![]() |
Metamath
Proof Explorer Theorem List (p. 209 of 480) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30439) |
![]() (30440-31962) |
![]() (31963-47940) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lmhmvsca 20801 | The pointwise scalar product of a linear function and a constant is linear, over a commutative ring. (Contributed by Mario Carneiro, 22-Sep-2015.) |
β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π½ = (Scalarβπ) & β’ πΎ = (Baseβπ½) β β’ ((π½ β CRing β§ π΄ β πΎ β§ πΉ β (π LMHom π)) β ((π Γ {π΄}) βf Β· πΉ) β (π LMHom π)) | ||
Theorem | lmhmf1o 20802 | A bijective module homomorphism is also converse homomorphic. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
β’ π = (Baseβπ) & β’ π = (Baseβπ) β β’ (πΉ β (π LMHom π) β (πΉ:πβ1-1-ontoβπ β β‘πΉ β (π LMHom π))) | ||
Theorem | lmhmima 20803 | The image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
β’ π = (LSubSpβπ) & β’ π = (LSubSpβπ) β β’ ((πΉ β (π LMHom π) β§ π β π) β (πΉ β π) β π) | ||
Theorem | lmhmpreima 20804 | The inverse image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
β’ π = (LSubSpβπ) & β’ π = (LSubSpβπ) β β’ ((πΉ β (π LMHom π) β§ π β π) β (β‘πΉ β π) β π) | ||
Theorem | lmhmlsp 20805 | Homomorphisms preserve spans. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
β’ π = (Baseβπ) & β’ πΎ = (LSpanβπ) & β’ πΏ = (LSpanβπ) β β’ ((πΉ β (π LMHom π) β§ π β π) β (πΉ β (πΎβπ)) = (πΏβ(πΉ β π))) | ||
Theorem | lmhmrnlss 20806 | The range of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
β’ (πΉ β (π LMHom π) β ran πΉ β (LSubSpβπ)) | ||
Theorem | lmhmkerlss 20807 | The kernel of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
β’ πΎ = (β‘πΉ β { 0 }) & β’ 0 = (0gβπ) & β’ π = (LSubSpβπ) β β’ (πΉ β (π LMHom π) β πΎ β π) | ||
Theorem | reslmhm 20808 | Restriction of a homomorphism to a subspace. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
β’ π = (LSubSpβπ) & β’ π = (π βΎs π) β β’ ((πΉ β (π LMHom π) β§ π β π) β (πΉ βΎ π) β (π LMHom π)) | ||
Theorem | reslmhm2 20809 | Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
β’ π = (π βΎs π) & β’ πΏ = (LSubSpβπ) β β’ ((πΉ β (π LMHom π) β§ π β LMod β§ π β πΏ) β πΉ β (π LMHom π)) | ||
Theorem | reslmhm2b 20810 | Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
β’ π = (π βΎs π) & β’ πΏ = (LSubSpβπ) β β’ ((π β LMod β§ π β πΏ β§ ran πΉ β π) β (πΉ β (π LMHom π) β πΉ β (π LMHom π))) | ||
Theorem | lmhmeql 20811 | The equalizer of two module homomorphisms is a subspace. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
β’ π = (LSubSpβπ) β β’ ((πΉ β (π LMHom π) β§ πΊ β (π LMHom π)) β dom (πΉ β© πΊ) β π) | ||
Theorem | lspextmo 20812* | A linear function is completely determined (or overdetermined) by its values on a spanning subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by NM, 17-Jun-2017.) |
β’ π΅ = (Baseβπ) & β’ πΎ = (LSpanβπ) β β’ ((π β π΅ β§ (πΎβπ) = π΅) β β*π β (π LMHom π)(π βΎ π) = πΉ) | ||
Theorem | pwsdiaglmhm 20813* | Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ ) & β’ πΉ = (π₯ β π΅ β¦ (πΌ Γ {π₯})) β β’ ((π β LMod β§ πΌ β π) β πΉ β (π LMHom π)) | ||
Theorem | pwssplit0 20814* | Splitting for structure powers, part 0: restriction is a function. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
β’ π = (π βs π) & β’ π = (π βs π) & β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) & β’ πΉ = (π₯ β π΅ β¦ (π₯ βΎ π)) β β’ ((π β π β§ π β π β§ π β π) β πΉ:π΅βΆπΆ) | ||
Theorem | pwssplit1 20815* | Splitting for structure powers, part 1: restriction is an onto function. The only actual monoid law we need here is that the base set is nonempty. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
β’ π = (π βs π) & β’ π = (π βs π) & β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) & β’ πΉ = (π₯ β π΅ β¦ (π₯ βΎ π)) β β’ ((π β Mnd β§ π β π β§ π β π) β πΉ:π΅βontoβπΆ) | ||
Theorem | pwssplit2 20816* | Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
β’ π = (π βs π) & β’ π = (π βs π) & β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) & β’ πΉ = (π₯ β π΅ β¦ (π₯ βΎ π)) β β’ ((π β Grp β§ π β π β§ π β π) β πΉ β (π GrpHom π)) | ||
