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Type | Label | Description |
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Statement | ||
Definition | df-frlm 21301* | Define the function associating with a ring and a set the direct sum indexed by that set of copies of that ring regarded as a left module over itself. Recall from df-dsmm 21286 that an element of a direct sum has finitely many nonzero coordinates. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ freeLMod = (π β V, π β V β¦ (π βm (π Γ {(ringLModβπ)}))) | ||
Theorem | frlmval 21302 | Value of the "free module" function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) β β’ ((π β π β§ πΌ β π) β πΉ = (π βm (πΌ Γ {(ringLModβπ )}))) | ||
Theorem | frlmlmod 21303 | The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) β β’ ((π β Ring β§ πΌ β π) β πΉ β LMod) | ||
Theorem | frlmpws 21304 | The free module as a restriction of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) β β’ ((π β π β§ πΌ β π) β πΉ = (((ringLModβπ ) βs πΌ) βΎs π΅)) | ||
Theorem | frlmlss 21305 | The base set of the free module is a subspace of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ π = (LSubSpβ((ringLModβπ ) βs πΌ)) β β’ ((π β Ring β§ πΌ β π) β π΅ β π) | ||
Theorem | frlmpwsfi 21306 | The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) β β’ ((π β π β§ πΌ β Fin) β πΉ = ((ringLModβπ ) βs πΌ)) | ||
Theorem | frlmsca 21307 | The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) β β’ ((π β π β§ πΌ β π) β π = (ScalarβπΉ)) | ||
Theorem | frlm0 21308 | Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 21305). (Contributed by Stefan O'Rear, 4-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) & β’ 0 = (0gβπ ) β β’ ((π β Ring β§ πΌ β π) β (πΌ Γ { 0 }) = (0gβπΉ)) | ||
Theorem | frlmbas 21309* | Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π΅ = {π β (π βm πΌ) β£ π finSupp 0 } β β’ ((π β π β§ πΌ β π) β π΅ = (BaseβπΉ)) | ||
Theorem | frlmelbas 21310 | Membership in the base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π΅ = (BaseβπΉ) β β’ ((π β π β§ πΌ β π) β (π β π΅ β (π β (π βm πΌ) β§ π finSupp 0 ))) | ||
Theorem | frlmrcl 21311 | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) β β’ (π β π΅ β π β V) | ||
Theorem | frlmbasfsupp 21312 | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ 0 = (0gβπ ) & β’ π΅ = (BaseβπΉ) β β’ ((πΌ β π β§ π β π΅) β π finSupp 0 ) | ||
Theorem | frlmbasmap 21313 | Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (Baseβπ ) & β’ π΅ = (BaseβπΉ) β β’ ((πΌ β π β§ π β π΅) β π β (π βm πΌ)) | ||
Theorem | frlmbasf 21314 | Elements of the free module are functions. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (Baseβπ ) & β’ π΅ = (BaseβπΉ) β β’ ((πΌ β π β§ π β π΅) β π:πΌβΆπ) | ||
Theorem | frlmlvec 21315 | The free module over a division ring is a left vector space. (Contributed by Steven Nguyen, 29-Apr-2023.) |
β’ πΉ = (π freeLMod πΌ) β β’ ((π β DivRing β§ πΌ β π) β πΉ β LVec) | ||
Theorem | frlmfibas 21316 | The base set of the finite free module as a set exponential. (Contributed by AV, 6-Dec-2018.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (Baseβπ ) β β’ ((π β π β§ πΌ β Fin) β (π βm πΌ) = (BaseβπΉ)) | ||
Theorem | elfrlmbasn0 21317 | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (Baseβπ ) & β’ π΅ = (BaseβπΉ) β β’ ((πΌ β π β§ πΌ β β ) β (π β π΅ β π β β )) | ||
Theorem | frlmplusgval 21318 | Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΅) & β’ + = (+gβπ ) & β’ β = (+gβπ) β β’ (π β (πΉ β πΊ) = (πΉ βf + πΊ)) | ||
Theorem | frlmsubgval 21319 | Subtraction in a free module. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) & β’ (π β π β Ring) & β’ (π β πΌ β π) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΅) & β’ β = (-gβπ ) & β’ π = (-gβπ) β β’ (π β (πΉππΊ) = (πΉ βf β πΊ)) | ||
Theorem | frlmvscafval 21320 | Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) & β’ πΎ = (Baseβπ ) & β’ (π β πΌ β π) & β’ (π β π΄ β πΎ) & β’ (π β π β π΅) & β’ β = ( Β·π βπ) & β’ Β· = (.rβπ ) β β’ (π β (π΄ β π) = ((πΌ Γ {π΄}) βf Β· π)) | ||
Theorem | frlmvplusgvalc 21321 | Coordinates of a sum with respect to a basis in a free module. (Contributed by AV, 16-Jan-2023.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π½ β πΌ) & β’ + = (+gβπ ) & β’ β = (+gβπΉ) β β’ (π β ((π β π)βπ½) = ((πβπ½) + (πβπ½))) | ||
Theorem | frlmvscaval 21322 | Coordinates of a scalar multiple with respect to a basis in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) & β’ πΎ = (Baseβπ ) & β’ (π β πΌ β π) & β’ (π β π΄ β πΎ) & β’ (π β π β π΅) & β’ (π β π½ β πΌ) & β’ β = ( Β·π βπ) & β’ Β· = (.rβπ ) β β’ (π β ((π΄ β π)βπ½) = (π΄ Β· (πβπ½))) | ||
Theorem | frlmplusgvalb 21323* | Addition in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ (π β πΌ β π) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β Ring) & β’ (π β π β π΅) & β’ + = (+gβπ ) & β’ β = (+gβπΉ) β β’ (π β (π = (π β π) β βπ β πΌ (πβπ) = ((πβπ) + (πβπ)))) | ||
Theorem | frlmvscavalb 21324* | Scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ (π β πΌ β π) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β Ring) & β’ πΎ = (Baseβπ ) & β’ (π β π΄ β πΎ) & β’ β = ( Β·π βπΉ) & β’ Β· = (.rβπ ) β β’ (π β (π = (π΄ β π) β βπ β πΌ (πβπ) = (π΄ Β· (πβπ)))) | ||
Theorem | frlmvplusgscavalb 21325* | Addition combined with scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ (π β πΌ β π) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β Ring) & β’ πΎ = (Baseβπ ) & β’ (π β π΄ β πΎ) & β’ β = ( Β·π βπΉ) & β’ Β· = (.rβπ ) & β’ (π β π β π΅) & β’ + = (+gβπ ) & β’ β = (+gβπΉ) & β’ (π β πΆ β πΎ) β β’ (π β (π = ((π΄ β π) β (πΆ β π)) β βπ β πΌ (πβπ) = ((π΄ Β· (πβπ)) + (πΆ Β· (πβπ))))) | ||
Theorem | frlmgsum 21326* | 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.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) & β’ 0 = (0gβπ) & β’ (π β πΌ β π) & β’ (π β π½ β π) & β’ (π β π β Ring) & β’ ((π β§ π¦ β π½) β (π₯ β πΌ β¦ π) β π΅) & β’ (π β (π¦ β π½ β¦ (π₯ β πΌ β¦ π)) finSupp 0 ) β β’ (π β (π Ξ£g (π¦ β π½ β¦ (π₯ β πΌ β¦ π))) = (π₯ β πΌ β¦ (π Ξ£g (π¦ β π½ β¦ π)))) | ||
Theorem | frlmsplit2 21327* | Restriction is homomorphic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
β’ π = (π freeLMod π) & β’ π = (π freeLMod π) & β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) & β’ πΉ = (π₯ β π΅ β¦ (π₯ βΎ π)) β β’ ((π β Ring β§ π β π β§ π β π) β πΉ β (π LMHom π)) | ||
