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Type | Label | Description |
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Statement | ||
Theorem | lincscm 47101* | A linear combinations multiplied with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 9-Apr-2019.) (Revised by AV, 28-Jul-2019.) |
β’ β = ( Β·π βπ) & β’ Β· = (.rβ(Scalarβπ)) & β’ π = (π΄( linC βπ)π) & β’ π = (Baseβ(Scalarβπ)) & β’ πΉ = (π₯ β π β¦ (π Β· (π΄βπ₯))) β β’ (((π β LMod β§ π β π« (Baseβπ)) β§ (π΄ β (π βm π) β§ π β π ) β§ π΄ finSupp (0gβ(Scalarβπ))) β (π β π) = (πΉ( linC βπ)π)) | ||
Theorem | lincsumcl 47102 | The sum of two linear combinations is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.) |
β’ + = (+gβπ) β β’ (((π β LMod β§ π β π« (Baseβπ)) β§ (πΆ β (π LinCo π) β§ π· β (π LinCo π))) β (πΆ + π·) β (π LinCo π)) | ||
Theorem | lincscmcl 47103 | The multiplication of a linear combination with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 11-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.) |
β’ Β· = ( Β·π βπ) & β’ π = (Baseβ(Scalarβπ)) β β’ (((π β LMod β§ π β π« (Baseβπ)) β§ πΆ β π β§ π· β (π LinCo π)) β (πΆ Β· π·) β (π LinCo π)) | ||
Theorem | lincsumscmcl 47104 | The sum of a linear combination and a multiplication of a linear combination with a scalar is a linear combination. (Contributed by AV, 11-Apr-2019.) |
β’ Β· = ( Β·π βπ) & β’ π = (Baseβ(Scalarβπ)) & β’ + = (+gβπ) β β’ (((π β LMod β§ π β π« (Baseβπ)) β§ (πΆ β π β§ π· β (π LinCo π) β§ π΅ β (π LinCo π))) β ((πΆ Β· π·) + π΅) β (π LinCo π)) | ||
Theorem | lincolss 47105 | According to the statement in [Lang] p. 129, the set (LSubSpβπ) of all linear combinations of a set of vectors V is a submodule (generated by V) of the module M. The elements of V are called generators of (LSubSpβπ). (Contributed by AV, 12-Apr-2019.) |
β’ ((π β LMod β§ π β π« (Baseβπ)) β (π LinCo π) β (LSubSpβπ)) | ||
Theorem | ellcoellss 47106* | Every linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
β’ ((π β LMod β§ π β (LSubSpβπ) β§ π β π) β βπ₯ β (π LinCo π)π₯ β π) | ||
Theorem | lcoss 47107 | A set of vectors of a module is a subset of the set of all linear combinations of the set. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
β’ ((π β LMod β§ π β π« (Baseβπ)) β π β (π LinCo π)) | ||
Theorem | lspsslco 47108 | Lemma for lspeqlco 47110. (Contributed by AV, 17-Apr-2019.) |
β’ π΅ = (Baseβπ) β β’ ((π β LMod β§ π β π« π΅) β ((LSpanβπ)βπ) β (π LinCo π)) | ||
Theorem | lcosslsp 47109 | Lemma for lspeqlco 47110. (Contributed by AV, 20-Apr-2019.) |
β’ π΅ = (Baseβπ) β β’ ((π β LMod β§ π β π« π΅) β (π LinCo π) β ((LSpanβπ)βπ)) | ||
Theorem | lspeqlco 47110 | Equivalence of a span of a set of vectors of a left module defined as the intersection of all linear subspaces which each contain every vector in that set (see df-lsp 20582) and as the set of all linear combinations of the vectors of the set with finite support. (Contributed by AV, 20-Apr-2019.) |
β’ π΅ = (Baseβπ) β β’ ((π β LMod β§ π β π« π΅) β (π LinCo π) = ((LSpanβπ)βπ)) | ||
According to the definition in [Lang] p. 129: "A subset S of a module M is said
to be linearly independent (over [the ring] A) if whenever we have a
linear combination βx βS axx which is equal to
0, then ax=0 for all xβS." This definition does not care for
the finiteness of the set S (because the definition of a linear combination
in [Lang] p.129 does already assure that only a finite number of coefficients
can be 0 in the sum). Our definition df-lininds 47113 does also neither claim that
the subset must be finite, nor that almost all coefficients within the linear
combination are 0. If this is required, it must be explicitly stated as
precondition in the corresponding theorems. | ||
Syntax | clininds 47111 | Extend class notation with the relation between a module and its linearly independent subsets. |
class linIndS | ||
Syntax | clindeps 47112 | Extend class notation with the relation between a module and its linearly dependent subsets. |
class linDepS | ||
Definition | df-lininds 47113* | Define the relation between a module and its linearly independent subsets. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
β’ linIndS = {β¨π , πβ© β£ (π β π« (Baseβπ) β§ βπ β ((Baseβ(Scalarβπ)) βm π )((π finSupp (0gβ(Scalarβπ)) β§ (π( linC βπ)π ) = (0gβπ)) β βπ₯ β π (πβπ₯) = (0gβ(Scalarβπ))))} | ||
Theorem | rellininds 47114 | The class defining the relation between a module and its linearly independent subsets is a relation. (Contributed by AV, 13-Apr-2019.) |
β’ Rel linIndS | ||
Definition | df-lindeps 47115* | Define the relation between a module and its linearly dependent subsets. (Contributed by AV, 26-Apr-2019.) |
β’ linDepS = {β¨π , πβ© β£ Β¬ π linIndS π} | ||
Theorem | linindsv 47116 | The classes of the module and its linearly independent subsets are sets. (Contributed by AV, 13-Apr-2019.) |
β’ (π linIndS π β (π β V β§ π β V)) | ||
Theorem | islininds 47117* | The property of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (0gβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ 0 = (0gβπ ) β β’ ((π β π β§ π β π) β (π linIndS π β (π β π« π΅ β§ βπ β (πΈ βm π)((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 )))) | ||
Theorem | linindsi 47118* | The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (0gβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ 0 = (0gβπ ) β β’ (π linIndS π β (π β π« π΅ β§ βπ β (πΈ βm π)((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 ))) | ||
Theorem | linindslinci 47119* | The implications of being a linearly independent subset and a linear combination of this subset being 0. (Contributed by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (0gβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ 0 = (0gβπ ) β β’ ((π linIndS π β§ (πΉ β (πΈ βm π) β§ πΉ finSupp 0 β§ (πΉ( linC βπ)π) = π)) β βπ₯ β π (πΉβπ₯) = 0 ) | ||
Theorem | islinindfis 47120* | The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (0gβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ 0 = (0gβπ ) β β’ ((π β Fin β§ π β π) β (π linIndS π β (π β π« π΅ β§ βπ β (πΈ βm π)((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 )))) | ||
Theorem | islinindfiss 47121* | The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (0gβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ 0 = (0gβπ ) β β’ ((π β π β§ π β Fin β§ π β π« π΅) β (π linIndS π β βπ β (πΈ βm π)((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 ))) | ||
Theorem | linindscl 47122 | A linearly independent set is a subset of (the base set of) a module. (Contributed by AV, 13-Apr-2019.) |
β’ (π linIndS π β π β π« (Baseβπ)) | ||
Theorem | lindepsnlininds 47123 | A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019.) |
β’ ((π β π β§ π β π) β (π linDepS π β Β¬ π linIndS π)) | ||
Theorem | islindeps 47124* | The property of being a linearly dependent subset. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (0gβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ 0 = (0gβπ ) β β’ ((π β π β§ π β π« π΅) β (π linDepS π β βπ β (πΈ βm π)(π finSupp 0 β§ (π( linC βπ)π) = π β§ βπ₯ β π (πβπ₯) β 0 ))) | ||
Theorem | lincext1 47125* | Property 1 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 29-Apr-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = (invgβπ ) & β’ πΉ = (π§ β π β¦ if(π§ = π, (πβπ), (πΊβπ§))) β β’ (((π β LMod β§ π β π« π΅) β§ (π β πΈ β§ π β π β§ πΊ β (πΈ βm (π β {π})))) β πΉ β (πΈ βm π)) | ||
Theorem | lincext2 47126* | Property 2 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = (invgβπ ) & β’ πΉ = (π§ β π β¦ if(π§ = π, (πβπ), (πΊβπ§))) β β’ (((π β LMod β§ π β π« π΅) β§ (π β πΈ β§ π β π β§ πΊ β (πΈ βm (π β {π}))) β§ πΊ finSupp 0 ) β πΉ finSupp 0 ) | ||
Theorem | lincext3 47127* | Property 3 of an extension of a linear combination. (Contributed by AV, 23-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = (invgβπ ) & β’ πΉ = (π§ β π β¦ if(π§ = π, (πβπ), (πΊβπ§))) β β’ (((π β LMod β§ π β π« π΅) β§ (π β πΈ β§ π β π β§ πΊ β (πΈ βm (π β {π}))) β§ (πΊ finSupp 0 β§ (π( Β·π βπ)π) = (πΊ( linC βπ)(π β {π})))) β (πΉ( linC βπ)π) = π) | ||
Theorem | lindslinindsimp1 47128* | Implication 1 for lindslininds 47135. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) (Proof shortened by II, 16-Feb-2023.) |
β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) β β’ ((π β π β§ π β LMod) β ((π β π« (Baseβπ) β§ βπ β (π΅ βm π)((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 )) β (π β (Baseβπ) β§ βπ β π βπ¦ β (π΅ β { 0 }) Β¬ (π¦( Β·π βπ)π ) β ((LSpanβπ)β(π β {π }))))) | ||
