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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | pell14qrrp 41901 | A positive Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·)) β π΄ β β+) | ||
Theorem | pell1234qrdich 41902 | A general Pell solution is either a positive solution, or its negation is. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell1234QRβπ·)) β (π΄ β (Pell14QRβπ·) β¨ -π΄ β (Pell14QRβπ·))) | ||
Theorem | elpell14qr2 41903 | A number is a positive Pell solution iff it is positive and a Pell solution, justifying our name choice. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
β’ (π· β (β β β»NN) β (π΄ β (Pell14QRβπ·) β (π΄ β (Pell1234QRβπ·) β§ 0 < π΄))) | ||
Theorem | pell14qrmulcl 41904 | Positive Pell solutions are closed under multiplication. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ π΅ β (Pell14QRβπ·)) β (π΄ Β· π΅) β (Pell14QRβπ·)) | ||
Theorem | pell14qrreccl 41905 | Positive Pell solutions are closed under reciprocal. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·)) β (1 / π΄) β (Pell14QRβπ·)) | ||
Theorem | pell14qrdivcl 41906 | Positive Pell solutions are closed under division. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ π΅ β (Pell14QRβπ·)) β (π΄ / π΅) β (Pell14QRβπ·)) | ||
Theorem | pell14qrexpclnn0 41907 | Lemma for pell14qrexpcl 41908. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ π΅ β β0) β (π΄βπ΅) β (Pell14QRβπ·)) | ||
Theorem | pell14qrexpcl 41908 | Positive Pell solutions are closed under integer powers. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ π΅ β β€) β (π΄βπ΅) β (Pell14QRβπ·)) | ||
Theorem | pell1qrss14 41909 | First-quadrant Pell solutions are a subset of the positive solutions. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ (π· β (β β β»NN) β (Pell1QRβπ·) β (Pell14QRβπ·)) | ||
Theorem | pell14qrdich 41910 | A positive Pell solution is either in the first quadrant, or its reciprocal is. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·)) β (π΄ β (Pell1QRβπ·) β¨ (1 / π΄) β (Pell1QRβπ·))) | ||
Theorem | pell1qrge1 41911 | A Pell solution in the first quadrant is at least 1. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell1QRβπ·)) β 1 β€ π΄) | ||
Theorem | pell1qr1 41912 | 1 is a Pell solution and in the first quadrant as one. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
β’ (π· β (β β β»NN) β 1 β (Pell1QRβπ·)) | ||
Theorem | elpell1qr2 41913 | The first quadrant solutions are precisely the positive Pell solutions which are at least one. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ (π· β (β β β»NN) β (π΄ β (Pell1QRβπ·) β (π΄ β (Pell14QRβπ·) β§ 1 β€ π΄))) | ||
Theorem | pell1qrgaplem 41914 | Lemma for pell1qrgap 41915. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ (((π· β β β§ (π΄ β β0 β§ π΅ β β0)) β§ (1 < (π΄ + ((ββπ·) Β· π΅)) β§ ((π΄β2) β (π· Β· (π΅β2))) = 1)) β ((ββ(π· + 1)) + (ββπ·)) β€ (π΄ + ((ββπ·) Β· π΅))) | ||
Theorem | pell1qrgap 41915 | First-quadrant Pell solutions are bounded away from 1. (This particular bound allows to prove exact values for the fundamental solution later.) (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell1QRβπ·) β§ 1 < π΄) β ((ββ(π· + 1)) + (ββπ·)) β€ π΄) | ||
Theorem | pell14qrgap 41916 | Positive Pell solutions are bounded away from 1. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ 1 < π΄) β ((ββ(π· + 1)) + (ββπ·)) β€ π΄) | ||