Theorem | frlmsslss 21328* | A subset of a free module obtained by restricting the support set is a submodule. π½ is the set of forbidden unit vectors. (Contributed by Stefan O'Rear, 4-Feb-2015.) |
β’ π = (π freeLMod πΌ) & β’ π = (LSubSpβπ) & β’ π΅ = (Baseβπ) & β’ 0 = (0gβπ ) & β’ πΆ = {π₯ β π΅ β£ (π₯ βΎ π½) = (π½ Γ { 0 })} β β’ ((π β Ring β§ πΌ β π β§ π½ β πΌ) β πΆ β π) | ||
Theorem | frlmsslss2 21329* | A subset of a free module obtained by restricting the support set is a submodule. π½ is the set of permitted unit vectors. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 23-Jun-2019.) |
β’ π = (π freeLMod πΌ) & β’ π = (LSubSpβπ) & β’ π΅ = (Baseβπ) & β’ 0 = (0gβπ ) & β’ πΆ = {π₯ β π΅ β£ (π₯ supp 0 ) β π½} β β’ ((π β Ring β§ πΌ β π β§ π½ β πΌ) β πΆ β π) | ||
Theorem | frlmbas3 21330 | 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.) |
β’ πΉ = (π freeLMod (π Γ π)) & β’ π΅ = (Baseβπ ) & β’ π = (BaseβπΉ) β β’ (((π β π β§ π β π) β§ (π β Fin β§ π β Fin) β§ (πΌ β π β§ π½ β π)) β (πΌππ½) β π΅) | ||
Theorem | mpofrlmd 21331* | Elements of the free module are mappings with two arguments defined by their operation values. (Contributed by AV, 20-Feb-2019.) (Proof shortened by AV, 3-Jul-2022.) |
β’ πΉ = (π freeLMod (π Γ π)) & β’ π = (BaseβπΉ) & β’ ((π = π β§ π = π) β π΄ = π΅) & β’ ((π β§ π β π β§ π β π) β π΄ β π) & β’ ((π β§ π β π β§ π β π) β π΅ β π) & β’ (π β (π β π β§ π β π β§ π β π)) β β’ (π β (π = (π β π, π β π β¦ π΅) β βπ β π βπ β π (πππ) = π΄)) | ||
Theorem | frlmip 21332* | The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ ) & β’ Β· = (.rβπ ) β β’ ((πΌ β π β§ π β π) β (π β (π΅ βm πΌ), π β (π΅ βm πΌ) β¦ (π Ξ£g (π₯ β πΌ β¦ ((πβπ₯) Β· (πβπ₯))))) = (Β·πβπ)) | ||
Theorem | frlmipval 21333 | The inner product of a free module. (Contributed by Thierry Arnoux, 21-Jun-2019.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ ) & β’ Β· = (.rβπ ) & β’ π = (Baseβπ) & β’ , = (Β·πβπ) β β’ (((πΌ β π β§ π β π) β§ (πΉ β π β§ πΊ β π)) β (πΉ , πΊ) = (π Ξ£g (πΉ βf Β· πΊ))) | ||
Theorem | frlmphllem 21334* | Lemma for frlmphl 21335. (Contributed by AV, 21-Jul-2019.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ ) & β’ Β· = (.rβπ ) & β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ π = (0gβπ) & β’ 0 = (0gβπ ) & β’ β = (*πβπ ) & β’ (π β π β Field) & β’ ((π β§ π β π β§ (π , π) = 0 ) β π = π) & β’ ((π β§ π₯ β π΅) β ( β βπ₯) = π₯) & β’ (π β πΌ β π) β β’ ((π β§ π β π β§ β β π) β (π₯ β πΌ β¦ ((πβπ₯) Β· (ββπ₯))) finSupp 0 ) | ||
Theorem | frlmphl 21335* | 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.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ ) & β’ Β· = (.rβπ ) & β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ π = (0gβπ) & β’ 0 = (0gβπ ) & β’ β = (*πβπ ) & β’ (π β π β Field) & β’ ((π β§ π β π β§ (π , π) = 0 ) β π = π) & β’ ((π β§ π₯ β π΅) β ( β βπ₯) = π₯) & β’ (π β πΌ β π) β β’ (π β π β 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 21336 | Class of basic unit vectors for an explicit free module. |
class unitVec | ||
Definition | df-uvc 21337* | ((π unitVec πΌ)βπ) is the unit vector in (π freeLMod πΌ) along the π axis. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ unitVec = (π β V, π β V β¦ (π β π β¦ (π β π β¦ if(π = π, (1rβπ), (0gβπ))))) | ||
Theorem | uvcfval 21338* | Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ π = (π unitVec πΌ) & β’ 1 = (1rβπ ) & β’ 0 = (0gβπ ) β β’ ((π β π β§ πΌ β π) β π = (π β πΌ β¦ (π β πΌ β¦ if(π = π, 1 , 0 )))) | ||
Theorem | uvcval 21339* | Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ π = (π unitVec πΌ) & β’ 1 = (1rβπ ) & β’ 0 = (0gβπ ) β β’ ((π β π β§ πΌ β π β§ π½ β πΌ) β (πβπ½) = (π β πΌ β¦ if(π = π½, 1 , 0 ))) | ||
Theorem | uvcvval 21340 | Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ π = (π unitVec πΌ) & β’ 1 = (1rβπ ) & β’ 0 = (0gβπ ) β β’ (((π β π β§ πΌ β π β§ π½ β πΌ) β§ πΎ β πΌ) β ((πβπ½)βπΎ) = if(πΎ = π½, 1 , 0 )) | ||
Theorem | uvcvvcl 21341 | A coordinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ π = (π unitVec πΌ) & β’ 1 = (1rβπ ) & β’ 0 = (0gβπ ) β β’ (((π β π β§ πΌ β π β§ π½ β πΌ) β§ πΎ β πΌ) β ((πβπ½)βπΎ) β { 0 , 1 }) | ||
Theorem | uvcvvcl2 21342 | A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ π = (π unitVec πΌ) & β’ π΅ = (Baseβπ ) & β’ (π β π β Ring) & β’ (π β πΌ β π) & β’ (π β π½ β πΌ) & β’ (π β πΎ β πΌ) β β’ (π β ((πβπ½)βπΎ) β π΅) | ||
Theorem | uvcvv1 21343 | The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ π = (π unitVec πΌ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β π½ β πΌ) & β’ 1 = (1rβπ ) β β’ (π β ((πβπ½)βπ½) = 1 ) | ||
Theorem | uvcvv0 21344 | The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ π = (π unitVec πΌ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β π½ β πΌ) & β’ (π β πΎ β πΌ) & β’ (π β π½ β πΎ) & β’ 0 = (0gβπ ) β β’ (π β ((πβπ½)βπΎ) = 0 ) | ||
Theorem | uvcff 21345 | Domain and codomain 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.) |
β’ π = (π unitVec πΌ) & β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) β β’ ((π β Ring β§ πΌ β π) β π:πΌβΆπ΅) | ||
Theorem | uvcf1 21346 | In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) |
β’ π = (π unitVec πΌ) & β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) β β’ ((π β NzRing β§ πΌ β π) β π:πΌβ1-1βπ΅) | ||
Theorem | uvcresum 21347 | 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.) |
β’ π = (π unitVec πΌ) & β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) & β’ Β· = ( Β·π βπ) β β’ ((π β Ring β§ πΌ β π β§ π β π΅) β π = (π Ξ£g (π βf Β· π))) | ||
Theorem | frlmssuvc1 21348* | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (π unitVec πΌ) & β’ π΅ = (BaseβπΉ) & β’ πΎ = (Baseβπ ) & β’ Β· = ( Β·π βπΉ) & β’ 0 = (0gβπ ) & β’ πΆ = {π₯ β π΅ β£ (π₯ supp 0 ) β π½} & β’ (π β π β Ring) & β’ (π β πΌ β π) & β’ (π β π½ β πΌ) & β’ (π β πΏ β π½) & β’ (π β π β πΎ) β β’ (π β (π Β· (πβπΏ)) β πΆ) | ||
Theorem | frlmssuvc2 21349* | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (π unitVec πΌ) & β’ π΅ = (BaseβπΉ) & β’ πΎ = (Baseβπ ) & β’ Β· = ( Β·π βπΉ) & β’ 0 = (0gβπ ) & β’ πΆ = {π₯ β π΅ β£ (π₯ supp 0 ) β π½} & β’ (π β π β Ring) & β’ (π β πΌ β π) & β’ (π β π½ β πΌ) & β’ (π β πΏ β (πΌ β π½)) & β’ (π β π β (πΎ β { 0 })) β β’ (π β Β¬ (π Β· (πβπΏ)) β πΆ) | ||
Theorem | frlmsslsp 21350* | 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.) |
β’ π = (π freeLMod πΌ) & β’ π = (π unitVec πΌ) & β’ πΎ = (LSpanβπ) & β’ π΅ = (Baseβπ) & β’ 0 = (0gβπ ) & β’ πΆ = {π₯ β π΅ β£ (π₯ supp 0 ) β π½} β β’ ((π β Ring β§ πΌ β π β§ π½ β πΌ) β (πΎβ(π β π½)) = πΆ) | ||
Theorem | frlmlbs 21351 | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (π unitVec πΌ) & β’ π½ = (LBasisβπΉ) β β’ ((π β Ring β§ πΌ β π) β ran π β π½) | ||
Theorem | frlmup1 21352* | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ πΆ = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΈ = (π₯ β π΅ β¦ (π Ξ£g (π₯ βf Β· π΄))) & β’ (π β π β LMod) & β’ (π β πΌ β π) & β’ (π β π = (Scalarβπ)) & β’ (π β π΄:πΌβΆπΆ) β β’ (π β πΈ β (πΉ LMHom π)) | ||
Theorem | frlmup2 21353* | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ πΆ = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΈ = (π₯ β π΅ β¦ (π Ξ£g (π₯ βf Β· π΄))) & β’ (π β π β LMod) & β’ (π β πΌ β π) & β’ (π β π = (Scalarβπ)) & β’ (π β π΄:πΌβΆπΆ) & β’ (π β π β πΌ) & β’ π = (π unitVec πΌ) β β’ (π β (πΈβ(πβπ)) = (π΄βπ)) | ||
Theorem | frlmup3 21354* | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ πΆ = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΈ = (π₯ β π΅ β¦ (π Ξ£g (π₯ βf Β· π΄))) & β’ (π β π β LMod) & β’ (π β πΌ β π) & β’ (π β π = (Scalarβπ)) & β’ (π β π΄:πΌβΆπΆ) & β’ πΎ = (LSpanβπ) β β’ (π β ran πΈ = (πΎβran π΄)) | ||
Theorem | frlmup4 21355* | Universal property of the free module by existential uniqueness. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
β’ π = (Scalarβπ) & β’ πΉ = (π freeLMod πΌ) & β’ π = (π unitVec πΌ) & β’ πΆ = (Baseβπ) β β’ ((π β LMod β§ πΌ β π β§ π΄:πΌβΆπΆ) β β!π β (πΉ LMHom π)(π β π) = π΄) | ||
Theorem | ellspd 21356* | 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.) (Revised by AV, 11-Apr-2024.) |
β’ π = (LSpanβπ) & β’ π΅ = (Baseβπ) & β’ πΎ = (Baseβπ) & β’ π = (Scalarβπ) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ (π β πΉ:πΌβΆπ΅) & β’ (π β π β LMod) & β’ (π β πΌ β π) β β’ (π β (π β (πβ(πΉ β πΌ)) β βπ β (πΎ βm πΌ)(π finSupp 0 β§ π = (π Ξ£g (π βf Β· πΉ))))) | ||
Theorem | elfilspd 21357* | Simplified version of ellspd 21356 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.) |
β’ π = (LSpanβπ) & β’ π΅ = (Baseβπ) & β’ πΎ = (Baseβπ) & β’ π = (Scalarβπ) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ (π β πΉ:πΌβΆπ΅) & β’ (π β π β LMod) & β’ (π β πΌ β Fin) β β’ (π β (π β (πβ(πΉ β πΌ)) β βπ β (πΎ βm πΌ)π = (π Ξ£g (π βf Β· πΉ)))) | ||
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 βxβSaxx which is equal to 0, then ax = 0 for all x β S", and according to the Definition in [Lang] p. 130: "a familiy {xi}iβI of elements of M is said to be linearly independent (over A) if whenever we have a linear combination βiβIaixi = 0, then ai = 0 for all i β I." These definitions correspond to Definitions df-linds 21361 and df-lindf 21360 respectively, where it is claimed that a nonzero summand can be extracted (βiβ{I\{j}}aixi = -ajxj) and be represented as a linear combination of the remaining elements of the family. | ||
Syntax | clindf 21358 | The class relationship of independent families in a module. |
class LIndF | ||
Syntax | clinds 21359 | The class generator of independent sets in a module. |
class LIndS | ||
Definition | df-lindf 21360* |