Theorem | lindslinindimp2lem1 47129* | Lemma 1 for lindslinindsimp2 47134. (Contributed by AV, 25-Apr-2019.) |
β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = ((invgβπ )β(πβπ₯)) & β’ πΊ = (π βΎ (π β {π₯})) β β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π β§ π β (π΅ βm π))) β π β π΅) | ||
Theorem | lindslinindimp2lem2 47130* | Lemma 2 for lindslinindsimp2 47134. (Contributed by AV, 25-Apr-2019.) |
β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = ((invgβπ )β(πβπ₯)) & β’ πΊ = (π βΎ (π β {π₯})) β β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π β§ π β (π΅ βm π))) β πΊ β (π΅ βm (π β {π₯}))) | ||
Theorem | lindslinindimp2lem3 47131* | Lemma 3 for lindslinindsimp2 47134. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = ((invgβπ )β(πβπ₯)) & β’ πΊ = (π βΎ (π β {π₯})) β β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π) β§ (π β (π΅ βm π) β§ π finSupp 0 )) β πΊ finSupp 0 ) | ||
Theorem | lindslinindimp2lem4 47132* | Lemma 4 for lindslinindsimp2 47134. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) (Proof shortened by II, 16-Feb-2023.) |
β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = ((invgβπ )β(πβπ₯)) & β’ πΊ = (π βΎ (π β {π₯})) β β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π) β§ (π β (π΅ βm π) β§ π finSupp 0 β§ (π( linC βπ)π) = π)) β (π Ξ£g (π¦ β (π β {π₯}) β¦ ((πβπ¦)( Β·π βπ)π¦))) = (π( Β·π βπ)π₯)) | ||
Theorem | lindslinindsimp2lem5 47133* | Lemma 5 for lindslinindsimp2 47134. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) β β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π)) β ((π β (π΅ βm π) β§ (π finSupp 0 β§ (π( linC βπ)π) = π)) β (βπ¦ β (π΅ β { 0 })βπ β (π΅ βm (π β {π₯}))(Β¬ π finSupp 0 β¨ Β¬ (π¦( Β·π βπ)π₯) = (π( linC βπ)(π β {π₯}))) β (πβπ₯) = 0 ))) | ||
Theorem | lindslinindsimp2 47134* | Implication 2 for lindslininds 47135. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) β β’ ((π β π β§ π β LMod) β ((π β (Baseβπ) β§ βπ β π βπ¦ β (π΅ β { 0 }) Β¬ (π¦( Β·π βπ)π ) β ((LSpanβπ)β(π β {π }))) β (π β π« (Baseβπ) β§ βπ β (π΅ βm π)((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 )))) | ||
Theorem | lindslininds 47135 | Equivalence of definitions df-linds 21361 and df-lininds 47113 for (linear) independence for (left) modules. (Contributed by AV, 26-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
β’ ((π β π β§ π β LMod) β (π linIndS π β π β (LIndSβπ))) | ||
Theorem | linds0 47136 | The empty set is always a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
β’ (π β π β β linIndS π) | ||
Theorem | el0ldep 47137 | A set containing the zero element of a module is always linearly dependent, if the underlying ring has at least two elements. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
β’ (((π β LMod β§ 1 < (β―β(Baseβ(Scalarβπ)))) β§ π β π« (Baseβπ) β§ (0gβπ) β π) β π linDepS π) | ||
Theorem | el0ldepsnzr 47138 | A set containing the zero element of a module over a nonzero ring is always linearly dependent. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.) |
β’ (((π β LMod β§ (Scalarβπ) β NzRing) β§ π β π« (Baseβπ) β§ (0gβπ) β π) β π linDepS π) | ||
Theorem | lindsrng01 47139 | Any subset of a module is always linearly independent if the underlying ring has at most one element. Since the underlying ring cannot be the empty set (see lmodsn0 20484), this means that the underlying ring has only one element, so it is a zero ring. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) β β’ ((π β LMod β§ ((β―βπΈ) = 0 β¨ (β―βπΈ) = 1) β§ π β π« π΅) β π linIndS π) | ||
Theorem | lindszr 47140 | Any subset of a module over a zero ring is always linearly independent. (Contributed by AV, 27-Apr-2019.) |
β’ ((π β LMod β§ Β¬ (Scalarβπ) β NzRing β§ π β π« (Baseβπ)) β π linIndS π) | ||
Theorem | snlindsntorlem 47141* | Lemma for snlindsntor 47142. (Contributed by AV, 15-Apr-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ π = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ Β· = ( Β·π βπ) β β’ ((π β LMod β§ π β π΅) β (βπ β (π βm {π})((π( linC βπ){π}) = π β (πβπ) = 0 ) β βπ β π ((π Β· π) = π β π = 0 ))) | ||
Theorem | snlindsntor 47142* | A singleton is linearly independent iff it does not contain a torsion element. According to Wikipedia ("Torsion (algebra)", 15-Apr-2019, https://en.wikipedia.org/wiki/Torsion_(algebra)): "An element m of a module M over a ring R is called a torsion element of the module if there exists a regular element r of the ring (an element that is neither a left nor a right zero divisor) that annihilates m, i.e., (π Β· π) = 0. In an integral domain (a commutative ring without zero divisors), every nonzero element is regular, so a torsion element of a module over an integral domain is one annihilated by a nonzero element of the integral domain." Analogously, the definition in [Lang] p. 147 states that "An element x of [a module] E [over a ring R] is called a torsion element if there exists π β π , π β 0, such that π Β· π₯ = 0. This definition includes the zero element of the module. Some authors, however, exclude the zero element from the definition of torsion elements. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ π = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ Β· = ( Β·π βπ) β β’ ((π β LMod β§ π β π΅) β (βπ β (π β { 0 })(π Β· π) β π β {π} linIndS π)) | ||
Theorem | ldepsprlem 47143 | Lemma for ldepspr 47144. (Contributed by AV, 16-Apr-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ π = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ 1 = (1rβπ ) & β’ π = (invgβπ ) β β’ ((π β LMod β§ (π β π΅ β§ π β π΅ β§ π΄ β π)) β (π = (π΄ Β· π) β (( 1 Β· π)(+gβπ)((πβπ΄) Β· π)) = π)) | ||
Theorem | ldepspr 47144 | If a vector is a scalar multiple of another vector, the (unordered pair containing the) two vectors are linearly dependent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ π = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ Β· = ( Β·π βπ) β β’ ((π β LMod β§ (π β π΅ β§ π β π΅ β§ π β π)) β ((π΄ β π β§ π = (π΄ Β· π)) β {π, π} linDepS π)) | ||
Theorem | lincresunit3lem3 47145 | Lemma 3 for lincresunit3 47152. (Contributed by AV, 18-May-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ π = (Unitβπ ) & β’ π = (invgβπ ) & β’ Β· = ( Β·π βπ) β β’ (((π β LMod β§ π β π΅ β§ π β π΅) β§ π΄ β π) β (((πβπ΄) Β· π) = ((πβπ΄) Β· π) β π = π)) | ||
Theorem | lincresunitlem1 47146 | Lemma 1 for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ π = (Unitβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = (invgβπ ) & β’ πΌ = (invrβπ ) & β’ Β· = (.rβπ ) & β’ πΊ = (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))) β β’ (((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π)) β (πΌβ(πβ(πΉβπ))) β πΈ) | ||
Theorem | lincresunitlem2 47147 | Lemma for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ π = (Unitβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = (invgβπ ) & β’ πΌ = (invrβπ ) & β’ Β· = (.rβπ ) & β’ πΊ = (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))) β β’ ((((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π)) β§ π β π) β ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ)) β πΈ) | ||
Theorem | lincresunit1 47148* | Property 1 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ π = (Unitβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = (invgβπ ) & β’ πΌ = (invrβπ ) & β’ Β· = (.rβπ ) & β’ πΊ = (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))) β β’ (((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π)) β πΊ β (πΈ βm (π β {π}))) | ||
Theorem | lincresunit2 47149* | Property 2 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ π = (Unitβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = (invgβπ ) & β’ πΌ = (invrβπ ) & β’ Β· = (.rβπ ) & β’ πΊ = (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))) β β’ (((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π β§ πΉ finSupp 0 )) β πΊ finSupp 0 ) | ||
Theorem | lincresunit3lem1 47150* | Lemma 1 for lincresunit3 47152. (Contributed by AV, 17-May-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ π = (Unitβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = (invgβπ ) & β’ πΌ = (invrβπ ) & β’ Β· = (.rβπ ) & β’ πΊ = (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))) β β’ (((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π β§ π§ β (π β {π}))) β ((πβ(πΉβπ))( Β·π βπ)((πΊβπ§)( Β·π βπ)π§)) = ((πΉβπ§)( Β·π βπ)π§)) | ||
Theorem | lincresunit3lem2 47151* | Lemma 2 for lincresunit3 47152. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ π = (Unitβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = (invgβπ ) & β’ πΌ = (invrβπ ) & β’ Β· = (.rβπ ) & β’ πΊ = (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))) β β’ (((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π β§ πΉ finSupp 0 )) β ((πβ(πΉβπ))( Β·π βπ)(π Ξ£g (π§ β (π β {π}) β¦ ((πΊβπ§)( Β·π βπ)π§)))) = ((πΉ βΎ (π β {π}))( linC βπ)(π β {π}))) | ||
Theorem | lincresunit3 47152* | Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ π = (Unitβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = (invgβπ ) & β’ πΌ = (invrβπ ) & β’ Β· = (.rβπ ) & β’ πΊ = (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))) β β’ (((π β π« π΅ β§ π β LMod β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β π β§ πΉ finSupp 0 ) β§ (πΉ( linC βπ)π) = π) β (πΊ( linC βπ)(π β {π})) = π) | ||
Theorem | lincreslvec3 47153* | Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ π = (Unitβπ ) & β’ 0 = (0gβπ ) & β’ π = (0gβπ) & β’ π = (invgβπ ) & β’ πΌ = (invrβπ ) & β’ Β· = (.rβπ ) & β’ πΊ = (π β (π β {π}) β¦ ((πΌβ(πβ(πΉβπ))) Β· (πΉβπ ))) β β’ (((π β π« π΅ β§ π β LVec β§ π β π) β§ (πΉ β (πΈ βm π) β§ (πΉβπ) β 0 β§ πΉ finSupp 0 ) β§ (πΉ( linC βπ)π) = π) β (πΊ( linC βπ)(π β {π})) = π) | ||
Theorem | islindeps2 47154* | Conditions for being a linearly dependent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (0gβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ 0 = (0gβπ ) β β’ ((π β LMod β§ π β π« π΅ β§ π β NzRing) β (βπ β π βπ β (πΈ βm (π β {π }))(π finSupp 0 β§ (π( linC βπ)(π β {π })) = π ) β π linDepS π)) | ||
Theorem | islininds2 47155* | Implication of being a linearly independent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (0gβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ 0 = (0gβπ ) β β’ ((π β LMod β§ π β π« π΅ β§ π β NzRing) β (π linIndS π β βπ β π βπ β (πΈ βm (π β {π }))(Β¬ π finSupp 0 β¨ (π( linC βπ)(π β {π })) β π ))) | ||
Theorem | isldepslvec2 47156* | Alternative definition of being a linearly dependent subset of a (left) vector space. In this case, the reverse implication of islindeps2 47154 holds, so that both definitions are equivalent (see theorem 1.6 in [Roman] p. 46 and the note in [Roman] p. 112: if a nontrivial linear combination of elements (where not all of the coefficients are 0) in an R-vector space is 0, then and only then each of the elements is a linear combination of the others. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
β’ π΅ = (Baseβπ) & β’ π = (0gβπ) & β’ π = (Scalarβπ) & β’ πΈ = (Baseβπ ) & β’ 0 = (0gβπ ) β β’ ((π β LVec β§ π β π« π΅) β (βπ β π βπ β (πΈ βm (π β {π }))(π finSupp 0 β§ (π( linC βπ)(π β {π })) = π ) β π linDepS π)) | ||
Theorem | lindssnlvec 47157 | A singleton not containing the zero element of a vector space is always linearly independent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 28-Apr-2019.) |
β’ ((π β LVec β§ π β (Baseβπ) β§ π β (0gβπ)) β {π} linIndS π) | ||
Theorem | lmod1lem1 47158* | Lemma 1 for lmod1 47163. (Contributed by AV, 28-Apr-2019.) |
β’ π = ({β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©, β¨(Scalarβndx), π β©} βͺ {β¨( Β·π βndx), (π₯ β (Baseβπ ), π¦ β {πΌ} β¦ π¦)β©}) β β’ ((πΌ β π β§ π β Ring β§ π β (Baseβπ )) β (π( Β·π βπ)πΌ) β {πΌ}) | ||
Theorem | lmod1lem2 47159* | Lemma 2 for lmod1 47163. (Contributed by AV, 28-Apr-2019.) |
β’ π = ({β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©, β¨(Scalarβndx), π β©} βͺ {β¨( Β·π βndx), (π₯ β (Baseβπ ), π¦ β {πΌ} β¦ π¦)β©}) β β’ ((πΌ β π β§ π β Ring β§ π β (Baseβπ )) β (π( Β·π βπ)(πΌ(+gβπ)πΌ)) = ((π( Β·π βπ)πΌ)(+gβπ)(π( Β·π βπ)πΌ))) | ||
Theorem | lmod1lem3 47160* | Lemma 3 for lmod1 47163. (Contributed by AV, 29-Apr-2019.) |
β’ π = ({β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©, β¨(Scalarβndx), π β©} βͺ {β¨( Β·π βndx), (π₯ β (Baseβπ ), π¦ β {πΌ} β¦ π¦)β©}) β β’ (((πΌ β π β§ π β Ring) β§ (π β (Baseβπ ) β§ π β (Baseβπ ))) β ((π(+gβ(Scalarβπ))π)( Β·π βπ)πΌ) = ((π( Β·π βπ)πΌ)(+gβπ)(π( Β·π βπ)πΌ))) | ||
Theorem | lmod1lem4 47161* | Lemma 4 for lmod1 47163. (Contributed by AV, 29-Apr-2019.) |
β’ π = ({β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©, β¨(Scalarβndx), π β©} βͺ {β¨( Β·π βndx), (π₯ β (Baseβπ ), π¦ β {πΌ} β¦ π¦)β©}) β β’ (((πΌ β π β§ π β Ring) β§ (π β (Baseβπ ) β§ π β (Baseβπ ))) β ((π(.rβ(Scalarβπ))π)( Β·π βπ)πΌ) = (π( Β·π βπ)(π( Β·π βπ)πΌ))) | ||
Theorem | lmod1lem5 47162* | Lemma 5 for lmod1 47163. (Contributed by AV, 28-Apr-2019.) |
β’ π = ({β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©, β¨(Scalarβndx), π β©} βͺ {β¨( Β·π βndx), (π₯ β (Baseβπ ), π¦ β {πΌ} β¦ π¦)β©}) β β’ ((πΌ β π β§ π β Ring) β ((1rβ(Scalarβπ))( Β·π βπ)πΌ) = πΌ) | ||
Theorem | lmod1 47163* | The (smallest) structure representing a zero module over an arbitrary ring. (Contributed by AV, 29-Apr-2019.) |
β’ π = ({β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©, β¨(Scalarβndx), π β©} βͺ {β¨( Β·π βndx), (π₯ β (Baseβπ ), π¦ β {πΌ} β¦ π¦)β©}) β β’ ((πΌ β π β§ π β Ring) β π β LMod) | ||
Theorem | lmod1zr 47164 | The (smallest) structure representing a zero module over a zero ring. (Contributed by AV, 29-Apr-2019.) |
β’ π = {β¨(Baseβndx), {π}β©, β¨(+gβndx), {β¨β¨π, πβ©, πβ©}β©, β¨(.rβndx), {β¨β¨π, πβ©, πβ©}β©} & β’ π = ({β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©, β¨(Scalarβndx), π β©} βͺ {β¨( Β·π βndx), {β¨β¨π, πΌβ©, πΌβ©}β©}) β β’ ((πΌ β π β§ π β π) β π β LMod) | ||
Theorem | lmod1zrnlvec 47165 | There is a (left) module (a zero module) which is not a (left) vector space. (Contributed by AV, 29-Apr-2019.) |
β’ π = {β¨(Baseβndx), {π}β©, β¨(+gβndx), {β¨β¨π, πβ©, πβ©}β©, β¨(.rβndx), {β¨β¨π, πβ©, πβ©}β©} & β’ π = ({β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©, β¨(Scalarβndx), π β©} βͺ {β¨( Β·π βndx), {β¨β¨π, πΌβ©, πΌβ©}β©}) β β’ ((πΌ β π β§ π β π) β π β LVec) | ||
Theorem | lmodn0 47166 | Left modules exist. (Contributed by AV, 29-Apr-2019.) |
β’ LMod β β | ||
Theorem | zlmodzxzequa 47167 | Example of an equation within the β€-module β€ Γ β€ (see example in [Roman] p. 112 for a linearly dependent set). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.) |
β’ π = (β€ring freeLMod {0, 1}) & β’ 0 = {β¨0, 0β©, β¨1, 0β©} & β’ β = ( Β·π βπ) & β’ β = (-gβπ) & β’ π΄ = {β¨0, 3β©, β¨1, 6β©} & β’ π΅ = {β¨0, 2β©, β¨1, 4β©} β β’ ((2 β π΄) β (3 β π΅)) = 0 | ||
Theorem | zlmodzxznm 47168 | Example of a linearly dependent set whose elements are not linear combinations of the others, see note in [Roman] p. 112). (Contributed by AV, 23-May-2019.) (Revised by AV, 10-Jun-2019.) |
β’ π = (β€ring freeLMod {0, 1}) & β’ 0 = {β¨0, 0β©, β¨1, 0β©} & β’ β = ( Β·π βπ) & β’ β = (-gβπ) & β’ π΄ = {β¨0, 3β©, β¨1, 6β©} & β’ π΅ = {β¨0, 2β©, β¨1, 4β©} β β’ βπ β β€ ((π β π΄) β π΅ β§ (π β π΅) β π΄) | ||
Theorem | zlmodzxzldeplem 47169 | A and B are not equal. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
β’ π = (β€ring freeLMod {0, 1}) & β’ π΄ = {β¨0, 3β©, β¨1, 6β©} & β’ π΅ = {β¨0, 2β©, β¨1, 4β©} β β’ π΄ β π΅ | ||
Theorem | zlmodzxzequap 47170 | Example of an equation within the β€-module β€ Γ β€ (see example in [Roman] p. 112 for a linearly dependent set), written as a sum. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
β’ π = (β€ring freeLMod {0, 1}) & β’ π΄ = {β¨0, 3β©, β¨1, 6β©} & β’ π΅ = {β¨0, 2β©, β¨1, 4β©} & β’ 0 = {β¨0, 0β©, β¨1, 0β©} & β’ + = (+gβπ) & β’ β = ( Β·π βπ) β β’ ((2 β π΄) + (-3 β π΅)) = 0 | ||
Theorem | zlmodzxzldeplem1 47171 | Lemma 1 for zlmodzxzldep 47175. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
β’ π = (β€ring freeLMod {0, 1}) & β’ π΄ = {β¨0, 3β©, β¨1, 6β©} & β’ π΅ = {β¨0, 2β©, β¨1, 4β©} & β’ πΉ = {β¨π΄, 2β©, β¨π΅, -3β©} β β’ πΉ β (β€ βm {π΄, π΅}) | ||
Theorem | zlmodzxzldeplem2 47172 | Lemma 2 for zlmodzxzldep 47175. (Contributed by AV, 24-May-2019.) (Revised by AV, 30-Jul-2019.) |
β’ π = (β€ring freeLMod {0, 1}) & β’ π΄ = {β¨0, 3β©, β¨1, 6β©} & β’ π΅ = {β¨0, 2β©, β¨1, 4β©} & β’ πΉ = {β¨π΄, 2β©, β¨π΅, -3β©} β β’ πΉ finSupp 0 | ||
Theorem | zlmodzxzldeplem3 47173 | Lemma 3 for zlmodzxzldep 47175. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
β’ π = (β€ring freeLMod {0, 1}) & β’ π΄ = {β¨0, 3β©, β¨1, 6β©} & β’ π΅ = {β¨0, 2β©, β¨1, 4β©} & β’ πΉ = {β¨π΄, 2β©, β¨π΅, -3β©} β β’ (πΉ( linC βπ){π΄, π΅}) = (0gβπ) | ||
Theorem | zlmodzxzldeplem4 47174* | Lemma 4 for zlmodzxzldep 47175. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
β’ π = (β€ring freeLMod {0, 1}) & β’ π΄ = {β¨0, 3β©, β¨1, 6β©} & β’ π΅ = {β¨0, 2β©, β¨1, 4β©} & β’ πΉ = {β¨π΄, 2β©, β¨π΅, -3β©} β β’ βπ¦ β {π΄, π΅} (πΉβπ¦) β 0 | ||
Theorem | zlmodzxzldep 47175 | { A , B } is a linearly dependent set within the β€-module β€ Γ β€ (see example in [Roman] p. 112). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.) |
β’ π = (β€ring freeLMod {0, 1}) & β’ π΄ = {β¨0, 3β©, β¨1, 6β©} & β’ π΅ = {β¨0, 2β©, β¨1, 4β©} β β’ {π΄, π΅} linDepS π | ||
Theorem | ldepsnlinclem1 47176 | Lemma 1 for ldepsnlinc 47179. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.) |
β’ π = (β€ring freeLMod {0, 1}) & β’ π΄ = {β¨0, 3β©, β¨1, 6β©} & β’ π΅ = {β¨0, 2β©, β¨1, 4β©} β β’ (πΉ β ((Baseββ€ring) βm {π΅}) β (πΉ( linC βπ){π΅}) β π΄) | ||
Theorem | ldepsnlinclem2 47177 | Lemma 2 for ldepsnlinc 47179. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.) |
β’ π = (β€ring freeLMod {0, 1}) & β’ π΄ = {β¨0, 3β©, β¨1, 6β©} & β’ π΅ = {β¨0, 2β©, β¨1, 4β©} β β’ (πΉ β ((Baseββ€ring) βm {π΄}) β (πΉ( linC βπ){π΄}) β π΅) | ||
Theorem | lvecpsslmod 47178 | The class of all (left) vector spaces is a proper subclass of the class of all (left) modules. Although it is obvious (and proven by lveclmod 20716) that every left vector space is a left module, there is (at least) one left module which is no left vector space, for example the zero module over the zero ring, see lmod1zrnlvec 47165. (Contributed by AV, 29-Apr-2019.) |
β’ LVec β LMod | ||
Theorem | ldepsnlinc 47179* | The reverse implication of islindeps2 47154 does not hold for arbitrary (left) modules, see note in [Roman] p. 112: "... if a nontrivial linear combination of the elements ... in an R-module M is 0, ... where not all of the coefficients are 0, then we cannot conclude ... that one of the elements ... is a linear combination of the others." This means that there is at least one left module having a linearly dependent subset in which there is at least one element which is not a linear combinantion of the other elements of this subset. Such a left module can be constructed by using zlmodzxzequa 47167 and zlmodzxznm 47168. (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.) |
β’ βπ β LMod βπ β π« (Baseβπ)(π linDepS π β§ βπ£ β π βπ β ((Baseβ(Scalarβπ)) βm (π β {π£}))(π finSupp (0gβ(Scalarβπ)) β (π( linC βπ)(π β {π£})) β π£)) | ||
Theorem | ldepslinc 47180* | For (left) vector spaces, isldepslvec2 47156 provides an alternative definition of being a linearly dependent subset, whereas ldepsnlinc 47179 indicates that there is not an analogous alternative definition for arbitrary (left) modules. (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.) |
β’ (βπ β LVec βπ β π« (Baseβπ)(π linDepS π β βπ£ β π βπ β ((Baseβ(Scalarβπ)) βm (π β {π£}))(π finSupp (0gβ(Scalarβπ)) β§ (π( linC βπ)(π β {π£})) = π£)) β§ Β¬ βπ β LMod βπ β π« (Baseβπ)(π linDepS π β βπ£ β π βπ β ((Baseβ(Scalarβπ)) βm (π β {π£}))(π finSupp (0gβ(Scalarβπ)) β§ (π( linC βπ)(π β {π£})) = π£))) | ||
Theorem | suppdm 47181 | If the range of a function does not contain the zero, the support of the function equals its domain. (Contributed by AV, 20-May-2020.) |
β’ (((Fun πΉ β§ πΉ β π β§ π β π) β§ π β ran πΉ) β (πΉ supp π) = dom πΉ) | ||
Theorem | eluz2cnn0n1 47182 | An integer greater than 1 is a complex number not equal to 0 or 1. (Contributed by AV, 23-May-2020.) |
β’ (π΅ β (β€β₯β2) β π΅ β (β β {0, 1})) | ||
Theorem | divge1b 47183 | The ratio of a real number to a positive real number is greater than or equal to 1 iff the divisor (the positive real number) is less than or equal to the dividend (the real number). (Contributed by AV, 26-May-2020.) |
β’ ((π΄ β β+ β§ π΅ β β) β (π΄ β€ π΅ β 1 β€ (π΅ / π΄))) | ||
Theorem | divgt1b 47184 | The ratio of a real number to a positive real number is greater than 1 iff the divisor (the positive real number) is less than the dividend (the real number). (Contributed by AV, 30-May-2020.) |
β’ ((π΄ β β+ β§ π΅ β β) β (π΄ < π΅ β 1 < (π΅ / π΄))) | ||
Theorem | ltsubaddb 47185 | Equivalence for the "less than" relation between differences and sums. (Contributed by AV, 6-Jun-2020.) |
β’ (((π΄ β β β§ π΅ β β) β§ (πΆ β β β§ π· β β)) β ((π΄ β πΆ) < (π΅ β π·) β (π΄ + π·) < (π΅ + πΆ))) | ||
Theorem | ltsubsubb 47186 | Equivalence for the "less than" relation between differences. (Contributed by AV, 6-Jun-2020.) |
β’ (((π΄ β β β§ π΅ β β) β§ (πΆ β β β§ π· β β)) β ((π΄ β πΆ) < (π΅ β π·) β (π΄ β π΅) < (πΆ β π·))) | ||
Theorem | ltsubadd2b 47187 | Equivalence for the "less than" relation between differences and sums. (Contributed by AV, 6-Jun-2020.) |
β’ (((π΄ β β β§ π΅ β β) β§ (πΆ β β β§ π· β β)) β ((π· β πΆ) < (π΅ β π΄) β (π΄ + π·) < (π΅ + πΆ))) | ||
Theorem | divsub1dir 47188 | Distribution of division over subtraction by 1. (Contributed by AV, 6-Jun-2020.) |
β’ ((π΄ β β β§ π΅ β β β§ π΅ β 0) β ((π΄ / π΅) β 1) = ((π΄ β π΅) / π΅)) | ||
Theorem | expnegico01 47189 | An integer greater than 1 to the power of a negative integer is in the closed-below, open-above interval between 0 and 1. (Contributed by AV, 24-May-2020.) |
β’ ((π΅ β (β€β₯β2) β§ π β β€ β§ π < 0) β (π΅βπ) β (0[,)1)) | ||
Theorem | elfzolborelfzop1 47190 | An element of a half-open integer interval is either equal to the left bound of the interval or an element of a half-open integer interval with a lower bound increased by 1. (Contributed by AV, 2-Jun-2020.) |
β’ (πΎ β (π..^π) β (πΎ = π β¨ πΎ β ((π + 1)..^π))) | ||
Theorem | pw2m1lepw2m1 47191 | 2 to the power of a positive integer decreased by 1 is less than or equal to 2 to the power of the integer minus 1. (Contributed by AV, 30-May-2020.) |
β’ (πΌ β β β (2β(πΌ β 1)) β€ ((2βπΌ) β 1)) | ||
Theorem | zgtp1leeq 47192 | If an integer is between another integer and its predecessor, the integer is equal to the other integer. (Contributed by AV, 7-Jun-2020.) |
β’ ((πΌ β β€ β§ π΄ β β€) β (((π΄ β 1) < πΌ β§ πΌ β€ π΄) β πΌ = π΄)) | ||
Theorem | flsubz 47193 | An integer can be moved in and out of the floor of a difference. (Contributed by AV, 29-May-2020.) |
β’ ((π΄ β β β§ π β β€) β (ββ(π΄ β π)) = ((ββπ΄) β π)) | ||
Theorem | fldivmod 47194 | Expressing the floor of a division by the modulo operator. (Contributed by AV, 6-Jun-2020.) |
β’ ((π΄ β β β§ π΅ β β+) β (ββ(π΄ / π΅)) = ((π΄ β (π΄ mod π΅)) / π΅)) | ||
Theorem | mod0mul 47195* | If an integer is 0 modulo a positive integer, this integer must be the product of another integer and the modulus. (Contributed by AV, 7-Jun-2020.) |
β’ ((π΄ β β€ β§ π β β) β ((π΄ mod π) = 0 β βπ₯ β β€ π΄ = (π₯ Β· π))) | ||
Theorem | modn0mul 47196* | If an integer is not 0 modulo a positive integer, this integer must be the sum of the product of another integer and the modulus and a positive integer less than the modulus. (Contributed by AV, 7-Jun-2020.) |
β’ ((π΄ β β€ β§ π β β) β ((π΄ mod π) β 0 β βπ₯ β β€ βπ¦ β (1..^π)π΄ = ((π₯ Β· π) + π¦))) | ||
Theorem | m1modmmod 47197 | An integer decreased by 1 modulo a positive integer minus the integer modulo the same modulus is either -1 or the modulus minus 1. (Contributed by AV, 7-Jun-2020.) |
β’ ((π΄ β β€ β§ π β β) β (((π΄ β 1) mod π) β (π΄ mod π)) = if((π΄ mod π) = 0, (π β 1), -1)) | ||
Theorem | difmodm1lt 47198 | The difference between an integer modulo a positive integer and the integer decreased by 1 modulo the same modulus is less than the modulus decreased by 1 (if the modulus is greater than 2). This theorem would not be valid for an odd π΄ and π = 2, since ((π΄ mod π) β ((π΄ β 1) mod π)) would be (1 β 0) = 1 which is not less than (π β 1) = 1. (Contributed by AV, 6-Jun-2012.) |
β’ ((π΄ β β€ β§ π β β β§ 2 < π) β ((π΄ mod π) β ((π΄ β 1) mod π)) < (π β 1)) | ||
Theorem | nn0onn0ex 47199* | For each odd nonnegative integer there is a nonnegative integer which, multiplied by 2 and increased by 1, results in the odd nonnegative integer. (Contributed by AV, 30-May-2020.) |
β’ ((π β β0 β§ ((π + 1) / 2) β β0) β βπ β β0 π = ((2 Β· π) + 1)) | ||
Theorem | nn0enn0ex 47200* | For each even nonnegative integer there is a nonnegative integer which, multiplied by 2, results in the even nonnegative integer. (Contributed by AV, 30-May-2020.) |
β’ ((π β β0 β§ (π / 2) β β0) β βπ β β0 π = (2 Β· π)) |
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