![]() |
Metamath
Proof Explorer Theorem List (p. 417 of 479) | < 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: | ![]() (1-30171) |
![]() (30172-31694) |
![]() (31695-47852) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | pell14qrmulcl 41601 | 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 41602 | 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 41603 | 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 41604 | Lemma for pell14qrexpcl 41605. (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 41605 | 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 41606 | 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 41607 | 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 41608 | 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 41609 | 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 41610 | 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 41611 | Lemma for pell1qrgap 41612. (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 41612 | 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 41613 | 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 41614 | 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 41615 | 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 41616* | 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 41617* | 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 41618* | 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 41619 | 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 41620 | 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 41621 | 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 41622 | 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 41623* | 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 41624 |
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 41614. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
β’ (π· β (β β β»NN) β (PellFundβπ·) β (Pell1QRβπ·)) | ||
Theorem | pellfund14gap 41625 | 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 41626 | 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 41627 | 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 41628 | General logarithm is a real number. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbcl 26278 instead. |
β’ ((π΄ β β+ β§ π΅ β β+ β§ π΅ β 1) β ((logβπ΄) / (logβπ΅)) β β) | ||
Theorem | reglogltb 41629 | General logarithm preserves "less than". (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logblt 26289 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 41630 | General logarithm preserves β€. (Contributed by Stefan O'Rear, 19-Oct-2014.) (New usage is discouraged.) Use logbleb 26288 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 41631 | Multiplication law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbmul 26282 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 41632 | Power law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbzexp 26281 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 41633 | General log of the base is 1. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logbid1 26273 instead. |
β’ ((πΆ β β+ β§ πΆ β 1) β ((logβπΆ) / (logβπΆ)) = 1) | ||
Theorem | reglog1 41634 | General log of 1 is 0. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logb1 26274 instead. |
β’ ((πΆ β β+ β§ πΆ β 1) β ((logβ1) / (logβπΆ)) = 0) | ||
Theorem | reglogexpbas 41635 | 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 26285 instead. |
β’ ((π β β€ β§ (πΆ β β+ β§ πΆ β 1)) β ((logβ(πΆβπ)) / (logβπΆ)) = π) | ||
Theorem | pellfund14 41636* | 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 41637* | 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 41638 | Extend class notation to include the Robertson-Matiyasevich X sequence. |
class Xrm | ||
Syntax | crmy 41639 | Extend class notation to include the Robertson-Matiyasevich Y sequence. |
class Yrm | ||
Definition | df-rmx 41640* | Define the X sequence as the rational part of some solution of a special Pell equation. See frmx 41652 and rmxyval 41654 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 41641* | Define the X sequence as the irrational part of some solution of a special Pell equation. See frmy 41653 and rmxyval 41654 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 41642* | Value of the X sequence. Not used after rmxyval 41654 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 41643* | Value of the Y sequence. Not used after rmxyval 41654 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 41644 | 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 41645 | 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 41646 | 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 41647 | 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 41648* | 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 41649* | Obsolete version of rmxyelqirr 41648 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 41650* | The function used to extract rational and irrational parts in df-rmx 41640 and df-rmy 41641 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 41651* | Lemma for frmx 41652 and frmy 41653. (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 41652 | The X sequence is a nonnegative integer. See rmxnn 41690 for a strengthening. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ Xrm :((β€β₯β2) Γ β€)βΆβ0 | ||
Theorem | frmy 41653 | The Y sequence is an integer. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ Yrm :((β€β₯β2) Γ β€)βΆβ€ | ||
Theorem | rmxyval 41654 | 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 41655 | 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 41656* | 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 41657 | 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 41658 | 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 41659 | 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 41660 | Addition formula for X and Y sequences. See rmxadd 41666 and rmyadd 41670 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 41661 | 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 41662 | 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 41663 | Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 41659, rmxyadd 41660, rmxy0 41662, and rmxy1 41661 via qirropth 41646 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 41664 | 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 41665 | 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 41666 | 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 41667 | 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 41668 | 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 41669 | 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 41670 | 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 41671 | 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 41672 | 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 41673 | 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 41674 | 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 41675 | 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 41676 | The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 41668 and rmy1 41669. 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 41677 | 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 41678 | "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 41679 | "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 41680* | 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 41681* | 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 41682* | 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 41683* | 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 41684* | 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 41685* | 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 41686 | 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 41687 | The Y-sequence is strictly monotonic on β0. Strengthened by ltrmy 41691. (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 41688 | 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 41689 | 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 41690 | 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 41691 | 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 41692 | 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 41693 | 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 41694 | 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 41695 | 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 41696 | 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 41697 | 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΅))) | ||
Theorem | jm2.24nn 41698 | X(n) is strictly greater than Y(n) + Y(n-1). Lemma 2.24 of [JonesMatijasevic] p. 697 restricted to β. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β) β ((π΄ Yrm (π β 1)) + (π΄ Yrm π)) < (π΄ Xrm π)) | ||
Theorem | jm2.17a 41699 | First half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 14-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0) β (((2 Β· π΄) β 1)βπ) β€ (π΄ Yrm (π + 1))) | ||
Theorem | jm2.17b 41700 | Weak form of the second half of lemma 2.17 of [JonesMatijasevic] p. 696, allowing induction to start lower. (Contributed by Stefan O'Rear, 15-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0) β (π΄ Yrm (π + 1)) β€ ((2 Β· π΄)βπ)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |