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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dsmmfi 19901 | For finite products, the direct sum is just the module product. See also the observation in [Lang] p. 129. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) → (𝑆 ⊕m 𝑅) = (𝑆Xs𝑅)) | ||
Theorem | dsmmelbas 19902* | Membership in the finitely supported hull of a structure product in terms of the index set. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ 𝐶 = (𝑆 ⊕m 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐻 = (Base‘𝐶) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 Fn 𝐼) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐻 ↔ (𝑋 ∈ 𝐵 ∧ {𝑎 ∈ 𝐼 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin))) | ||
Theorem | dsmm0cl 19903 | The all-zero vector is contained in the finite hull, since its support is empty and therefore finite. This theorem along with the next one effectively proves that the finite hull is a "submonoid", although that does not exist as a defined concept yet. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅)) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) & ⊢ 0 = (0g‘𝑃) ⇒ ⊢ (𝜑 → 0 ∈ 𝐻) | ||
Theorem | dsmmacl 19904 | The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅)) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) & ⊢ (𝜑 → 𝐽 ∈ 𝐻) & ⊢ (𝜑 → 𝐾 ∈ 𝐻) & ⊢ + = (+g‘𝑃) ⇒ ⊢ (𝜑 → (𝐽 + 𝐾) ∈ 𝐻) | ||
Theorem | prdsinvgd2 19905 | Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Grp) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑁 = (invg‘𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝑁‘𝑋)‘𝐽) = ((invg‘(𝑅‘𝐽))‘(𝑋‘𝐽))) | ||
Theorem | dsmmsubg 19906 | The finite hull of a product of groups is additionally closed under negation and thus is a subgroup of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅)) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Grp) ⇒ ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝑃)) | ||
Theorem | dsmmlss 19907* | The finite hull of a product of modules is additionally closed under scalar multiplication and thus is a linear subspace of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ Ring) & ⊢ (𝜑 → 𝑅:𝐼⟶LMod) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆) & ⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ 𝑈 = (LSubSp‘𝑃) & ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅)) ⇒ ⊢ (𝜑 → 𝐻 ∈ 𝑈) | ||
Theorem | dsmmlmod 19908* | The direct sum of a family of modules is a module. See also the remark in [Lang] p. 128. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ Ring) & ⊢ (𝜑 → 𝑅:𝐼⟶LMod) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆) & ⊢ 𝐶 = (𝑆 ⊕m 𝑅) ⇒ ⊢ (𝜑 → 𝐶 ∈ LMod) | ||
According to Wikipedia ("Free module", 03-Mar-2019, https://en.wikipedia.org/wiki/Free_module) "In mathematics, a free module is a module that has a basis - that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules.". The same definition is used in [Lang] p. 135: "By a free module we shall mean a module which admits a basis, or the zero module.". In the following, however, a free module is defined as direct sum of a family consisting of the same ring regarded as a (left) module over itself, see df-frlm 19910. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 19910 (see lmisfree 20000), the two definitions are essentially equivalent. The free modules as defined by df-frlm 19910 are also taken for the motivation of free modules by [Lang] p. 135. | ||
Syntax | cfrlm 19909 | Class of free module generator. |
class freeLMod | ||
Definition | df-frlm 19910* | The 𝑖-dimensional free module over a ring 𝑟 is the product of 𝑖-many copies of the ring with componentwise addition and multiplication. If 𝑖 is infinite, the allowed vectors are restricted to those with finitely many nonzero coordinates; this ensures that the resulting module is actually spanned by its unit vectors. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)}))) | ||
Theorem | frlmval 19911 | Value of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) | ||
Theorem | frlmlmod 19912 | The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod) | ||
Theorem | frlmpws 19913 | The free module as a restriction of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) | ||
Theorem | frlmlss 19914 | The base set of the free module is a subspace of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝑈 = (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼)) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐵 ∈ 𝑈) | ||
Theorem | frlmpwsfi 19915 | The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ Fin) → 𝐹 = ((ringLMod‘𝑅) ↑s 𝐼)) | ||
Theorem | frlmsca 19916 | The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝐹)) | ||
Theorem | frlm0 19917 | Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 19914). (Contributed by Stefan O'Rear, 4-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝐼 × { 0 }) = (0g‘𝐹)) | ||
Theorem | frlmbas 19918* | Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑁 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐵 = {𝑘 ∈ (𝑁 ↑𝑚 𝐼) ∣ 𝑘 finSupp 0 } ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐵 = (Base‘𝐹)) | ||