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 21380, 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 21392) and only one representation for each element of the range (islindf5 21393). (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ LIndF = {β¨π, π€β© β£ (π:dom πβΆ(Baseβπ€) β§ [(Scalarβπ€) / π ]βπ₯ β dom πβπ β ((Baseβπ ) β {(0gβπ )}) Β¬ (π( Β·π βπ€)(πβπ₯)) β ((LSpanβπ€)β(π β (dom π β {π₯}))))} | ||
Definition | df-linds 21361* | An independent set is a set which is independent as a family. See also islinds3 21388 and islinds4 21389. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ LIndS = (π€ β V β¦ {π β π« (Baseβπ€) β£ ( I βΎ π ) LIndF π€}) | ||
Theorem | rellindf 21362 | The independent-family predicate is a proper relation and can be used with brrelex1i 5732. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ Rel LIndF | ||
Theorem | islinds 21363 | Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π΅ = (Baseβπ) β β’ (π β π β (π β (LIndSβπ) β (π β π΅ β§ ( I βΎ π) LIndF π))) | ||
Theorem | linds1 21364 | An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π΅ = (Baseβπ) β β’ (π β (LIndSβπ) β π β π΅) | ||
Theorem | linds2 21365 | An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ (π β (LIndSβπ) β ( I βΎ π) LIndF π) | ||
Theorem | islindf 21366* | Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π΅ = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΎ = (LSpanβπ) & β’ π = (Scalarβπ) & β’ π = (Baseβπ) & β’ 0 = (0gβπ) β β’ ((π β π β§ πΉ β π) β (πΉ LIndF π β (πΉ:dom πΉβΆπ΅ β§ βπ₯ β dom πΉβπ β (π β { 0 }) Β¬ (π Β· (πΉβπ₯)) β (πΎβ(πΉ β (dom πΉ β {π₯})))))) | ||
Theorem | islinds2 21367* | Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π΅ = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΎ = (LSpanβπ) & β’ π = (Scalarβπ) & β’ π = (Baseβπ) & β’ 0 = (0gβπ) β β’ (π β π β (πΉ β (LIndSβπ) β (πΉ β π΅ β§ βπ₯ β πΉ βπ β (π β { 0 }) Β¬ (π Β· π₯) β (πΎβ(πΉ β {π₯}))))) | ||
Theorem | islindf2 21368* | Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
β’ π΅ = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΎ = (LSpanβπ) & β’ π = (Scalarβπ) & β’ π = (Baseβπ) & β’ 0 = (0gβπ) β β’ ((π β π β§ πΌ β π β§ πΉ:πΌβΆπ΅) β (πΉ LIndF π β βπ₯ β πΌ βπ β (π β { 0 }) Β¬ (π Β· (πΉβπ₯)) β (πΎβ(πΉ β (πΌ β {π₯}))))) | ||
Theorem | lindff 21369 | Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π΅ = (Baseβπ) β β’ ((πΉ LIndF π β§ π β π) β πΉ:dom πΉβΆπ΅) | ||
Theorem | lindfind 21370 | 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.) |
β’ Β· = ( Β·π βπ) & β’ π = (LSpanβπ) & β’ πΏ = (Scalarβπ) & β’ 0 = (0gβπΏ) & β’ πΎ = (BaseβπΏ) β β’ (((πΉ LIndF π β§ πΈ β dom πΉ) β§ (π΄ β πΎ β§ π΄ β 0 )) β Β¬ (π΄ Β· (πΉβπΈ)) β (πβ(πΉ β (dom πΉ β {πΈ})))) | ||
Theorem | lindsind 21371 | 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.) |
β’ Β· = ( Β·π βπ) & β’ π = (LSpanβπ) & β’ πΏ = (Scalarβπ) & β’ 0 = (0gβπΏ) & β’ πΎ = (BaseβπΏ) β β’ (((πΉ β (LIndSβπ) β§ πΈ β πΉ) β§ (π΄ β πΎ β§ π΄ β 0 )) β Β¬ (π΄ Β· πΈ) β (πβ(πΉ β {πΈ}))) | ||
Theorem | lindfind2 21372 | 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.) |
β’ πΎ = (LSpanβπ) & β’ πΏ = (Scalarβπ) β β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π β§ πΈ β dom πΉ) β Β¬ (πΉβπΈ) β (πΎβ(πΉ β (dom πΉ β {πΈ})))) | ||
Theorem | lindsind2 21373 | 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.) |