Theorem | pell14qrgapw 41917 | Positive Pell solutions are bounded away from 1, with a friendlier bound. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ 1 < π΄) β 2 < π΄) | ||
Theorem | pellqrexplicit 41918 | Condition for a calculated real to be a Pell solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
β’ (((π· β (β β β»NN) β§ π΄ β β0 β§ π΅ β β0) β§ ((π΄β2) β (π· Β· (π΅β2))) = 1) β (π΄ + ((ββπ·) Β· π΅)) β (Pell1QRβπ·)) | ||
Theorem | infmrgelbi 41919* | Any lower bound of a nonempty set of real numbers is less than or equal to its infimum, one-direction version. (Contributed by Stefan O'Rear, 1-Sep-2013.) (Revised by AV, 17-Sep-2020.) |
β’ (((π΄ β β β§ π΄ β β β§ π΅ β β) β§ βπ₯ β π΄ π΅ β€ π₯) β π΅ β€ inf(π΄, β, < )) | ||
Theorem | pellqrex 41920* | There is a nontrivial solution of a Pell equation in the first quadrant. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ (π· β (β β β»NN) β βπ₯ β (Pell1QRβπ·)1 < π₯) | ||
Theorem | pellfundval 41921* | Value of the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 18-Sep-2014.) (Revised by AV, 17-Sep-2020.) |
β’ (π· β (β β β»NN) β (PellFundβπ·) = inf({π₯ β (Pell14QRβπ·) β£ 1 < π₯}, β, < )) | ||
Theorem | pellfundre 41922 | The fundamental solution of a Pell equation exists as a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ (π· β (β β β»NN) β (PellFundβπ·) β β) | ||
Theorem | pellfundge 41923 | Lower bound on the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
β’ (π· β (β β β»NN) β ((ββ(π· + 1)) + (ββπ·)) β€ (PellFundβπ·)) | ||
Theorem | pellfundgt1 41924 | Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
β’ (π· β (β β β»NN) β 1 < (PellFundβπ·)) | ||
Theorem | pellfundlb 41925 | A nontrivial first quadrant solution is at least as large as the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Proof shortened by AV, 15-Sep-2020.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ 1 < π΄) β (PellFundβπ·) β€ π΄) | ||
Theorem | pellfundglb 41926* | If a real is larger than the fundamental solution, there is a nontrivial solution less than it. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β β β§ (PellFundβπ·) < π΄) β βπ₯ β (Pell1QRβπ·)((PellFundβπ·) β€ π₯ β§ π₯ < π΄)) | ||
Theorem | pellfundex 41927 |
The fundamental solution as an infimum is itself a solution, showing
that the solution set is discrete.
Since the fundamental solution is an infimum, there must be an element ge to Fund and lt 2*Fund. If this element is equal to the fundamental solution we're done, otherwise use the infimum again to find another element which must be ge Fund and lt the first element; their ratio is a group element in (1,2), contradicting pell14qrgapw 41917. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ (π· β (β β β»NN) β (PellFundβπ·) β (Pell1QRβπ·)) | ||
Theorem | pellfund14gap 41928 | There are no solutions between 1 and the fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β π΄ = 1) | ||
Theorem | pellfundrp 41929 | The fundamental Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
β’ (π· β (β β β»NN) β (PellFundβπ·) β β+) | ||
Theorem | pellfundne1 41930 | The fundamental Pell solution is never 1. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
β’ (π· β (β β β»NN) β (PellFundβπ·) β 1) | ||
Section should be obsolete because its contents are covered by section "Logarithms to an arbitrary base" now. | ||