Theorem | frlmelbas 19919 | Membership in the base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑁 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ (𝑁 ↑𝑚 𝐼) ∧ 𝑋 finSupp 0 ))) | ||
Theorem | frlmrcl 19920 | If a free module is inhabited, this is sufficient to conclude that the ring expression defines a set. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ (𝑋 ∈ 𝐵 → 𝑅 ∈ V) | ||
Theorem | frlmbasfsupp 19921 | Elements of the free module are finitely supported. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 finSupp 0 ) | ||
Theorem | frlmbasmap 19922 | Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑁 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (𝑁 ↑𝑚 𝐼)) | ||
Theorem | frlmbasf 19923 | Elements of the free module are functions. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑁 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋:𝐼⟶𝑁) | ||
Theorem | frlmfibas 19924 | The base set of the finite free module as a set exponential. (Contributed by AV, 6-Dec-2018.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑁 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ Fin) → (𝑁 ↑𝑚 𝐼) = (Base‘𝐹)) | ||
Theorem | elfrlmbasn0 19925 | If the dimension of a free module over a ring is not 0, every element of its base set is not empty. (Contributed by AV, 10-Feb-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑁 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ≠ ∅) → (𝑋 ∈ 𝐵 → 𝑋 ≠ ∅)) | ||
Theorem | frlmplusgval 19926 | Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ + = (+g‘𝑅) & ⊢ ✚ = (+g‘𝑌) ⇒ ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹 ∘𝑓 + 𝐺)) | ||
Theorem | frlmsubgval 19927 | Subtraction in a free module. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ − = (-g‘𝑅) & ⊢ 𝑀 = (-g‘𝑌) ⇒ ⊢ (𝜑 → (𝐹𝑀𝐺) = (𝐹 ∘𝑓 − 𝐺)) | ||
Theorem | frlmvscafval 19928 | Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ ∙ = ( ·𝑠 ‘𝑌) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) | ||
Theorem | frlmvscaval 19929 | Scalar multiplication in a free module at a coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ ∙ = ( ·𝑠 ‘𝑌) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) | ||
Theorem | frlmgsum 19930* | Finite commutative sums in a free module are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 5-Jul-2015.) (Revised by AV, 23-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑥 ∈ 𝐼 ↦ 𝑈) ∈ 𝐵) & ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝑌 Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (𝑥 ∈ 𝐼 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))) | ||
Theorem | frlmsplit2 19931* | Restriction is homomorphic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝑈) & ⊢ 𝑍 = (𝑅 freeLMod 𝑉) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐶 = (Base‘𝑍) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍)) | ||
Theorem | frlmsslss 19932* | A subset of a free module obtained by restricting the support set is a submodule. 𝐽 is the set of forbidden unit vectors. (Contributed by Stefan O'Rear, 4-Feb-2015.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (LSubSp‘𝑌) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ 𝐽) = (𝐽 × { 0 })} ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ 𝑈) | ||
Theorem | frlmsslss2 19933* | A subset of a free module obtained by restricting the support set is a submodule. 𝐽 is the set of permitted unit vectors. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 23-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (LSubSp‘𝑌) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ 𝑈) | ||
Theorem | frlmbas3 19934 | An element of the base set of a finite free module with a Cartesian product as index set as operation value. (Contributed by AV, 14-Feb-2019.) |
⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑀)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑉 = (Base‘𝐹) ⇒ ⊢ (((𝑅 ∈ 𝑊 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑀)) → (𝐼𝑋𝐽) ∈ 𝐵) | ||
Theorem | mpt2frlmd 19935* | Elements of the free module are mappings with two arguments defined by their operation values. (Contributed by AV, 20-Feb-2019.) |
⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑀)) & ⊢ 𝑉 = (Base‘𝐹) & ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑀) → 𝐴 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑀) → 𝐵 ∈ 𝑌) & ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉)) ⇒ ⊢ (𝜑 → (𝑍 = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑀 ↦ 𝐵) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑀 (𝑖𝑍𝑗) = 𝐴)) | ||
Theorem | frlmip 19936* | The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑓 ∈ (𝐵 ↑𝑚 𝐼), 𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))) = (·𝑖‘𝑌)) | ||
Theorem | frlmipval 19937 | The inner product of a free module. (Contributed by Thierry Arnoux, 21-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑉 = (Base‘𝑌) & ⊢ , = (·𝑖‘𝑌) ⇒ ⊢ (((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉)) → (𝐹 , 𝐺) = (𝑅 Σg (𝐹 ∘𝑓 · 𝐺))) | ||
Theorem | frlmphllem 19938* | Lemma for frlmphl 19939. (Contributed by AV, 21-Jul-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑉 = (Base‘𝑌) & ⊢ , = (·𝑖‘𝑌) & ⊢ 𝑂 = (0g‘𝑌) & ⊢ 0 = (0g‘𝑅) & ⊢ ∗ = (*𝑟‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Field) & ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) ⇒ ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) finSupp 0 ) | ||
Theorem | frlmphl 19939* | Conditions for a free module to be a pre-Hilbert space. (Contributed by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑉 = (Base‘𝑌) & ⊢ , = (·𝑖‘𝑌) & ⊢ 𝑂 = (0g‘𝑌) & ⊢ 0 = (0g‘𝑅) & ⊢ ∗ = (*𝑟‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Field) & ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝑌 ∈ PreHil) | ||
According to Wikipedia ("Standard basis", 16-Mar-2019, https://en.wikipedia.org/wiki/Standard_basis) "In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.", and ("Unit vector", 16-Mar-2019, https://en.wikipedia.org/wiki/Unit_vector) "In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.". In the following, the term "unit vector" (or more specific "basic unit vector") is used for the (special) unit vectors forming the standard basis of free modules. However, the length of the unit vectors is not considered here, so it is not required to regard normed spaces. | ||
Syntax | cuvc 19940 | Class of basic unit vectors for an explicit free module. |
class unitVec | ||
Definition | df-uvc 19941* | ((𝑅 unitVec 𝐼)‘𝑗) is the unit vector in (𝑅 freeLMod 𝐼) along the 𝑗 axis. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟))))) | ||
Theorem | uvcfval 19942* | Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑈 = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))) | ||
Theorem | uvcval 19943* | Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))) | ||
Theorem | uvcvval 19944 | Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 )) | ||
Theorem | uvcvvcl 19945 | A coodinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) ∈ { 0 , 1 }) | ||
Theorem | uvcvvcl2 19946 | A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ (𝜑 → 𝐾 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) ∈ 𝐵) | ||
Theorem | uvcvv1 19947 | The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 ) | ||
Theorem | uvcvv0 19948 | The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ (𝜑 → 𝐾 ∈ 𝐼) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 ) | ||
Theorem | uvcff 19949 | Domain and range of the unit vector generator; ring condition required to be sure 1 and 0 are actually in the ring. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼⟶𝐵) | ||
Theorem | uvcf1 19950 | In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼–1-1→𝐵) | ||
Theorem | uvcresum 19951 | Any element of a free module can be expressed as a finite linear combination of unit vectors. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Jul-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = ( ·𝑠 ‘𝑌) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝑌 Σg (𝑋 ∘𝑓 · 𝑈))) | ||
Theorem | frlmssuvc1 19952* | A scalar multiple of a unit vector included in a support-restriction subspace is included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝐹) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝐿 ∈ 𝐽) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) ⇒ ⊢ (𝜑 → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶) | ||
Theorem | frlmssuvc2 19953* | A nonzero scalar multiple of a unit vector not included in a support-restriction subspace is not included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝐹) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝐿 ∈ (𝐼 ∖ 𝐽)) & ⊢ (𝜑 → 𝑋 ∈ (𝐾 ∖ { 0 })) ⇒ ⊢ (𝜑 → ¬ (𝑋 · (𝑈‘𝐿)) ∈ 𝐶) | ||
Theorem | frlmsslsp 19954* | A subset of a free module obtained by restricting the support set is spanned by the relevant unit vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by AV, 24-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝐾 = (LSpan‘𝑌) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) = 𝐶) | ||
Theorem | frlmlbs 19955 | The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝐽 = (LBasis‘𝐹) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ∈ 𝐽) | ||
Theorem | frlmup1 19956* | Any assignment of unit vectors to target vectors can be extended (uniquely) to a homomorphism from a free module to an arbitrary other module on the same base ring. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) | ||
Theorem | frlmup2 19957* | The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ 𝑈 = (𝑅 unitVec 𝐼) ⇒ ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐴‘𝑌)) | ||
Theorem | frlmup3 19958* | The range of such an evaluation map is the finite linear combinations of the target vectors and also the span of the target vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) & ⊢ 𝐾 = (LSpan‘𝑇) ⇒ ⊢ (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴)) | ||
Theorem | frlmup4 19959* | Universal property of the free module by existential uniquenes. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ 𝑅 = (Scalar‘𝑇) & ⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ ((𝑇 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐴:𝐼⟶𝐶) → ∃!𝑚 ∈ (𝐹 LMHom 𝑇)(𝑚 ∘ 𝑈) = 𝐴) | ||
Theorem | ellspd 19960* | The elements of the span of an indexed collection of basic vectors are those vectors which can be written as finite linear combinations of basic vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by AV, 24-Jun-2019.) |
⊢ 𝑁 = (LSpan‘𝑀) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 0 = (0g‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ V) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑𝑚 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹))))) | ||
Theorem | elfilspd 19961* | Simplified version of ellspd 19960 when the spanning set is finite: all linear combinations are then acceptable. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝑁 = (LSpan‘𝑀) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 0 = (0g‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ Fin) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑𝑚 𝐼)𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹)))) | ||
According to the definition in [Lang] p. 129: "A subset S of a module M is said
to be linearly independent (over A) if whenever we have a linear
combination ∑x ∈ S axx which is equal to 0, then
ax = 0 for all x ∈ S.", and according to the Definition in
[Lang] p. 130: "a familiy {xi}i ∈ I of elements of M
is said to be linearly independent (over A) if whenever we have a linear
combination ∑i ∈ I aixi = 0, then
ai = 0 for all i.". These definitions correspond to the definitions
df-linds 19965 resp. df-lindf 19964, where it is claimed that a nonzero summand can be
extracted ( ∑i ∈ {I \ { j } }aixi =
-ajxj ) and be represented as a linear combination of the
remaining elements of the family.
| ||
Syntax | clindf 19962 | The class relationship of independent families in a module. |
class LIndF | ||
Syntax | clinds 19963 | The class generator of independent sets in a module. |
class LIndS | ||
Definition | df-lindf 19964* |
An independent family is a family of vectors, no nonzero multiple of
which can be expressed as a linear combination of other elements of the
family. This is almost, but not quite, the same as a function into an
independent set.
This is a defined concept because it matters in many cases whether independence is taken at a set or family level. For instance, a number is transcedental iff its nonzero powers are linearly independent. Is 1 transcedental? It has only one nonzero power. We can almost define family independence as a family of unequal elements with independent range, as islindf3 19984, but taking that as primitive would lead to unpleasant corner case behavior with the zero ring. This is equivalent to the common definition of having no nontrivial representations of zero (islindf4 19996) and only one representation for each element of the range (islindf5 19997). (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ LIndF = {〈𝑓, 𝑤〉 ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))} | ||
Definition | df-linds 19965* | An independent set is a set which is independent as a family. See also islinds3 19992 and islinds4 19993. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤}) | ||
Theorem | rellindf 19966 | The independent-family predicate is a proper relation and can be used with brrelexi 5082. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ Rel LIndF | ||
Theorem | islinds 19967 | Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊))) | ||
Theorem | linds1 19968 | An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑋 ⊆ 𝐵) | ||
Theorem | linds2 19969 | An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ (𝑋 ∈ (LIndS‘𝑊) → ( I ↾ 𝑋) LIndF 𝑊) | ||
Theorem | islindf 19970* | Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝑁 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ∈ 𝑋) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))) | ||
Theorem | islinds2 19971* | Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝑁 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ (𝑊 ∈ 𝑌 → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))) | ||
Theorem | islindf2 19972* | Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝑁 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐼 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥}))))) | ||
Theorem | lindff 19973 | Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → 𝐹:dom 𝐹⟶𝐵) | ||
Theorem | lindfind 19974 | A linearly independent family is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐿 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐿) & ⊢ 𝐾 = (Base‘𝐿) ⇒ ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))) | ||
Theorem | lindsind 19975 | A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐿 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐿) & ⊢ 𝐾 = (Base‘𝐿) ⇒ ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))) | ||
Theorem | lindfind2 19976 | In a linearly independent family in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝐿 = (Scalar‘𝑊) ⇒ ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → ¬ (𝐹‘𝐸) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))) | ||
Theorem | lindsind2 19977 | In a linearly independent set in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝐿 = (Scalar‘𝑊) ⇒ ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) → ¬ 𝐸 ∈ (𝐾‘(𝐹 ∖ {𝐸}))) | ||
Theorem | lindff1 19978 | A linearly independent family over a nonzero ring has no repeated elements. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐿 = (Scalar‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹–1-1→𝐵) | ||
Theorem | lindfrn 19979 | The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ∈ (LIndS‘𝑊)) | ||
Theorem | f1lindf 19980 | Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺) LIndF 𝑊) | ||
Theorem | lindfres 19981 | Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (𝐹 ↾ 𝑋) LIndF 𝑊) | ||
Theorem | lindsss 19982 | Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ ((𝑊 ∈ LMod ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐺 ⊆ 𝐹) → 𝐺 ∈ (LIndS‘𝑊)) | ||
Theorem | f1linds 19983 | A family constructed from non-repeated elements of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → 𝐹 LIndF 𝑊) | ||
Theorem | islindf3 19984 | In a nonzero ring, independent families can be equivalently characterized as renamings of independent sets. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐿 = (Scalar‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹–1-1→V ∧ ran 𝐹 ∈ (LIndS‘𝑊)))) | ||
Theorem | lindfmm 19985 | Linear independence of a family is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑆 ↔ (𝐺 ∘ 𝐹) LIndF 𝑇)) | ||
Theorem | lindsmm 19986 | Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹 ⊆ 𝐵) → (𝐹 ∈ (LIndS‘𝑆) ↔ (𝐺 “ 𝐹) ∈ (LIndS‘𝑇))) | ||
Theorem | lindsmm2 19987 | The monomorphic image of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹 ∈ (LIndS‘𝑆)) → (𝐺 “ 𝐹) ∈ (LIndS‘𝑇)) | ||
Theorem | lsslindf 19988 | Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑈 = (LSubSp‘𝑊) & ⊢ 𝑋 = (𝑊 ↾s 𝑆) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊)) | ||
Theorem | lsslinds 19989 | Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝑈 = (LSubSp‘𝑊) & ⊢ 𝑋 = (𝑊 ↾s 𝑆) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ∈ (LIndS‘𝑋) ↔ 𝐹 ∈ (LIndS‘𝑊))) | ||
Theorem | islbs4 19990 | A basis is an independent spanning set. This could have been used as alternative definition of a basis: LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ (((LSpan‘𝑤) ‘𝑏) = (Base‘𝑤) ∧ 𝑏 ∈ (LIndS‘𝑤))}). (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) ⇒ ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵)) | ||
Theorem | lbslinds 19991 | A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ 𝐽 ⊆ (LIndS‘𝑊) | ||
Theorem | islinds3 19992 | A subset is linearly independent iff it is a basis of its span. (Contributed by Stefan O'Rear, 25-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝑋 = (𝑊 ↾s (𝐾‘𝑌)) & ⊢ 𝐽 = (LBasis‘𝑋) ⇒ ⊢ (𝑊 ∈ LMod → (𝑌 ∈ (LIndS‘𝑊) ↔ 𝑌 ∈ 𝐽)) | ||
Theorem | islinds4 19993* | A set is independent in a vector space iff it is a subset of some basis. (AC equivalent) (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec → (𝑌 ∈ (LIndS‘𝑊) ↔ ∃𝑏 ∈ 𝐽 𝑌 ⊆ 𝑏)) | ||
Theorem | lmimlbs 19994 | The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑆) & ⊢ 𝐾 = (LBasis‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ 𝐵) ∈ 𝐾) | ||
Theorem | lmiclbs 19995 | Having a basis is an isomorphism invariant. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑆) & ⊢ 𝐾 = (LBasis‘𝑇) ⇒ ⊢ (𝑆 ≃𝑚 𝑇 → (𝐽 ≠ ∅ → 𝐾 ≠ ∅)) | ||
Theorem | islindf4 19996* | A family is independent iff it has no nontrivial representations of zero. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑌 = (0g‘𝑅) & ⊢ 𝐿 = (Base‘(𝑅 freeLMod 𝐼)) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → 𝑥 = (𝐼 × {𝑌})))) | ||
Theorem | islindf5 19997* | A family is independent iff the linear combinations homomorphism is injective. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) ⇒ ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶)) | ||
Theorem | indlcim 19998* | An independent, spanning family extends to an isomorphism from a free module. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝑁 = (LSpan‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼–onto→𝐽) & ⊢ (𝜑 → 𝐴 LIndF 𝑇) & ⊢ (𝜑 → (𝑁‘𝐽) = 𝐶) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMIso 𝑇)) | ||
Theorem | lbslcic 19999 | A module with a basis is isomorphic to a free module with the same cardinality. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝑊 ≃𝑚 (𝐹 freeLMod 𝐼)) | ||
Theorem | lmisfree 20000* | A module has a basis iff it is isomorphic to a free module. In settings where isomorphic objects are not distinguished, it is common to define "free module" as any module with a basis; thus for instance lbsex 18986 might be described as "every vector space is free." (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → (𝐽 ≠ ∅ ↔ ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘))) |
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