β’ πΎ = (LSpanβπ) & β’ πΏ = (Scalarβπ) β β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ β (LIndSβπ) β§ πΈ β πΉ) β Β¬ πΈ β (πΎβ(πΉ β {πΈ}))) | ||
Theorem | lindff1 21374 | A linearly independent family over a nonzero ring has no repeated elements. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π΅ = (Baseβπ) & β’ πΏ = (Scalarβπ) β β’ ((π β LMod β§ πΏ β NzRing β§ πΉ LIndF π) β πΉ:dom πΉβ1-1βπ΅) | ||
Theorem | lindfrn 21375 | The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ ((π β LMod β§ πΉ LIndF π) β ran πΉ β (LIndSβπ)) | ||
Theorem | f1lindf 21376 | Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ ((π β LMod β§ πΉ LIndF π β§ πΊ:πΎβ1-1βdom πΉ) β (πΉ β πΊ) LIndF π) | ||
Theorem | lindfres 21377 | Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ ((π β LMod β§ πΉ LIndF π) β (πΉ βΎ π) LIndF π) | ||
Theorem | lindsss 21378 | Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ ((π β LMod β§ πΉ β (LIndSβπ) β§ πΊ β πΉ) β πΊ β (LIndSβπ)) | ||
Theorem | f1linds 21379 | A family constructed from non-repeated elements of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
β’ ((π β LMod β§ π β (LIndSβπ) β§ πΉ:π·β1-1βπ) β πΉ LIndF π) | ||
Theorem | islindf3 21380 | In a nonzero ring, independent families can be equivalently characterized as renamings of independent sets. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
β’ πΏ = (Scalarβπ) β β’ ((π β LMod β§ πΏ β NzRing) β (πΉ LIndF π β (πΉ:dom πΉβ1-1βV β§ ran πΉ β (LIndSβπ)))) | ||
Theorem | lindfmm 21381 | 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.) |
β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) β β’ ((πΊ β (π LMHom π) β§ πΊ:π΅β1-1βπΆ β§ πΉ:πΌβΆπ΅) β (πΉ LIndF π β (πΊ β πΉ) LIndF π)) | ||
Theorem | lindsmm 21382 | Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) β β’ ((πΊ β (π LMHom π) β§ πΊ:π΅β1-1βπΆ β§ πΉ β π΅) β (πΉ β (LIndSβπ) β (πΊ β πΉ) β (LIndSβπ))) | ||
Theorem | lindsmm2 21383 | The monomorphic image of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) β β’ ((πΊ β (π LMHom π) β§ πΊ:π΅β1-1βπΆ β§ πΉ β (LIndSβπ)) β (πΊ β πΉ) β (LIndSβπ)) | ||
Theorem | lsslindf 21384 | 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.) |
β’ π = (LSubSpβπ) & β’ π = (π βΎs π) β β’ ((π β LMod β§ π β π β§ ran πΉ β π) β (πΉ LIndF π β πΉ LIndF π)) | ||
Theorem | lsslinds 21385 | Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π = (LSubSpβπ) & β’ π = (π βΎs π) β β’ ((π β LMod β§ π β π β§ πΉ β π) β (πΉ β (LIndSβπ) β πΉ β (LIndSβπ))) | ||
Theorem | islbs4 21386 | A basis is an independent spanning set. This could have been used as alternative definition of a basis: LBasis = (π€ β V β¦ {π β π« (Baseβπ€) β£ (((LSpanβπ€) βπ) = (Baseβπ€) β§ π β (LIndSβπ€))}). (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π΅ = (Baseβπ) & β’ π½ = (LBasisβπ) & β’ πΎ = (LSpanβπ) β β’ (π β π½ β (π β (LIndSβπ) β§ (πΎβπ) = π΅)) | ||
Theorem | lbslinds 21387 | A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π½ = (LBasisβπ) β β’ π½ β (LIndSβπ) | ||
Theorem | islinds3 21388 | A subset is linearly independent iff it is a basis of its span. (Contributed by Stefan O'Rear, 25-Feb-2015.) |
β’ π΅ = (Baseβπ) & β’ πΎ = (LSpanβπ) & β’ π = (π βΎs (πΎβπ)) & β’ π½ = (LBasisβπ) β β’ (π β LMod β (π β (LIndSβπ) β π β π½)) | ||
