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Type | Label | Description |
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Statement | ||
Theorem | lsppreli 20701 | A vector expressed as a sum belongs to the span of its components. (Contributed by NM, 9-Apr-2015.) |
β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ Β· = ( Β·π βπ) & β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) & β’ π = (LSpanβπ) & β’ (π β π β LMod) & β’ (π β π΄ β πΎ) & β’ (π β π΅ β πΎ) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β ((π΄ Β· π) + (π΅ Β· π)) β (πβ{π, π})) | ||
Theorem | lsmelpr 20702 | Two ways to say that a vector belongs to the span of a pair of vectors. (Contributed by NM, 14-Jan-2015.) |
β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ β = (LSSumβπ) & β’ (π β π β LMod) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β (π β (πβ{π, π}) β (πβ{π}) β ((πβ{π}) β (πβ{π})))) | ||
Theorem | lsppr0 20703 | The span of a vector paired with zero equals the span of the singleton of the vector. (Contributed by NM, 29-Aug-2014.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LMod) & β’ (π β π β π) β β’ (π β (πβ{π, 0 }) = (πβ{π})) | ||
Theorem | lsppr 20704* | Span of a pair of vectors. (Contributed by NM, 22-Aug-2014.) |
β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) & β’ Β· = ( Β·π βπ) & β’ π = (LSpanβπ) & β’ (π β π β LMod) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β (πβ{π, π}) = {π£ β£ βπ β πΎ βπ β πΎ π£ = ((π Β· π) + (π Β· π))}) | ||
Theorem | lspprel 20705* | Member of the span of a pair of vectors. (Contributed by NM, 10-Apr-2015.) |
β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) & β’ Β· = ( Β·π βπ) & β’ π = (LSpanβπ) & β’ (π β π β LMod) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β (π β (πβ{π, π}) β βπ β πΎ βπ β πΎ π = ((π Β· π) + (π Β· π)))) | ||
Theorem | lspprabs 20706 | Absorption of vector sum into span of pair. (Contributed by NM, 27-Apr-2015.) |
β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LMod) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β (πβ{π, (π + π)}) = (πβ{π, π})) | ||
Theorem | lspvadd 20707 | The span of a vector sum is included in the span of its arguments. (Contributed by NM, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.) |
β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ π = (LSpanβπ) β β’ ((π β LMod β§ π β π β§ π β π) β (πβ{(π + π)}) β (πβ{π, π})) | ||
Theorem | lspsntri 20708 | Triangle-type inequality for span of a singleton. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) |
β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ π = (LSpanβπ) & β’ β = (LSSumβπ) β β’ ((π β LMod β§ π β π β§ π β π) β (πβ{(π + π)}) β ((πβ{π}) β (πβ{π}))) | ||
Theorem | lspsntrim 20709 | Triangle-type inequality for span of a singleton of vector difference. (Contributed by NM, 25-Apr-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) |
β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ β = (LSSumβπ) & β’ π = (LSpanβπ) β β’ ((π β LMod β§ π β π β§ π β π) β (πβ{(π β π)}) β ((πβ{π}) β (πβ{π}))) | ||
Theorem | lbspropd 20710* | If two structures have the same components (properties), they have the same set of bases. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.) |
β’ (π β π΅ = (BaseβπΎ)) & β’ (π β π΅ = (BaseβπΏ)) & β’ (π β π΅ β π) & β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) & β’ ((π β§ (π₯ β π β§ π¦ β π΅)) β (π₯( Β·π βπΎ)π¦) β π) & β’ ((π β§ (π₯ β π β§ π¦ β π΅)) β (π₯( Β·π βπΎ)π¦) = (π₯( Β·π βπΏ)π¦)) & β’ πΉ = (ScalarβπΎ) & β’ πΊ = (ScalarβπΏ) & β’ (π β π = (BaseβπΉ)) & β’ (π β π = (BaseβπΊ)) & β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯(+gβπΉ)π¦) = (π₯(+gβπΊ)π¦)) & β’ (π β πΎ β π) & β’ (π β πΏ β π) β β’ (π β (LBasisβπΎ) = (LBasisβπΏ)) | ||
Theorem | pj1lmhm 20711 | The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
β’ πΏ = (LSubSpβπ) & β’ β = (LSSumβπ) & β’ 0 = (0gβπ) & β’ π = (proj1βπ) & β’ (π β π β LMod) & β’ (π β π β πΏ) & β’ (π β π β πΏ) & β’ (π β (π β© π) = { 0 }) β β’ (π β (πππ) β ((π βΎs (π β π)) LMHom π)) | ||
