20807* | The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) |
| ⊢ 𝐶 = (𝑈 ∩ DivRing) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) & ⊢ 𝐷 = (𝑈 ∩ Field) & ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → (((RingCat‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat) | ||
| Theorem | fldhmsubc 20808* | According to df-subc 17873, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17904 and subcss2 17907). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.) |
| ⊢ 𝐶 = (𝑈 ∩ DivRing) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) & ⊢ 𝐷 = (𝑈 ∩ Field) & ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾cat 𝐽))) | ||
| Syntax | csdrg 20809 | Syntax for subfields (sub-division-rings). |
| class SubDRing | ||
| Definition | df-sdrg 20810* | 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 20821), 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 20811 | 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 20812 | Reverse closure for a sub-division-ring predicate. (Contributed by SN, 19-Feb-2025.) |
| ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing) | ||
| Theorem | sdrgdrng 20813 | 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 20814 | A sub-division-ring is a subring. (Contributed by SN, 19-Feb-2025.) |
| ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 ∈ (SubRing‘𝑅)) | ||
| Theorem | sdrgid 20815 | 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 20816 | 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 20817 | 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 20818* | 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 20819 | 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 20820 | 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 20821 | 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 20822* | 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 20823 | Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵)) | ||
| Theorem | sdrgacs 20824 | Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) ∈ (ACS‘𝐵)) | ||
| Theorem | cntzsdrg 20825 | 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 20826* | 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 20827 | 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 20828 | 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 20829 | 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 20830 | 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)) | ||
| Syntax | cabv 20831 | The set of absolute values on a ring. |
| class AbsVal | ||
| Definition | df-abv 20832* | Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 15285 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 20833* | 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 20834* | 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 20835* | 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 20836 | Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.) |
| ⊢ 𝐴 = (AbsVal‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) | ||
| Theorem | abvfge0 20837 | 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 20838 | 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 20839 | 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 20840 | 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 20841 | 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 20842 | 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 20843 | 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 20844 | 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 20845 | 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 20846 | 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 20847 | 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 20848 | 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 20849 | 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 20850 | 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 20851 | 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 20852 | 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 20853 | 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 20854 | 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 20855* | 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 | abvtrivg 20856* | The trivial absolute value. This theorem is not true for rings with zero divisors, which violate the multiplication axiom; abvdom 20853 is the converse of this theorem. (Contributed by SN, 25-Jun-2025.) |
| ⊢ 𝐴 = (AbsVal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, 1)) ⇒ ⊢ (𝑅 ∈ Domn → 𝐹 ∈ 𝐴) | ||
| Theorem | abvtriv 20857* | The trivial absolute value. (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 20858* | 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)) | ||
| Theorem | abvn0b 20859 | Another characterization of domains, hinted at in abvtrivg 20856: a nonzero ring is a domain iff it has an absolute value. (Contributed by Mario Carneiro, 6-May-2015.) |
| ⊢ 𝐴 = (AbsVal‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ 𝐴 ≠ ∅)) | ||
| Syntax | cstf 20860 | Extend class notation with the functionalization of the *-ring involution. |
| class *rf | ||
| Syntax | csr 20861 | Extend class notation with class of all *-rings. |
| class *-Ring | ||
| Definition | df-staf 20862* | 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 20863* | 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 20864* | 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 20865 | 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 20866 | 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 20867 | 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 20868 | 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 20869 | A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.) |
| ⊢ (𝑅 ∈ *-Ring → 𝑅 ∈ Ring) | ||
| Theorem | srngcnv 20870 | 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 20871 | 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 20872 | 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 20873 | 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 20874 | 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 20875 | 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 20876 | 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 20878.) (Contributed by Mario Carneiro, 6-Oct-2015.) |
| ⊢ ∗ = (*𝑟‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ *-Ring → ( ∗ ‘ 1 ) = 1 ) | ||
| Theorem | srng0 20877 | 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 20878* | 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 20879* | 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 20880 | Extend class notation with class of all left modules. |
| class LMod | ||
| Syntax | cscaf 20881 | The functionalization of the scalar multiplication operation. |
| class ·sf | ||
| Definition | df-lmod 20882* | 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 20883* | 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 20884* | 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 20885 | 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 20886* | Properties that determine a left module. See note in isgrpd2 18996 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 20887 | 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 20888 | 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 20889 | 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 | lmodgrpd 20890 | A left module is a group. (Contributed by SN, 16-May-2024.) |
| ⊢ (𝜑 → 𝑊 ∈ LMod) ⇒ ⊢ (𝜑 → 𝑊 ∈ Grp) | ||
| Theorem | lmodbn0 20891 | 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 20892 | 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 20893 | 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 20894 | 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 20895 | 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 20896 | 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 20897 | 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 20898 | Closure of scalar product for a left module. (hvmulcl 31045 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 | lmodvscld 20899 | Closure of scalar product for a left module. (Contributed by SN, 15-Mar-2025.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑅 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑅 · 𝑋) ∈ 𝑉) | ||
| Theorem | scaffval 20900* | 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)) | ||
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