Theorem | islinds4 21389* | A set is independent in a vector space iff it is a subset of some basis. This is an axiom of choice equivalent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π½ = (LBasisβπ) β β’ (π β LVec β (π β (LIndSβπ) β βπ β π½ π β π)) | ||
Theorem | lmimlbs 21390 | The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
β’ π½ = (LBasisβπ) & β’ πΎ = (LBasisβπ) β β’ ((πΉ β (π LMIso π) β§ π΅ β π½) β (πΉ β π΅) β πΎ) | ||
Theorem | lmiclbs 21391 | Having a basis is an isomorphism invariant. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
β’ π½ = (LBasisβπ) & β’ πΎ = (LBasisβπ) β β’ (π βπ π β (π½ β β β πΎ β β )) | ||
Theorem | islindf4 21392* | A family is independent iff it has no nontrivial representations of zero. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (0gβπ ) & β’ πΏ = (Baseβ(π freeLMod πΌ)) β β’ ((π β LMod β§ πΌ β π β§ πΉ:πΌβΆπ΅) β (πΉ LIndF π β βπ₯ β πΏ ((π Ξ£g (π₯ βf Β· πΉ)) = 0 β π₯ = (πΌ Γ {π})))) | ||
Theorem | islindf5 21393* | A family is independent iff the linear combinations homomorphism is injective. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ πΆ = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΈ = (π₯ β π΅ β¦ (π Ξ£g (π₯ βf Β· π΄))) & β’ (π β π β LMod) & β’ (π β πΌ β π) & β’ (π β π = (Scalarβπ)) & β’ (π β π΄:πΌβΆπΆ) β β’ (π β (π΄ LIndF π β πΈ:π΅β1-1βπΆ)) | ||
Theorem | indlcim 21394* | An independent, spanning family extends to an isomorphism from a free module. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ πΆ = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (LSpanβπ) & β’ πΈ = (π₯ β π΅ β¦ (π Ξ£g (π₯ βf Β· π΄))) & β’ (π β π β LMod) & β’ (π β πΌ β π) & β’ (π β π = (Scalarβπ)) & β’ (π β π΄:πΌβontoβπ½) & β’ (π β π΄ LIndF π) & β’ (π β (πβπ½) = πΆ) β β’ (π β πΈ β (πΉ LMIso π)) | ||
Theorem | lbslcic 21395 | A module with a basis is isomorphic to a free module with the same cardinality. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
β’ πΉ = (Scalarβπ) & β’ π½ = (LBasisβπ) β β’ ((π β LMod β§ π΅ β π½ β§ πΌ β π΅) β π βπ (πΉ freeLMod πΌ)) | ||
Theorem | lmisfree 21396* | 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 20777 might be described as "every vector space is free". (Contributed by Stefan O'Rear, 26-Feb-2015.) |
β’ π½ = (LBasisβπ) & β’ πΉ = (Scalarβπ) β β’ (π β LMod β (π½ β β β βπ π βπ (πΉ freeLMod π))) | ||
Theorem | lvecisfrlm 21397* | Every vector space is isomorphic to a free module. (Contributed by AV, 7-Mar-2019.) |
β’ πΉ = (Scalarβπ) β β’ (π β LVec β βπ π βπ (πΉ freeLMod π)) | ||
Theorem | lmimco 21398 | The composition of two isomorphisms of modules is an isomorphism of modules. (Contributed by AV, 10-Mar-2019.) |
β’ ((πΉ β (π LMIso π) β§ πΊ β (π LMIso π)) β (πΉ β πΊ) β (π LMIso π)) | ||
Theorem | lmictra 21399 | Module isomorphism is transitive. (Contributed by AV, 10-Mar-2019.) |
β’ ((π βπ π β§ π βπ π) β π βπ π) | ||
Theorem | uvcf1o 21400 | In a nonzero ring, the mapping of the index set of a free module onto the unit vectors of the free module is a 1-1 onto function. (Contributed by AV, 10-Mar-2019.) |
β’ π = (π unitVec πΌ) β β’ ((π β NzRing β§ πΌ β π) β π:πΌβ1-1-ontoβran π) |
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