Theorem | pj1lmhm2 20712 | The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
β’ πΏ = (LSubSpβπ) & β’ β = (LSSumβπ) & β’ 0 = (0gβπ) & β’ π = (proj1βπ) & β’ (π β π β LMod) & β’ (π β π β πΏ) & β’ (π β π β πΏ) & β’ (π β (π β© π) = { 0 }) β β’ (π β (πππ) β ((π βΎs (π β π)) LMHom (π βΎs π))) | ||
Syntax | clvec 20713 | Extend class notation with class of all left vector spaces. |
class LVec | ||
Definition | df-lvec 20714 | Define the class of all left vector spaces. A left vector space over a division ring is an Abelian group (vectors) together with a division ring (scalars) and a left scalar product connecting them. Some authors call this a "left module over a division ring", reserving "vector space" for those where the division ring is commutative, i.e., is a field. (Contributed by NM, 11-Nov-2013.) |
β’ LVec = {π β LMod β£ (Scalarβπ) β DivRing} | ||
Theorem | islvec 20715 | The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.) |
β’ πΉ = (Scalarβπ) β β’ (π β LVec β (π β LMod β§ πΉ β DivRing)) | ||
Theorem | lvecdrng 20716 | The set of scalars of a left vector space is a division ring. (Contributed by NM, 17-Apr-2014.) |
β’ πΉ = (Scalarβπ) β β’ (π β LVec β πΉ β DivRing) | ||
Theorem | lveclmod 20717 | A left vector space is a left module. (Contributed by NM, 9-Dec-2013.) |
β’ (π β LVec β π β LMod) | ||
Theorem | lveclmodd 20718 | A vector space is a left module. (Contributed by SN, 16-May-2024.) |
β’ (π β π β LVec) β β’ (π β π β LMod) | ||
Theorem | lsslvec 20719 | A vector subspace is a vector space. (Contributed by NM, 14-Mar-2015.) |
β’ π = (π βΎs π) & β’ π = (LSubSpβπ) β β’ ((π β LVec β§ π β π) β π β LVec) | ||
Theorem | lmhmlvec 20720 | The property for modules to be vector spaces is invariant under module isomorphism. (Contributed by Steven Nguyen, 15-Aug-2023.) |
β’ (πΉ β (π LMHom π) β (π β LVec β π β LVec)) | ||
Theorem | lvecvs0or 20721 | If a scalar product is zero, one of its factors must be zero. (hvmul0or 30278 analog.) (Contributed by NM, 2-Jul-2014.) |
β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) & β’ π = (0gβπΉ) & β’ 0 = (0gβπ) & β’ (π β π β LVec) & β’ (π β π΄ β πΎ) & β’ (π β π β π) β β’ (π β ((π΄ Β· π) = 0 β (π΄ = π β¨ π = 0 ))) | ||
Theorem | lvecvsn0 20722 | A scalar product is nonzero iff both of its factors are nonzero. (Contributed by NM, 3-Jan-2015.) |
β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) & β’ π = (0gβπΉ) & β’ 0 = (0gβπ) & β’ (π β π β LVec) & β’ (π β π΄ β πΎ) & β’ (π β π β π) β β’ (π β ((π΄ Β· π) β 0 β (π΄ β π β§ π β 0 ))) | ||
Theorem | lssvs0or 20723 | If a scalar product belongs to a subspace, either the scalar component is zero or the vector component also belongs to the subspace. (Contributed by NM, 5-Apr-2015.) |
β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) & β’ 0 = (0gβπΉ) & β’ π = (LSubSpβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π΄ β πΎ) β β’ (π β ((π΄ Β· π) β π β (π΄ = 0 β¨ π β π))) | ||
Theorem | lvecvscan 20724 | Cancellation law for scalar multiplication. (hvmulcan 30325 analog.) (Contributed by NM, 2-Jul-2014.) |
β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) & β’ 0 = (0gβπΉ) & β’ (π β π β LVec) & β’ (π β π΄ β πΎ) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π΄ β 0 ) β β’ (π β ((π΄ Β· π) = (π΄ Β· π) β π = π)) | ||
Theorem | lvecvscan2 20725 | Cancellation law for scalar multiplication. (hvmulcan2 30326 analog.) (Contributed by NM, 2-Jul-2014.) |
β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) & β’ 0 = (0gβπ) & β’ (π β π β LVec) & β’ (π β π΄ β πΎ) & β’ (π β π΅ β πΎ) & β’ (π β π β π) & β’ (π β π β 0 ) β β’ (π β ((π΄ Β· π) = (π΅ Β· π) β π΄ = π΅)) | ||
Theorem | lvecinv 20726 | Invert coefficient of scalar product. (Contributed by NM, 11-Apr-2015.) |
β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) & β’ 0 = (0gβπΉ) & β’ πΌ = (invrβπΉ) & β’ (π β π β LVec) & β’ (π β π΄ β (πΎ β { 0 })) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β (π = (π΄ Β· π) β π = ((πΌβπ΄) Β· π))) | ||
Theorem | lspsnvs 20727 | A nonzero scalar product does not change the span of a singleton. (spansncol 30821 analog.) (Contributed by NM, 23-Apr-2014.) |