Theorem | pwssplit3 20817* | Splitting for structure powers, part 3: restriction is a module homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
β’ π = (π βs π) & β’ π = (π βs π) & β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) & β’ πΉ = (π₯ β π΅ β¦ (π₯ βΎ π)) β β’ ((π β LMod β§ π β π β§ π β π) β πΉ β (π LMHom π)) | ||
Theorem | islmim 20818 | An isomorphism of left modules is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
β’ π΅ = (Baseβπ ) & β’ πΆ = (Baseβπ) β β’ (πΉ β (π LMIso π) β (πΉ β (π LMHom π) β§ πΉ:π΅β1-1-ontoβπΆ)) | ||
Theorem | lmimf1o 20819 | An isomorphism of left modules is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
β’ π΅ = (Baseβπ ) & β’ πΆ = (Baseβπ) β β’ (πΉ β (π LMIso π) β πΉ:π΅β1-1-ontoβπΆ) | ||
Theorem | lmimlmhm 20820 | An isomorphism of modules is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
β’ (πΉ β (π LMIso π) β πΉ β (π LMHom π)) | ||
Theorem | lmimgim 20821 | An isomorphism of modules is an isomorphism of groups. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
β’ (πΉ β (π LMIso π) β πΉ β (π GrpIso π)) | ||
Theorem | islmim2 20822 | An isomorphism of left modules is a homomorphism whose converse is a homomorphism. (Contributed by Mario Carneiro, 6-May-2015.) |
β’ (πΉ β (π LMIso π) β (πΉ β (π LMHom π) β§ β‘πΉ β (π LMHom π ))) | ||
Theorem | lmimcnv 20823 | The converse of a bijective module homomorphism is a bijective module homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
β’ (πΉ β (π LMIso π) β β‘πΉ β (π LMIso π)) | ||
Theorem | brlmic 20824 | The relation "is isomorphic to" for modules. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
β’ (π βπ π β (π LMIso π) β β ) | ||
Theorem | brlmici 20825 | Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
β’ (πΉ β (π LMIso π) β π βπ π) | ||
Theorem | lmiclcl 20826 | Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
β’ (π βπ π β π β LMod) | ||
Theorem | lmicrcl 20827 | Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015.) |
β’ (π βπ π β π β LMod) | ||
Theorem | lmicsym 20828 | Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
β’ (π βπ π β π βπ π ) | ||
Theorem | lmhmpropd 20829* | Module homomorphism depends only on the module attributes of structures. (Contributed by Mario Carneiro, 8-Oct-2015.) |
β’ (π β π΅ = (Baseβπ½)) & β’ (π β πΆ = (BaseβπΎ)) & β’ (π β π΅ = (BaseβπΏ)) & β’ (π β πΆ = (Baseβπ)) & β’ (π β πΉ = (Scalarβπ½)) & β’ (π β πΊ = (ScalarβπΎ)) & β’ (π β πΉ = (ScalarβπΏ)) & β’ (π β πΊ = (Scalarβπ)) & β’ π = (BaseβπΉ) & β’ π = (BaseβπΊ) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπ½)π¦) = (π₯(+gβπΏ)π¦)) & β’ ((π β§ (π₯ β πΆ β§ π¦ β πΆ)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπ)π¦)) & β’ ((π β§ (π₯ β π β§ π¦ β π΅)) β (π₯( Β·π βπ½)π¦) = (π₯( Β·π βπΏ)π¦)) & β’ ((π β§ (π₯ β π β§ π¦ β πΆ)) β (π₯( Β·π βπΎ)π¦) = (π₯( Β·π βπ)π¦)) β β’ (π β (π½ LMHom πΎ) = (πΏ LMHom π)) | ||
Syntax | clbs 20830 | Extend class notation with the set of bases for a vector space. |
class LBasis | ||
Definition | df-lbs 20831* | Define the set of bases to a left module or left vector space. (Contributed by Mario Carneiro, 24-Jun-2014.) |
β’ LBasis = (π€ β V β¦ {π β π« (Baseβπ€) β£ [(LSpanβπ€) / π][(Scalarβπ€) / π ]((πβπ) = (Baseβπ€) β§ βπ₯ β π βπ¦ β ((Baseβπ ) β {(0gβπ )}) Β¬ (π¦( Β·π βπ€)π₯) β (πβ(π β {π₯})))}) | ||
Theorem | islbs 20832* | The predicate "π΅ is a basis for the left module or vector space π". A subset of the base set is a basis if zero is not in the set, it spans the set, and no nonzero multiple of an element of the basis is in the span of the rest of the family. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 14-Jan-2015.) |
β’ π = (Baseβπ) & β’ πΉ = (Scalarβπ) & β’ Β· = ( Β·π βπ) & β’ πΎ = (BaseβπΉ) & β’ π½ = (LBasisβπ) & β’ π = (LSpanβπ) & β’ 0 = (0gβπΉ) β β’ (π β π β (π΅ β π½ β (π΅ β π β§ (πβπ΅) = π β§ βπ₯ β π΅ βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π΅ β {π₯}))))) | ||
Theorem | lbsss 20833 | A basis is a set of vectors. (Contributed by Mario Carneiro, 24-Jun-2014.) |