Theorem | reglogcl 41931 | General logarithm is a real number. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbcl 26511 instead. |
β’ ((π΄ β β+ β§ π΅ β β+ β§ π΅ β 1) β ((logβπ΄) / (logβπ΅)) β β) | ||
Theorem | reglogltb 41932 | General logarithm preserves "less than". (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logblt 26522 instead. |
β’ (((π΄ β β+ β§ π΅ β β+) β§ (πΆ β β+ β§ 1 < πΆ)) β (π΄ < π΅ β ((logβπ΄) / (logβπΆ)) < ((logβπ΅) / (logβπΆ)))) | ||
Theorem | reglogleb 41933 | General logarithm preserves β€. (Contributed by Stefan O'Rear, 19-Oct-2014.) (New usage is discouraged.) Use logbleb 26521 instead. |
β’ (((π΄ β β+ β§ π΅ β β+) β§ (πΆ β β+ β§ 1 < πΆ)) β (π΄ β€ π΅ β ((logβπ΄) / (logβπΆ)) β€ ((logβπ΅) / (logβπΆ)))) | ||
Theorem | reglogmul 41934 | Multiplication law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbmul 26515 instead. |
β’ ((π΄ β β+ β§ π΅ β β+ β§ (πΆ β β+ β§ πΆ β 1)) β ((logβ(π΄ Β· π΅)) / (logβπΆ)) = (((logβπ΄) / (logβπΆ)) + ((logβπ΅) / (logβπΆ)))) | ||
Theorem | reglogexp 41935 | Power law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbzexp 26514 instead. |
β’ ((π΄ β β+ β§ π β β€ β§ (πΆ β β+ β§ πΆ β 1)) β ((logβ(π΄βπ)) / (logβπΆ)) = (π Β· ((logβπ΄) / (logβπΆ)))) | ||
Theorem | reglogbas 41936 | General log of the base is 1. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logbid1 26506 instead. |
β’ ((πΆ β β+ β§ πΆ β 1) β ((logβπΆ) / (logβπΆ)) = 1) | ||
Theorem | reglog1 41937 | General log of 1 is 0. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logb1 26507 instead. |
β’ ((πΆ β β+ β§ πΆ β 1) β ((logβ1) / (logβπΆ)) = 0) | ||
Theorem | reglogexpbas 41938 | General log of a power of the base is the exponent. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbexp 26518 instead. |
β’ ((π β β€ β§ (πΆ β β+ β§ πΆ β 1)) β ((logβ(πΆβπ)) / (logβπΆ)) = π) | ||
Theorem | pellfund14 41939* | Every positive Pell solution is a power of the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·)) β βπ₯ β β€ π΄ = ((PellFundβπ·)βπ₯)) | ||
Theorem | pellfund14b 41940* | The positive Pell solutions are precisely the integer powers of the fundamental solution. To get the general solution set (which we will not be using), throw in a copy of Z/2Z. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
β’ (π· β (β β β»NN) β (π΄ β (Pell14QRβπ·) β βπ₯ β β€ π΄ = ((PellFundβπ·)βπ₯))) | ||
Syntax | crmx 41941 | Extend class notation to include the Robertson-Matiyasevich X sequence. |
class Xrm | ||
Syntax | crmy 41942 | Extend class notation to include the Robertson-Matiyasevich Y sequence. |
class Yrm | ||
Definition | df-rmx 41943* | Define the X sequence as the rational part of some solution of a special Pell equation. See frmx 41955 and rmxyval 41957 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
β’ Xrm = (π β (β€β₯β2), π β β€ β¦ (1st β(β‘(π β (β0 Γ β€) β¦ ((1st βπ) + ((ββ((πβ2) β 1)) Β· (2nd βπ))))β((π + (ββ((πβ2) β 1)))βπ)))) | ||
Definition | df-rmy 41944* | Define the X sequence as the irrational part of some solution of a special Pell equation. See frmy 41956 and rmxyval 41957 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
β’ Yrm = (π β (β€β₯β2), π β β€ β¦ (2nd β(β‘(π β (β0 Γ β€) β¦ ((1st βπ) + ((ββ((πβ2) β 1)) Β· (2nd βπ))))β((π + (ββ((πβ2) β 1)))βπ)))) | ||
Theorem | rmxfval 41945* | Value of the X sequence. Not used after rmxyval 41957 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm π) = (1st β(β‘(π β (β0 Γ β€) β¦ ((1st βπ) + ((ββ((π΄β2) β 1)) Β· (2nd βπ))))β((π΄ + (ββ((π΄β2) β 1)))βπ)))) | ||
Theorem | rmyfval 41946* | Value of the Y sequence. Not used after rmxyval 41957 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm π) = (2nd β(β‘(π β (β0 Γ β€) β¦ ((1st βπ) + ((ββ((π΄β2) β 1)) Β· (2nd βπ))))β((π΄ + (ββ((π΄β2) β 1)))βπ)))) | ||
Theorem | rmspecsqrtnq 41947 | The discriminant used to define the X and Y sequences has an irrational square root. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof shortened by AV, 2-Aug-2021.) |
β’ (π΄ β (β€β₯β2) β (ββ((π΄β2) β 1)) β (β β β)) | ||
Theorem | rmspecnonsq 41948 | The discriminant used to define the X and Y sequences is a nonsquare positive integer and thus a valid Pell equation discriminant. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β ((π΄β2) β 1) β (β β β»NN)) | ||
Theorem | qirropth 41949 | This lemma implements the concept of "equate rational and irrational parts", used to prove many arithmetical properties of the X and Y sequences. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
β’ ((π΄ β (β β β) β§ (π΅ β β β§ πΆ β β) β§ (π· β β β§ πΈ β β)) β ((π΅ + (π΄ Β· πΆ)) = (π· + (π΄ Β· πΈ)) β (π΅ = π· β§ πΆ = πΈ))) | ||
Theorem | rmspecfund 41950 | The base of exponent used to define the X and Y sequences is the fundamental solution of the corresponding Pell equation. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β (PellFundβ((π΄β2) β 1)) = (π΄ + (ββ((π΄β2) β 1)))) | ||
Theorem | rmxyelqirr 41951* | The solutions used to construct the X and Y sequences are quadratic irrationals. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof shortened by SN, 23-Dec-2024.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ + (ββ((π΄β2) β 1)))βπ) β {π β£ βπ β β0 βπ β β€ π = (π + ((ββ((π΄β2) β 1)) Β· π))}) | ||
Theorem | rmxyelqirrOLD 41952* | Obsolete version of rmxyelqirr 41951 as of 23-Dec-2024. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ + (ββ((π΄β2) β 1)))βπ) β {π β£ βπ β β0 βπ β β€ π = (π + ((ββ((π΄β2) β 1)) Β· π))}) | ||
Theorem | rmxypairf1o 41953* | The function used to extract rational and irrational parts in df-rmx 41943 and df-rmy 41944 in fact achieves a one-to-one mapping from the quadratic irrationals to pairs of integers. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β (π β (β0 Γ β€) β¦ ((1st βπ) + ((ββ((π΄β2) β 1)) Β· (2nd βπ)))):(β0 Γ β€)β1-1-ontoβ{π β£ βπ β β0 βπ β β€ π = (π + ((ββ((π΄β2) β 1)) Β· π))}) | ||
Theorem | rmxyelxp 41954* | Lemma for frmx 41955 and frmy 41956. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (β‘(π β (β0 Γ β€) β¦ ((1st βπ) + ((ββ((π΄β2) β 1)) Β· (2nd βπ))))β((π΄ + (ββ((π΄β2) β 1)))βπ)) β (β0 Γ β€)) | ||
Theorem | frmx 41955 | The X sequence is a nonnegative integer. See rmxnn 41993 for a strengthening. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ Xrm :((β€β₯β2) Γ β€)βΆβ0 | ||
Theorem | frmy 41956 | The Y sequence is an integer. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ Yrm :((β€β₯β2) Γ β€)βΆβ€ | ||
Theorem | rmxyval 41957 | Main definition of the X and Y sequences. Compare definition 2.3 of [JonesMatijasevic] p. 694. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Xrm π) + ((ββ((π΄β2) β 1)) Β· (π΄ Yrm π))) = ((π΄ + (ββ((π΄β2) β 1)))βπ)) | ||
Theorem | rmspecpos 41958 | The discriminant used to define the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β ((π΄β2) β 1) β β+) | ||
Theorem | rmxycomplete 41959* | The X and Y sequences taken together enumerate all solutions to the corresponding Pell equation in the right half-plane. This is Metamath 100 proof #39. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0 β§ π β β€) β (((πβ2) β (((π΄β2) β 1) Β· (πβ2))) = 1 β βπ β β€ (π = (π΄ Xrm π) β§ π = (π΄ Yrm π)))) | ||
Theorem | rmxynorm 41960 | The X and Y sequences define a solution to the corresponding Pell equation. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (((π΄ Xrm π)β2) β (((π΄β2) β 1) Β· ((π΄ Yrm π)β2))) = 1) | ||
Theorem | rmbaserp 41961 | The base of exponentiation for the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β (π΄ + (ββ((π΄β2) β 1))) β β+) | ||
Theorem | rmxyneg 41962 | Negation law for X and Y sequences. JonesMatijasevic is inconsistent on whether the X and Y sequences have domain β0 or β€; we use β€ consistently to avoid the need for a separate subtraction law. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Xrm -π) = (π΄ Xrm π) β§ (π΄ Yrm -π) = -(π΄ Yrm π))) | ||
Theorem | rmxyadd 41963 | Addition formula for X and Y sequences. See rmxadd 41969 and rmyadd 41973 for most uses. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β ((π΄ Xrm (π + π)) = (((π΄ Xrm π) Β· (π΄ Xrm π)) + (((π΄β2) β 1) Β· ((π΄ Yrm π) Β· (π΄ Yrm π)))) β§ (π΄ Yrm (π + π)) = (((π΄ Yrm π) Β· (π΄ Xrm π)) + ((π΄ Xrm π) Β· (π΄ Yrm π))))) | ||
Theorem | rmxy1 41964 | Value of the X and Y sequences at 1. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β ((π΄ Xrm 1) = π΄ β§ (π΄ Yrm 1) = 1)) | ||
Theorem | rmxy0 41965 | Value of the X and Y sequences at 0. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β ((π΄ Xrm 0) = 1 β§ (π΄ Yrm 0) = 0)) | ||
Theorem | rmxneg 41966 | Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 41962, rmxyadd 41963, rmxy0 41965, and rmxy1 41964 via qirropth 41949 results in two theorems at once, but typical use requires only one, so this group of theorems serves to separate the cases. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm -π) = (π΄ Xrm π)) | ||
Theorem | rmx0 41967 | Value of X sequence at 0. Part 1 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β (π΄ Xrm 0) = 1) | ||
Theorem | rmx1 41968 | Value of X sequence at 1. Part 2 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β (π΄ Xrm 1) = π΄) | ||
Theorem | rmxadd 41969 | Addition formula for X sequence. Equation 2.7 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β (π΄ Xrm (π + π)) = (((π΄ Xrm π) Β· (π΄ Xrm π)) + (((π΄β2) β 1) Β· ((π΄ Yrm π) Β· (π΄ Yrm π))))) | ||
Theorem | rmyneg 41970 | Negation formula for Y sequence (odd function). (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm -π) = -(π΄ Yrm π)) | ||
Theorem | rmy0 41971 | Value of Y sequence at 0. Part 1 of equation 2.12 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β (π΄ Yrm 0) = 0) | ||
Theorem | rmy1 41972 | Value of Y sequence at 1. Part 2 of equation 2.12 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β (π΄ Yrm 1) = 1) | ||
Theorem | rmyadd 41973 | Addition formula for Y sequence. Equation 2.8 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β (π΄ Yrm (π + π)) = (((π΄ Yrm π) Β· (π΄ Xrm π)) + ((π΄ Xrm π) Β· (π΄ Yrm π)))) | ||
Theorem | rmxp1 41974 | Special addition-of-1 formula for X sequence. Part 1 of equation 2.9 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm (π + 1)) = (((π΄ Xrm π) Β· π΄) + (((π΄β2) β 1) Β· (π΄ Yrm π)))) | ||
Theorem | rmyp1 41975 | Special addition of 1 formula for Y sequence. Part 2 of equation 2.9 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm (π + 1)) = (((π΄ Yrm π) Β· π΄) + (π΄ Xrm π))) | ||
Theorem | rmxm1 41976 | Subtraction of 1 formula for X sequence. Part 1 of equation 2.10 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 14-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm (π β 1)) = ((π΄ Β· (π΄ Xrm π)) β (((π΄β2) β 1) Β· (π΄ Yrm π)))) | ||
Theorem | rmym1 41977 | Subtraction of 1 formula for Y sequence. Part 2 of equation 2.10 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm (π β 1)) = (((π΄ Yrm π) Β· π΄) β (π΄ Xrm π))) | ||
Theorem | rmxluc 41978 | The X sequence is a Lucas (second-order integer recurrence) sequence. Part 3 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 14-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm (π + 1)) = (((2 Β· π΄) Β· (π΄ Xrm π)) β (π΄ Xrm (π β 1)))) | ||
Theorem | rmyluc 41979 | The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 41971 and rmy1 41972. Part 3 of equation 2.12 of [JonesMatijasevic] p. 695. JonesMatijasevic uses this theorem to redefine the X and Y sequences to have domain (β€ Γ β€), which simplifies some later theorems. It may shorten the derivation to use this as our initial definition. Incidentally, the X sequence satisfies the exact same recurrence. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm (π + 1)) = ((2 Β· ((π΄ Yrm π) Β· π΄)) β (π΄ Yrm (π β 1)))) | ||
Theorem | rmyluc2 41980 | Lucas sequence property of Y with better output ordering. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm (π + 1)) = (((2 Β· π΄) Β· (π΄ Yrm π)) β (π΄ Yrm (π β 1)))) | ||
Theorem | rmxdbl 41981 | "Double-angle formula" for X-values. Equation 2.13 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm (2 Β· π)) = ((2 Β· ((π΄ Xrm π)β2)) β 1)) | ||
Theorem | rmydbl 41982 | "Double-angle formula" for Y-values. Equation 2.14 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm (2 Β· π)) = ((2 Β· (π΄ Xrm π)) Β· (π΄ Yrm π))) | ||
Theorem | monotuz 41983* | A function defined on an upper set of integers which increases at every adjacent pair is globally strictly monotonic by induction. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
β’ ((π β§ π¦ β π») β πΉ < πΊ) & β’ ((π β§ π₯ β π») β πΆ β β) & β’ π» = (β€β₯βπΌ) & β’ (π₯ = (π¦ + 1) β πΆ = πΊ) & β’ (π₯ = π¦ β πΆ = πΉ) & β’ (π₯ = π΄ β πΆ = π·) & β’ (π₯ = π΅ β πΆ = πΈ) β β’ ((π β§ (π΄ β π» β§ π΅ β π»)) β (π΄ < π΅ β π· < πΈ)) | ||
Theorem | monotoddzzfi 41984* | A function which is odd and monotonic on β0 is monotonic on β€. This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.) |
β’ ((π β§ π₯ β β€) β (πΉβπ₯) β β) & β’ ((π β§ π₯ β β€) β (πΉβ-π₯) = -(πΉβπ₯)) & β’ ((π β§ π₯ β β0 β§ π¦ β β0) β (π₯ < π¦ β (πΉβπ₯) < (πΉβπ¦))) β β’ ((π β§ π΄ β β€ β§ π΅ β β€) β (π΄ < π΅ β (πΉβπ΄) < (πΉβπ΅))) | ||
Theorem | monotoddzz 41985* | A function (given implicitly) which is odd and monotonic on β0 is monotonic on β€. This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.) |
β’ ((π β§ π₯ β β0 β§ π¦ β β0) β (π₯ < π¦ β πΈ < πΉ)) & β’ ((π β§ π₯ β β€) β πΈ β β) & β’ ((π β§ π¦ β β€) β πΊ = -πΉ) & β’ (π₯ = π΄ β πΈ = πΆ) & β’ (π₯ = π΅ β πΈ = π·) & β’ (π₯ = π¦ β πΈ = πΉ) & β’ (π₯ = -π¦ β πΈ = πΊ) β β’ ((π β§ π΄ β β€ β§ π΅ β β€) β (π΄ < π΅ β πΆ < π·)) | ||
Theorem | oddcomabszz 41986* | An odd function which takes nonnegative values on nonnegative arguments commutes with abs. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
β’ ((π β§ π₯ β β€) β π΄ β β) & β’ ((π β§ π₯ β β€ β§ 0 β€ π₯) β 0 β€ π΄) & β’ ((π β§ π¦ β β€) β πΆ = -π΅) & β’ (π₯ = π¦ β π΄ = π΅) & β’ (π₯ = -π¦ β π΄ = πΆ) & β’ (π₯ = π· β π΄ = πΈ) & β’ (π₯ = (absβπ·) β π΄ = πΉ) β β’ ((π β§ π· β β€) β (absβπΈ) = πΉ) | ||
Theorem | 2nn0ind 41987* | Induction on nonnegative integers with two base cases, for use with Lucas-type sequences. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
β’ π & β’ π & β’ (π¦ β β β ((π β§ π) β π)) & β’ (π₯ = 0 β (π β π)) & β’ (π₯ = 1 β (π β π)) & β’ (π₯ = (π¦ β 1) β (π β π)) & β’ (π₯ = π¦ β (π β π)) & β’ (π₯ = (π¦ + 1) β (π β π)) & β’ (π₯ = π΄ β (π β π)) β β’ (π΄ β β0 β π) | ||
Theorem | zindbi 41988* | Inductively transfer a property to the integers if it holds for zero and passes between adjacent integers in either direction. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
β’ (π¦ β β€ β (π β π)) & β’ (π₯ = π¦ β (π β π)) & β’ (π₯ = (π¦ + 1) β (π β π)) & β’ (π₯ = 0 β (π β π)) & β’ (π₯ = π΄ β (π β π)) β β’ (π΄ β β€ β (π β π)) | ||
Theorem | rmxypos 41989 | For all nonnegative indices, X is positive and Y is nonnegative. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0) β (0 < (π΄ Xrm π) β§ 0 β€ (π΄ Yrm π))) | ||
Theorem | ltrmynn0 41990 | The Y-sequence is strictly monotonic on β0. Strengthened by ltrmy 41994. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0 β§ π β β0) β (π < π β (π΄ Yrm π) < (π΄ Yrm π))) | ||
Theorem | ltrmxnn0 41991 | The X-sequence is strictly monotonic on β0. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0 β§ π β β0) β (π < π β (π΄ Xrm π) < (π΄ Xrm π))) | ||
Theorem | lermxnn0 41992 | The X-sequence is monotonic on β0. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0 β§ π β β0) β (π β€ π β (π΄ Xrm π) β€ (π΄ Xrm π))) | ||
Theorem | rmxnn 41993 | The X-sequence is defined to range over β0 but never actually takes the value 0. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm π) β β) | ||
Theorem | ltrmy 41994 | The Y-sequence is strictly monotonic over β€. (Contributed by Stefan O'Rear, 25-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β (π < π β (π΄ Yrm π) < (π΄ Yrm π))) | ||
Theorem | rmyeq0 41995 | Y is zero only at zero. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π = 0 β (π΄ Yrm π) = 0)) | ||
Theorem | rmyeq 41996 | Y is one-to-one. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β (π = π β (π΄ Yrm π) = (π΄ Yrm π))) | ||
Theorem | lermy 41997 | Y is monotonic (non-strict). (Contributed by Stefan O'Rear, 3-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β (π β€ π β (π΄ Yrm π) β€ (π΄ Yrm π))) | ||
Theorem | rmynn 41998 | Yrm is positive for positive arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β) β (π΄ Yrm π) β β) | ||
Theorem | rmynn0 41999 | Yrm is nonnegative for nonnegative arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0) β (π΄ Yrm π) β β0) | ||
Theorem | rmyabs 42000 | Yrm commutes with abs. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π΅ β β€) β (absβ(π΄ Yrm π΅)) = (π΄ Yrm (absβπ΅))) |
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