β’ π = (Baseβπ) & β’ πΉ = (Scalarβπ) & β’ Β· = ( Β·π βπ) & β’ πΎ = (BaseβπΉ) & β’ 0 = (0gβπΉ) & β’ π = (LSpanβπ) β β’ ((π β LVec β§ (π β πΎ β§ π β 0 ) β§ π β π) β (πβ{(π Β· π)}) = (πβ{π})) | ||
Theorem | lspsneleq 20728 | Membership relation that implies equality of spans. (spansneleq 30823 analog.) (Contributed by NM, 4-Jul-2014.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β (πβ{π})) & β’ (π β π β 0 ) β β’ (π β (πβ{π}) = (πβ{π})) | ||
Theorem | lspsncmp 20729 | Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β (π β { 0 })) & β’ (π β π β π) β β’ (π β ((πβ{π}) β (πβ{π}) β (πβ{π}) = (πβ{π}))) | ||
Theorem | lspsnne1 20730 | Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β (π β { 0 })) & β’ (π β π β π) & β’ (π β (πβ{π}) β (πβ{π})) β β’ (π β Β¬ π β (πβ{π})) | ||
Theorem | lspsnne2 20731 | Two ways to express that vectors have different spans. (Contributed by NM, 20-May-2015.) |
β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ (π β π β LMod) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β Β¬ π β (πβ{π})) β β’ (π β (πβ{π}) β (πβ{π})) | ||
Theorem | lspsnnecom 20732 | Swap two vectors with different spans. (Contributed by NM, 20-May-2015.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β (π β { 0 })) & β’ (π β Β¬ π β (πβ{π})) β β’ (π β Β¬ π β (πβ{π})) | ||
Theorem | lspabs2 20733 | Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.) |
β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β (π β { 0 })) & β’ (π β (πβ{π}) = (πβ{(π + π)})) β β’ (π β (πβ{π}) = (πβ{π})) | ||
Theorem | lspabs3 20734 | Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.) |
β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β (π + π) β 0 ) & β’ (π β (πβ{π}) = (πβ{π})) β β’ (π β (πβ{π}) = (πβ{(π + π)})) | ||
Theorem | lspsneq 20735* | Equal spans of singletons must have proportional vectors. See lspsnss2 20616 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.) |
β’ π = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΎ = (Baseβπ) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β ((πβ{π}) = (πβ{π}) β βπ β (πΎ β { 0 })π = (π Β· π))) | ||
Theorem | lspsneu 20736* | Nonzero vectors with equal singleton spans have a unique proportionality constant. (Contributed by NM, 31-May-2015.) |
β’ π = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΎ = (Baseβπ) & β’ π = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β (π β { 0 })) β β’ (π β ((πβ{π}) = (πβ{π}) β β!π β (πΎ β {π})π = (π Β· π))) | ||
Theorem | lspsnel4 20737 | A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 30826 analog.) (Contributed by NM, 4-Jul-2014.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSubSpβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β (πβ{π})) & β’ (π β π β 0 ) β β’ (π β (π β π β π β π)) | ||
Theorem | lspdisj 20738 | The span of a vector not in a subspace is disjoint with the subspace. (Contributed by NM, 6-Apr-2015.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ π = (LSubSpβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β Β¬ π β π) β β’ (π β ((πβ{π}) β© π) = { 0 }) | ||
Theorem | lspdisjb 20739 | A nonzero vector is not in a subspace iff its span is disjoint with the subspace. (Contributed by NM, 23-Apr-2015.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ π = (LSubSpβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β (π β { 0 })) β β’ (π β (Β¬ π β π β ((πβ{π}) β© π) = { 0 })) | ||
Theorem | lspdisj2 20740 | Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β (πβ{π}) β (πβ{π})) β β’ (π β ((πβ{π}) β© (πβ{π})) = { 0 }) | ||
Theorem | lspfixed 20741* | Show membership in the span of the sum of two vectors, one of which (π) is fixed in advance. (Contributed by NM, 27-May-2015.) (Revised by AV, 12-Jul-2022.) |
β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β Β¬ π β (πβ{π})) & β’ (π β Β¬ π β (πβ{π})) & β’ (π β π β (πβ{π, π})) β β’ (π β βπ§ β ((πβ{π}) β { 0 })π β (πβ{(π + π§)})) | ||
Theorem | lspexch 20742 | Exchange property for span of a pair. TODO: see if a version with Y,Z and X,Z reversed will shorten proofs (analogous to lspexchn1 20743 versus lspexchn2 20744); look for lspexch 20742 and prcom 4737 in same proof. TODO: would a hypothesis of Β¬ π β (πβ{π}) instead of (πβ{π}) β (πβ{π}) be better overall? This would be shorter and also satisfy the π β 0 condition. Here and also lspindp* and all proofs affected by them (all in NM's mathbox); there are 58 hypotheses with the β pattern as of 24-May-2015. (Contributed by NM, 11-Apr-2015.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β (π β { 0 })) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β (πβ{π}) β (πβ{π})) & β’ (π β π β (πβ{π, π})) β β’ (π β π β (πβ{π, π})) | ||
Theorem | lspexchn1 20743 | Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 20742 to see if this will shorten proofs. (Contributed by NM, 20-May-2015.) |
β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β Β¬ π β (πβ{π})) & β’ (π β Β¬ π β (πβ{π, π})) β β’ (π β Β¬ π β (πβ{π, π})) | ||
Theorem | lspexchn2 20744 | Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 20742 to see if this will shorten proofs. (Contributed by NM, 24-May-2015.) |
β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β Β¬ π β (πβ{π})) & β’ (π β Β¬ π β (πβ{π, π})) β β’ (π β Β¬ π β (πβ{π, π})) | ||
Theorem | lspindpi 20745 | Partial independence property. (Contributed by NM, 23-Apr-2015.) |
β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β Β¬ π β (πβ{π, π})) β β’ (π β ((πβ{π}) β (πβ{π}) β§ (πβ{π}) β (πβ{π}))) | ||
Theorem | lspindp1 20746 | Alternate way to say 3 vectors are mutually independent (swap 1st and 2nd). (Contributed by NM, 11-Apr-2015.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β (π β { 0 })) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β (πβ{π}) β (πβ{π})) & β’ (π β Β¬ π β (πβ{π, π})) β β’ (π β ((πβ{π}) β (πβ{π}) β§ Β¬ π β (πβ{π, π}))) | ||
Theorem | lspindp2l 20747 | Alternate way to say 3 vectors are mutually independent (rotate left). (Contributed by NM, 10-May-2015.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β (π β { 0 })) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β (πβ{π}) β (πβ{π})) & β’ (π β Β¬ π β (πβ{π, π})) β β’ (π β ((πβ{π}) β (πβ{π}) β§ Β¬ π β (πβ{π, π}))) | ||
Theorem | lspindp2 20748 | Alternate way to say 3 vectors are mutually independent (rotate right). (Contributed by NM, 12-Apr-2015.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β (π β { 0 })) & β’ (π β π β π) & β’ (π β (πβ{π}) β (πβ{π})) & β’ (π β Β¬ π β (πβ{π, π})) β β’ (π β ((πβ{π}) β (πβ{π}) β§ Β¬ π β (πβ{π, π}))) | ||
Theorem | lspindp3 20749 | Independence of 2 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.) |
β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β (π β { 0 })) & β’ (π β (πβ{π}) β (πβ{π})) β β’ (π β (πβ{π}) β (πβ{(π + π)})) | ||
Theorem | lspindp4 20750 | (Partial) independence of 3 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.) |
β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LMod) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β Β¬ π β (πβ{π, π})) β β’ (π β Β¬ π β (πβ{π, (π + π)})) | ||
Theorem | lvecindp 20751 | Compute the π coefficient in a sum with an independent vector π (first conjunct), which can then be removed to continue with the remaining vectors summed in expressions π and π (second conjunct). Typically, π is the span of the remaining vectors. (Contributed by NM, 5-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 19-Jul-2022.) |
β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) & β’ Β· = ( Β·π βπ) & β’ π = (LSubSpβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β Β¬ π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π΄ β πΎ) & β’ (π β π΅ β πΎ) & β’ (π β ((π΄ Β· π) + π) = ((π΅ Β· π) + π)) β β’ (π β (π΄ = π΅ β§ π = π)) | ||
Theorem | lvecindp2 20752 | Sums of independent vectors must have equal coefficients. (Contributed by NM, 22-Mar-2015.) |
β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β (π β { 0 })) & β’ (π β π β (π β { 0 })) & β’ (π β π΄ β πΎ) & β’ (π β π΅ β πΎ) & β’ (π β πΆ β πΎ) & β’ (π β π· β πΎ) & β’ (π β (πβ{π}) β (πβ{π})) & β’ (π β ((π΄ Β· π) + (π΅ Β· π)) = ((πΆ Β· π) + (π· Β· π))) β β’ (π β (π΄ = πΆ β§ π΅ = π·)) | ||
Theorem | lspsnsubn0 20753 | Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ β = (-gβπ) & β’ (π β π β LMod) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β (πβ{π}) β (πβ{π})) β β’ (π β (π β π) β 0 ) | ||
Theorem | lsmcv 20754 | Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 30905 analog.) TODO: ugly proof; can it be shortened? (Contributed by NM, 2-Oct-2014.) |
β’ π = (Baseβπ) & β’ π = (LSubSpβπ) & β’ π = (LSpanβπ) & β’ β = (LSSumβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) β β’ ((π β§ π β π β§ π β (π β (πβ{π}))) β π = (π β (πβ{π}))) | ||
Theorem | lspsolvlem 20755* | Lemma for lspsolv 20756. (Contributed by Mario Carneiro, 25-Jun-2014.) |
β’ π = (Baseβπ) & β’ π = (LSubSpβπ) & β’ π = (LSpanβπ) & β’ πΉ = (Scalarβπ) & β’ π΅ = (BaseβπΉ) & β’ + = (+gβπ) & β’ Β· = ( Β·π βπ) & β’ π = {π§ β π β£ βπ β π΅ (π§ + (π Β· π)) β (πβπ΄)} & β’ (π β π β LMod) & β’ (π β π΄ β π) & β’ (π β π β π) & β’ (π β π β (πβ(π΄ βͺ {π}))) β β’ (π β βπ β π΅ (π + (π Β· π)) β (πβπ΄)) | ||
Theorem | lspsolv 20756 | If π is in the span of π΄ βͺ {π} but not π΄, then π is in the span of π΄ βͺ {π}. (Contributed by Mario Carneiro, 25-Jun-2014.) |
β’ π = (Baseβπ) & β’ π = (LSubSpβπ) & β’ π = (LSpanβπ) β β’ ((π β LVec β§ (π΄ β π β§ π β π β§ π β ((πβ(π΄ βͺ {π})) β (πβπ΄)))) β π β (πβ(π΄ βͺ {π}))) | ||
Theorem | lssacsex 20757* | In a vector space, subspaces form an algebraic closure system whose closure operator has the exchange property. Strengthening of lssacs 20578 by lspsolv 20756. (Contributed by David Moews, 1-May-2017.) |
β’ π΄ = (LSubSpβπ) & β’ π = (mrClsβπ΄) & β’ π = (Baseβπ) β β’ (π β LVec β (π΄ β (ACSβπ) β§ βπ β π« πβπ¦ β π βπ§ β ((πβ(π βͺ {π¦})) β (πβπ ))π¦ β (πβ(π βͺ {π§})))) | ||
Theorem | lspsnat 20758 | There is no subspace strictly between the zero subspace and the span of a vector (i.e. a 1-dimensional subspace is an atom). (h1datomi 30834 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 22-Jun-2014.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSubSpβπ) & β’ π = (LSpanβπ) β β’ (((π β LVec β§ π β π β§ π β π) β§ π β (πβ{π})) β (π = (πβ{π}) β¨ π = { 0 })) | ||
Theorem | lspsncv0 20759* | The span of a singleton covers the zero subspace, using Definition 3.2.18 of [PtakPulmannova] p. 68 for "covers".) (Contributed by NM, 12-Aug-2014.) (Revised by AV, 13-Jul-2022.) |
β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (LSubSpβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) β β’ (π β Β¬ βπ¦ β π ({ 0 } β π¦ β§ π¦ β (πβ{π}))) | ||
Theorem | lsppratlem1 20760 | Lemma for lspprat 20766. Let π₯ β (π β {0}) (if there is no such π₯ then π is the zero subspace), and let π¦ β (π β (πβ{π₯})) (assuming the conclusion is false). The goal is to write π, π in terms of π₯, π¦, which would normally be done by solving the system of linear equations. The span equivalent of this process is lspsolv 20756 (hence the name), which we use extensively below. In this lemma, we show that since π₯ β (πβ{π, π}), either π₯ β (πβ{π}) or π β (πβ{π₯, π}). (Contributed by NM, 29-Aug-2014.) |
β’ π = (Baseβπ) & β’ π = (LSubSpβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β (πβ{π, π})) & β’ 0 = (0gβπ) & β’ (π β π₯ β (π β { 0 })) & β’ (π β π¦ β (π β (πβ{π₯}))) β β’ (π β (π₯ β (πβ{π}) β¨ π β (πβ{π₯, π}))) | ||
Theorem | lsppratlem2 20761 | Lemma for lspprat 20766. Show that if π and π are both in (πβ{π₯, π¦}) (which will be our goal for each of the two cases above), then (πβ{π, π}) β π, contradicting the hypothesis for π. (Contributed by NM, 29-Aug-2014.) (Revised by Mario Carneiro, 5-Sep-2014.) |
β’ π = (Baseβπ) & β’ π = (LSubSpβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β (πβ{π, π})) & β’ 0 = (0gβπ) & β’ (π β π₯ β (π β { 0 })) & β’ (π β π¦ β (π β (πβ{π₯}))) & β’ (π β π β (πβ{π₯, π¦})) & β’ (π β π β (πβ{π₯, π¦})) β β’ (π β (πβ{π, π}) β π) | ||
Theorem | lsppratlem3 20762 | Lemma for lspprat 20766. In the first case of lsppratlem1 20760, since π₯ β (πββ ), also π β (πβ{π₯}), and since π¦ β (πβ{π, π}) β (πβ{π, π₯}) and π¦ β (πβ{π₯}), we have π β (πβ{π₯, π¦}) as desired. (Contributed by NM, 29-Aug-2014.) |
β’ π = (Baseβπ) & β’ π = (LSubSpβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β (πβ{π, π})) & β’ 0 = (0gβπ) & β’ (π β π₯ β (π β { 0 })) & β’ (π β π¦ β (π β (πβ{π₯}))) & β’ (π β π₯ β (πβ{π})) β β’ (π β (π β (πβ{π₯, π¦}) β§ π β (πβ{π₯, π¦}))) | ||
Theorem | lsppratlem4 20763 | Lemma for lspprat 20766. In the second case of lsppratlem1 20760, π¦ β (πβ{π, π}) β (πβ{π₯, π}) and π¦ β (πβ{π₯}) implies π β (πβ{π₯, π¦}) and thus π β (πβ{π₯, π}) β (πβ{π₯, π¦}) as well. (Contributed by NM, 29-Aug-2014.) |
β’ π = (Baseβπ) & β’ π = (LSubSpβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β (πβ{π, π})) & β’ 0 = (0gβπ) & β’ (π β π₯ β (π β { 0 })) & β’ (π β π¦ β (π β (πβ{π₯}))) & β’ (π β π β (πβ{π₯, π})) β β’ (π β (π β (πβ{π₯, π¦}) β§ π β (πβ{π₯, π¦}))) | ||
Theorem | lsppratlem5 20764 | Lemma for lspprat 20766. Combine the two cases and show a contradiction to π β (πβ{π, π}) under the assumptions on π₯ and π¦. (Contributed by NM, 29-Aug-2014.) |
β’ π = (Baseβπ) & β’ π = (LSubSpβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β (πβ{π, π})) & β’ 0 = (0gβπ) & β’ (π β π₯ β (π β { 0 })) & β’ (π β π¦ β (π β (πβ{π₯}))) β β’ (π β (πβ{π, π}) β π) | ||
Theorem | lsppratlem6 20765 | Lemma for lspprat 20766. Negating the assumption on π¦, we arrive close to the desired conclusion. (Contributed by NM, 29-Aug-2014.) |
β’ π = (Baseβπ) & β’ π = (LSubSpβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β (πβ{π, π})) & β’ 0 = (0gβπ) β β’ (π β (π₯ β (π β { 0 }) β π = (πβ{π₯}))) | ||
Theorem | lspprat 20766* | A proper subspace of the span of a pair of vectors is the span of a singleton (an atom) or the zero subspace (if π§ is zero). Proof suggested by Mario Carneiro, 28-Aug-2014. (Contributed by NM, 29-Aug-2014.) |
β’ π = (Baseβπ) & β’ π = (LSubSpβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β (πβ{π, π})) β β’ (π β βπ§ β π π = (πβ{π§})) | ||
Theorem | islbs2 20767* | An equivalent formulation of the basis predicate in a vector space: a subset is a basis iff no element is in the span of the rest of the set. (Contributed by Mario Carneiro, 14-Jan-2015.) |
β’ π = (Baseβπ) & β’ π½ = (LBasisβπ) & β’ π = (LSpanβπ) β β’ (π β LVec β (π΅ β π½ β (π΅ β π β§ (πβπ΅) = π β§ βπ₯ β π΅ Β¬ π₯ β (πβ(π΅ β {π₯}))))) | ||
Theorem | islbs3 20768* | An equivalent formulation of the basis predicate: a subset is a basis iff it is a minimal spanning set. (Contributed by Mario Carneiro, 25-Jun-2014.) |
β’ π = (Baseβπ) & β’ π½ = (LBasisβπ) & β’ π = (LSpanβπ) β β’ (π β LVec β (π΅ β π½ β (π΅ β π β§ (πβπ΅) = π β§ βπ (π β π΅ β (πβπ ) β π)))) | ||
Theorem | lbsacsbs 20769 | Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 20767. (Contributed by David Moews, 1-May-2017.) |
β’ π΄ = (LSubSpβπ) & β’ π = (mrClsβπ΄) & β’ π = (Baseβπ) & β’ πΌ = (mrIndβπ΄) & β’ π½ = (LBasisβπ) β β’ (π β LVec β (π β π½ β (π β πΌ β§ (πβπ) = π))) | ||
Theorem | lvecdim 20770 | The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 20757 and lbsacsbs 20769 to show that being a basis for a vector space is equivalent to being a basis for the associated algebraic closure system, and then using acsexdimd 18512. (Contributed by David Moews, 1-May-2017.) |
β’ π½ = (LBasisβπ) β β’ ((π β LVec β§ π β π½ β§ π β π½) β π β π) | ||
Theorem | lbsextlem1 20771* | Lemma for lbsext 20776. The set π is the set of all linearly independent sets containing πΆ; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014.) |
β’ π = (Baseβπ) & β’ π½ = (LBasisβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β πΆ β π) & β’ (π β βπ₯ β πΆ Β¬ π₯ β (πβ(πΆ β {π₯}))) & β’ π = {π§ β π« π β£ (πΆ β π§ β§ βπ₯ β π§ Β¬ π₯ β (πβ(π§ β {π₯})))} β β’ (π β π β β ) | ||
Theorem | lbsextlem2 20772* | Lemma for lbsext 20776. Since π΄ is a chain (actually, we only need it to be closed under binary union), the union π of the spans of each individual element of π΄ is a subspace, and it contains all of βͺ π΄ (except for our target vector π₯- we are trying to make π₯ a linear combination of all the other vectors in some set from π΄). (Contributed by Mario Carneiro, 25-Jun-2014.) |
β’ π = (Baseβπ) & β’ π½ = (LBasisβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β πΆ β π) & β’ (π β βπ₯ β πΆ Β¬ π₯ β (πβ(πΆ β {π₯}))) & β’ π = {π§ β π« π β£ (πΆ β π§ β§ βπ₯ β π§ Β¬ π₯ β (πβ(π§ β {π₯})))} & β’ π = (LSubSpβπ) & β’ (π β π΄ β π) & β’ (π β π΄ β β ) & β’ (π β [β] Or π΄) & β’ π = βͺ π’ β π΄ (πβ(π’ β {π₯})) β β’ (π β (π β π β§ (βͺ π΄ β {π₯}) β π)) | ||
Theorem | lbsextlem3 20773* | Lemma for lbsext 20776. A chain in π has an upper bound in π. (Contributed by Mario Carneiro, 25-Jun-2014.) |
β’ π = (Baseβπ) & β’ π½ = (LBasisβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β πΆ β π) & β’ (π β βπ₯ β πΆ Β¬ π₯ β (πβ(πΆ β {π₯}))) & β’ π = {π§ β π« π β£ (πΆ β π§ β§ βπ₯ β π§ Β¬ π₯ β (πβ(π§ β {π₯})))} & β’ π = (LSubSpβπ) & β’ (π β π΄ β π) & β’ (π β π΄ β β ) & β’ (π β [β] Or π΄) & β’ π = βͺ π’ β π΄ (πβ(π’ β {π₯})) β β’ (π β βͺ π΄ β π) | ||
Theorem | lbsextlem4 20774* | Lemma for lbsext 20776. lbsextlem3 20773 satisfies the conditions for the application of Zorn's lemma zorn 10502 (thus invoking AC), and so there is a maximal linearly independent set extending πΆ. Here we prove that such a set is a basis. (Contributed by Mario Carneiro, 25-Jun-2014.) |
β’ π = (Baseβπ) & β’ π½ = (LBasisβπ) & β’ π = (LSpanβπ) & β’ (π β π β LVec) & β’ (π β πΆ β π) & β’ (π β βπ₯ β πΆ Β¬ π₯ β (πβ(πΆ β {π₯}))) & β’ π = {π§ β π« π β£ (πΆ β π§ β§ βπ₯ β π§ Β¬ π₯ β (πβ(π§ β {π₯})))} & β’ (π β π« π β dom card) β β’ (π β βπ β π½ πΆ β π ) | ||
Theorem | lbsextg 20775* | For any linearly independent subset πΆ of π, there is a basis containing the vectors in πΆ. (Contributed by Mario Carneiro, 17-May-2015.) |
β’ π½ = (LBasisβπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) β β’ (((π β LVec β§ π« π β dom card) β§ πΆ β π β§ βπ₯ β πΆ Β¬ π₯ β (πβ(πΆ β {π₯}))) β βπ β π½ πΆ β π ) | ||
Theorem | lbsext 20776* | For any linearly independent subset πΆ of π, there is a basis containing the vectors in πΆ. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
β’ π½ = (LBasisβπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) β β’ ((π β LVec β§ πΆ β π β§ βπ₯ β πΆ Β¬ π₯ β (πβ(πΆ β {π₯}))) β βπ β π½ πΆ β π ) | ||
Theorem | lbsexg 20777 | Every vector space has a basis. This theorem is an AC equivalent; this is the forward implication. (Contributed by Mario Carneiro, 17-May-2015.) |
β’ π½ = (LBasisβπ) β β’ ((CHOICE β§ π β LVec) β π½ β β ) | ||
Theorem | lbsex 20778 | Every vector space has a basis. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 25-Jun-2014.) |
β’ π½ = (LBasisβπ) β β’ (π β LVec β π½ β β ) | ||
Theorem | lvecprop2d 20779* | If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 20780 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.) |
β’ (π β π΅ = (BaseβπΎ)) & β’ (π β π΅ = (BaseβπΏ)) & β’ πΉ = (ScalarβπΎ) & β’ πΊ = (ScalarβπΏ) & β’ (π β π = (BaseβπΉ)) & β’ (π β π = (BaseβπΊ)) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) & β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯(+gβπΉ)π¦) = (π₯(+gβπΊ)π¦)) & β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯(.rβπΉ)π¦) = (π₯(.rβπΊ)π¦)) & β’ ((π β§ (π₯ β π β§ π¦ β π΅)) β (π₯( Β·π βπΎ)π¦) = (π₯( Β·π βπΏ)π¦)) β β’ (π β (πΎ β LVec β πΏ β LVec)) | ||
Theorem | lvecpropd 20780* | If two structures have the same components (properties), one is a left vector space iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.) |
β’ (π β π΅ = (BaseβπΎ)) & β’ (π β π΅ = (BaseβπΏ)) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) & β’ (π β πΉ = (ScalarβπΎ)) & β’ (π β πΉ = (ScalarβπΏ)) & β’ π = (BaseβπΉ) & β’ ((π β§ (π₯ β π β§ π¦ β π΅)) β (π₯( Β·π βπΎ)π¦) = (π₯( Β·π βπΏ)π¦)) β β’ (π β (πΎ β LVec β πΏ β LVec)) | ||
Syntax | csra 20781 | Extend class notation with the subring algebra generator. |
class subringAlg | ||
Syntax | crglmod 20782 | Extend class notation with the left module induced by a ring over itself. |
class ringLMod | ||
Syntax | clidl 20783 | Ring left-ideal function. |
class LIdeal | ||
Syntax | crsp 20784 | Ring span function. |
class RSpan | ||
Definition | df-sra 20785* | Any ring can be regarded as a left algebra over any of its subrings. The function subringAlg associates with any ring and any of its subrings the left algebra consisting in the ring itself regarded as a left algebra over the subring. It has an inner product which is simply the ring product. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
β’ subringAlg = (π€ β V β¦ (π β π« (Baseβπ€) β¦ (((π€ sSet β¨(Scalarβndx), (π€ βΎs π )β©) sSet β¨( Β·π βndx), (.rβπ€)β©) sSet β¨(Β·πβndx), (.rβπ€)β©))) | ||
Definition | df-rgmod 20786 | Any ring can be regarded as a left algebra over itself. The function ringLMod associates with any ring the left algebra consisting in the ring itself regarded as a left algebra over itself. It has an inner product which is simply the ring product. (Contributed by Stefan O'Rear, 6-Dec-2014.) |
β’ ringLMod = (π€ β V β¦ ((subringAlg βπ€)β(Baseβπ€))) | ||
Definition | df-lidl 20787 | Define the class of left ideals of a given ring. An ideal is a submodule of the ring viewed as a module over itself. For the usual textbook definition of a (left) ideal of a ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring, see dflidl2 20843. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
β’ LIdeal = (LSubSp β ringLMod) | ||
Definition | df-rsp 20788 | Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.) |
β’ RSpan = (LSpan β ringLMod) | ||
Theorem | sraval 20789 | Lemma for srabase 20792 through sravsca 20800. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
β’ ((π β π β§ π β (Baseβπ)) β ((subringAlg βπ)βπ) = (((π sSet β¨(Scalarβndx), (π βΎs π)β©) sSet β¨( Β·π βndx), (.rβπ)β©) sSet β¨(Β·πβndx), (.rβπ)β©)) | ||
Theorem | sralem 20790 | Lemma for srabase 20792 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) |
β’ (π β π΄ = ((subringAlg βπ)βπ)) & β’ (π β π β (Baseβπ)) & β’ πΈ = Slot (πΈβndx) & β’ (Scalarβndx) β (πΈβndx) & β’ ( Β·π βndx) β (πΈβndx) & β’ (Β·πβndx) β (πΈβndx) β β’ (π β (πΈβπ) = (πΈβπ΄)) | ||
Theorem | sralemOLD 20791 | Obsolete version of sralem 20790 as of 29-Oct-2024. Lemma for srabase 20792 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β π΄ = ((subringAlg βπ)βπ)) & β’ (π β π β (Baseβπ)) & β’ πΈ = Slot π & β’ π β β & β’ (π < 5 β¨ 8 < π) β β’ (π β (πΈβπ) = (πΈβπ΄)) | ||
Theorem | srabase 20792 | Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) |
β’ (π β π΄ = ((subringAlg βπ)βπ)) & β’ (π β π β (Baseβπ)) β β’ (π β (Baseβπ) = (Baseβπ΄)) | ||
Theorem | srabaseOLD 20793 | Obsolete proof of srabase 20792 as of 29-Oct-2024. Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β π΄ = ((subringAlg βπ)βπ)) & β’ (π β π β (Baseβπ)) β β’ (π β (Baseβπ) = (Baseβπ΄)) | ||
Theorem | sraaddg 20794 | Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) |
β’ (π β π΄ = ((subringAlg βπ)βπ)) & β’ (π β π β (Baseβπ)) β β’ (π β (+gβπ) = (+gβπ΄)) | ||
Theorem | sraaddgOLD 20795 | Obsolete proof of sraaddg 20794 as of 29-Oct-2024. Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β π΄ = ((subringAlg βπ)βπ)) & β’ (π β π β (Baseβπ)) β β’ (π β (+gβπ) = (+gβπ΄)) | ||
Theorem | sramulr 20796 | Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) |
β’ (π β π΄ = ((subringAlg βπ)βπ)) & β’ (π β π β (Baseβπ)) β β’ (π β (.rβπ) = (.rβπ΄)) | ||
Theorem | sramulrOLD 20797 | Obsolete proof of sramulr 20796 as of 29-Oct-2024. Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β π΄ = ((subringAlg βπ)βπ)) & β’ (π β π β (Baseβπ)) β β’ (π β (.rβπ) = (.rβπ΄)) | ||
Theorem | srasca 20798 | The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.) |
β’ (π β π΄ = ((subringAlg βπ)βπ)) & β’ (π β π β (Baseβπ)) β β’ (π β (π βΎs π) = (Scalarβπ΄)) | ||
Theorem | srascaOLD 20799 | Obsolete proof of srasca 20798 as of 12-Nov-2024. The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 12-Nov-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β π΄ = ((subringAlg βπ)βπ)) & β’ (π β π β (Baseβπ)) β β’ (π β (π βΎs π) = (Scalarβπ΄)) | ||
Theorem | sravsca 20800 | The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.) |
β’ (π β π΄ = ((subringAlg βπ)βπ)) & β’ (π β π β (Baseβπ)) β β’ (π β (.rβπ) = ( Β·π βπ΄)) |
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