β’ π = (Baseβπ) & β’ π½ = (LBasisβπ) β β’ (π΅ β π½ β π΅ β π) | ||
Theorem | lbsel 20834 | An element of a basis is a vector. (Contributed by Mario Carneiro, 24-Jun-2014.) |
β’ π = (Baseβπ) & β’ π½ = (LBasisβπ) β β’ ((π΅ β π½ β§ πΈ β π΅) β πΈ β π) | ||
Theorem | lbssp 20835 | The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.) |
β’ π = (Baseβπ) & β’ π½ = (LBasisβπ) & β’ π = (LSpanβπ) β β’ (π΅ β π½ β (πβπ΅) = π) | ||
Theorem | lbsind 20836 | A basis is linearly independent; that is, every element has a span which trivially intersects the span of the remainder of the basis. (Contributed by Mario Carneiro, 12-Jan-2015.) |
β’ π = (Baseβπ) & β’ π½ = (LBasisβπ) & β’ π = (LSpanβπ) & β’ πΉ = (Scalarβπ) & β’ Β· = ( Β·π βπ) & β’ πΎ = (BaseβπΉ) & β’ 0 = (0gβπΉ) β β’ (((π΅ β π½ β§ πΈ β π΅) β§ (π΄ β πΎ β§ π΄ β 0 )) β Β¬ (π΄ Β· πΈ) β (πβ(π΅ β {πΈ}))) | ||
Theorem | lbsind2 20837 | A basis is linearly independent; that is, every element is not in the span of the remainder of the basis. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 12-Jan-2015.) |
β’ π½ = (LBasisβπ) & β’ π = (LSpanβπ) & β’ πΉ = (Scalarβπ) & β’ 1 = (1rβπΉ) & β’ 0 = (0gβπΉ) β β’ (((π β LMod β§ 1 β 0 ) β§ π΅ β π½ β§ πΈ β π΅) β Β¬ πΈ β (πβ(π΅ β {πΈ}))) | ||
Theorem | lbspss 20838 | No proper subset of a basis spans the space. (Contributed by Mario Carneiro, 25-Jun-2014.) |
β’ π½ = (LBasisβπ) & β’ π = (LSpanβπ) & β’ πΉ = (Scalarβπ) & β’ 1 = (1rβπΉ) & β’ 0 = (0gβπΉ) & β’ π = (Baseβπ) β β’ (((π β LMod β§ 1 β 0 ) β§ π΅ β π½ β§ πΆ β π΅) β (πβπΆ) β π) | ||
Theorem | lsmcl 20839 | The sum of two subspaces is a subspace. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
β’ π = (LSubSpβπ) & β’ β = (LSSumβπ) β β’ ((π β LMod β§ π β π β§ π β π) β (π β π) β π) | ||
Theorem | lsmspsn 20840* | Member of subspace sum of spans of singletons. (Contributed by NM, 8-Apr-2015.) |
β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) & β’ Β· = ( Β·π βπ) & β’ β = (LSSumβπ) & 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}) β (πβ{π})) β βπ β πΎ βπ β πΎ π = ((π Β· π) + (π Β· π)))) | ||
Theorem | lsmelval2 20841* | Subspace sum membership in terms of a sum of 1-dim subspaces (atoms), which can be useful for treating subspaces as projective lattice elements. (Contributed by NM, 9-Aug-2014.) |
β’ π = (Baseβπ) & β’ π = (LSubSpβπ) & β’ β = (LSSumβπ) & 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 (π β π) β (π β π β§ βπ¦ β π βπ§ β π (πβ{π}) β ((πβ{π¦}) β (πβ{π§}))))) | ||
Theorem | lsmsp 20842 | Subspace sum in terms of span. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.) |
β’ π = (LSubSpβπ) & β’ π = (LSpanβπ) & β’ β = (LSSumβπ) β β’ ((π β LMod β§ π β π β§ π β π) β (π β π) = (πβ(π βͺ π))) | ||
Theorem | lsmsp2 20843 | Subspace sum of spans of subsets is the span of their union. (spanuni 31065 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) |
β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ β = (LSSumβπ) β β’ ((π β LMod β§ π β π β§ π β π) β ((πβπ) β (πβπ)) = (πβ(π βͺ π))) | ||
Theorem | lsmssspx 20844 | Subspace sum (in its extended domain) is a subset of the span of the union of its arguments. (Contributed by NM, 6-Aug-2014.) |
β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ β = (LSSumβπ) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β LMod) β β’ (π β (π β π) β (πβ(π βͺ π))) | ||
Theorem | lsmpr 20845 | The span of a pair of vectors equals the sum of the spans of their singletons. (Contributed by NM, 13-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 β’ (π β (πβ{π, π}) = ((πβ{π}) β (πβ{π}))) | ||
Theorem | lsppreli 20846 | 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 20847 | 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 20848 | 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 20849* | 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 20850* | 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 20851 | 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 20852 | 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 20853 | 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 20854 | 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 20855* | 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 20856 | 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 20857 | 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 20858 | Extend class notation with class of all left vector spaces. |
class LVec | ||
Definition | df-lvec 20859 | 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 20860 | 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 20861 | 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 20862 | A left vector space is a left module. (Contributed by NM, 9-Dec-2013.) |
β’ (π β LVec β π β LMod) | ||
Theorem | lveclmodd 20863 | 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 | lvecgrpd 20864 | A vector space is a group. (Contributed by SN, 16-May-2024.) |
β’ (π β π β LVec) β β’ (π β π β Grp) | ||
Theorem | lsslvec 20865 | 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 20866 | 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 20867 | If a scalar product is zero, one of its factors must be zero. (hvmul0or 30546 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 20868 | 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 20869 | 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 20870 | Cancellation law for scalar multiplication. (hvmulcan 30593 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 20871 | Cancellation law for scalar multiplication. (hvmulcan2 30594 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 20872 | 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 20873 | A nonzero scalar product does not change the span of a singleton. (spansncol 31089 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 20874 | Membership relation that implies equality of spans. (spansneleq 31091 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 20875 | 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 20876 | 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 20877 | 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 20878 | 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 20879 | 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 20880 | 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 20881* | Equal spans of singletons must have proportional vectors. See lspsnss2 20761 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 20882* | 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 20883 | A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 31094 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 20884 | 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 20885 | 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 20886 | 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 20887* | 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 20888 | 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 20889 versus lspexchn2 20890); look for lspexch 20888 and prcom 4736 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 20889 | Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 20888 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 20890 | Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 20888 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 20891 | 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 20892 | 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 20893 | 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 20894 | 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 20895 | 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 20896 | (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 20897 | 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 20898 | 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 20899 | 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 20900 | Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 31173 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}))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |