P. 358 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35701-35800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pellfundgt1 35701	Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
|- ( D e. ( NN \ ◻NN ) -> 1 < (PellFund ` D ) )
Theorem	pellfundlb 35702	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.)
|- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ 1 < A ) -> (PellFund ` D ) <_ A )
Theorem	pellfundglb 35703*	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.)
|- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> E. x e. (Pell1QR ` D ) ( (PellFund ` D ) <_ x /\ x < A ) )
Theorem	pellfundex 35704	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 35692. (Contributed by Stefan O'Rear, 18-Sep-2014.)
|- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) e. (Pell1QR ` D ) )
Theorem	pellfund14gap 35705	There are no solutions between 1 and the fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
|- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ ( 1 <_ A /\ A < (PellFund ` D ) ) ) -> A = 1 )
Theorem	pellfundrp 35706	The fundamental Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
|- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) e. RR+ )
Theorem	pellfundne1 35707	The fundamental Pell solution is never 1. (Contributed by Stefan O'Rear, 19-Sep-2014.)
|- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) =/= 1 )
21.23.25  Logarithm laws generalized to an arbitrary base Section should be obsolete because its contents are covered by section "Logarithms to an arbitrary base" now.
Theorem	reglogcl 35708	General logarithm is a real number. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbcl 23708 instead.
|- ( ( A e. RR+ /\ B e. RR+ /\ B =/= 1 ) -> ( ( log ` A ) / ( log ` B ) ) e. RR )
Theorem	reglogltb 35709	General logarithm preserves "less than". (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logblt 23719 instead.
|- ( ( ( A e. RR+ /\ B e. RR+ ) /\ ( C e. RR+ /\ 1 < C ) ) -> ( A < B <-> ( ( log ` A ) / ( log ` C ) ) < ( ( log ` B ) / ( log ` C ) ) ) )
Theorem	reglogleb 35710	General logarithm preserves <_. (Contributed by Stefan O'Rear, 19-Oct-2014.) (New usage is discouraged.) Use logbleb 23718 instead.
|- ( ( ( A e. RR+ /\ B e. RR+ ) /\ ( C e. RR+ /\ 1 < C ) ) -> ( A <_ B <-> ( ( log ` A ) / ( log ` C ) ) <_ ( ( log ` B ) / ( log ` C ) ) ) )
Theorem	reglogmul 35711	Multiplication law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbmul 23712 instead.
|- ( ( A e. RR+ /\ B e. RR+ /\ ( C e. RR+ /\ C =/= 1 ) ) -> ( ( log ` ( A x. B ) ) / ( log ` C ) ) = ( ( ( log ` A ) / ( log ` C ) ) + ( ( log ` B ) / ( log ` C ) ) ) )
Theorem	reglogexp 35712	Power law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbzexp 23711 instead.
|- ( ( A e. RR+ /\ N e. ZZ /\ ( C e. RR+ /\ C =/= 1 ) ) -> ( ( log ` ( A ^ N ) ) / ( log ` C ) ) = ( N x. ( ( log ` A ) / ( log ` C ) ) ) )
Theorem	reglogbas 35713	General log of the base is 1. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logbid1 23703 instead.
|- ( ( C e. RR+ /\ C =/= 1 ) -> ( ( log ` C ) / ( log ` C ) ) = 1 )
Theorem	reglog1 35714	General log of 1 is 0. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logb1 23704 instead.
|- ( ( C e. RR+ /\ C =/= 1 ) -> ( ( log ` 1 ) / ( log ` C ) ) = 0 )
Theorem	reglogexpbas 35715	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 23715 instead.
|- ( ( N e. ZZ /\ ( C e. RR+ /\ C =/= 1 ) ) -> ( ( log ` ( C ^ N ) ) / ( log ` C ) ) = N )
21.23.26  Pell equations 4: the positive solution group is infinite cyclic
Theorem	pellfund14 35716*	Every positive Pell solution is a power of the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
|- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> E. x e. ZZ A = ( (PellFund ` D ) ^ x ) )
Theorem	pellfund14b 35717*	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.)
|- ( D e. ( NN \ ◻NN ) -> ( A e. (Pell14QR ` D ) <-> E. x e. ZZ A = ( (PellFund ` D ) ^ x ) ) )
21.23.27  X and Y sequences 1: Definition and recurrence laws
Syntax	crmx 35718	Extend class notation to include the Robertson-Matiyasevich X sequence.
class Xrm
Syntax	crmy 35719	Extend class notation to include the Robertson-Matiyasevich Y sequence.
class Yrm
Definition	df-rmx 35720*	Define the X sequence as the rational part of some solution of a special Pell equation. See frmx 35731 and rmxyval 35733 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.)
|- Xrm = ( a e. ( ZZ>= ` 2 ) , n e. ZZ |-> ( 1st ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( a + ( sqr ` ( ( a ^ 2 ) - 1 ) ) ) ^ n ) ) ) )
Definition	df-rmy 35721*	Define the X sequence as the irrational part of some solution of a special Pell equation. See frmy 35732 and rmxyval 35733 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.)
|- Yrm = ( a e. ( ZZ>= ` 2 ) , n e. ZZ |-> ( 2nd ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( a + ( sqr ` ( ( a ^ 2 ) - 1 ) ) ) ^ n ) ) ) )
Theorem	rmxfval 35722*	Value of the X sequence. Not used after rmxyval 35733 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Xrm N ) = ( 1st ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) ) ) )
Theorem	rmyfval 35723*	Value of the Y sequence. Not used after rmxyval 35733 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Yrm N ) = ( 2nd ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) ) ) )
Theorem	rmspecsqrtnq 35724	The discriminant used to define the X and Y sequences has an irrational square root. (Contributed by Stefan O'Rear, 21-Sep-2014.)
|- ( A e. ( ZZ>= ` 2 ) -> ( sqr ` ( ( A ^ 2 ) - 1 ) ) e. ( CC \ QQ ) )
Theorem	rmspecnonsq 35725	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.)
|- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - 1 ) e. ( NN \ ◻NN ) )
Theorem	qirropth 35726	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.)
|- ( ( A e. ( CC \ QQ ) /\ ( B e. QQ /\ C e. QQ ) /\ ( D e. QQ /\ E e. QQ ) ) -> ( ( B + ( A x. C ) ) = ( D + ( A x. E ) ) <-> ( B = D /\ C = E ) ) )
Theorem	rmspecfund 35727	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.)
|- ( A e. ( ZZ>= ` 2 ) -> (PellFund ` ( ( A ^ 2 ) - 1 ) ) = ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) )
Theorem	rmxyelqirr 35728*	The solutions used to construct the X and Y sequences are quadratic irrationals. (Contributed by Stefan O'Rear, 21-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) e. { a | E. c e. NN0 E. d e. ZZ a = ( c + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. d ) ) } )
Theorem	rmxypairf1o 35729*	The function used to extract rational and irrational parts in df-rmx 35720 and df-rmy 35721 in fact achieves a one-to-one mapping from the quadratic irrationals to pairs of integers. (Contributed by Stefan O'Rear, 21-Sep-2014.)
|- ( A e. ( ZZ>= ` 2 ) -> ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) : ( NN0 X. ZZ ) -1-1-onto-> { a | E. c e. NN0 E. d e. ZZ a = ( c + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. d ) ) } )
Theorem	rmxyelxp 35730*	Lemma for frmx 35731 and frmy 35732. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) ) e. ( NN0 X. ZZ ) )
Theorem	frmx 35731	The X sequence is a nonnegative integer. See rmxnn 35771 for a strengthening. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- Xrm : ( ( ZZ>= ` 2 ) X. ZZ ) --> NN0
Theorem	frmy 35732	The Y sequence is an integer. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- Yrm : ( ( ZZ>= ` 2 ) X. ZZ ) --> ZZ
Theorem	rmxyval 35733	Main definition of the X and Y sequences. Compare definition 2.3 of [JonesMatijasevic] p. 694. (Contributed by Stefan O'Rear, 19-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( ( A Xrm N ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( A Yrm N ) ) ) = ( ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) )
Theorem	rmspecpos 35734	The discriminant used to define the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - 1 ) e. RR+ )
Theorem	rmxycomplete 35735*	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.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ X e. NN0 /\ Y e. ZZ ) -> ( ( ( X ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( Y ^ 2 ) ) ) = 1 <-> E. n e. ZZ ( X = ( A Xrm n ) /\ Y = ( A Yrm n ) ) ) )
Theorem	rmxynorm 35736	The X and Y sequences define a solution to the corresponding Pell equation. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( ( ( A Xrm N ) ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( ( A Yrm N ) ^ 2 ) ) ) = 1 )
Theorem	rmbaserp 35737	The base of exponentiation for the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( A e. ( ZZ>= ` 2 ) -> ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) e. RR+ )
Theorem	rmxyneg 35738	Negation law for X and Y sequences. JonesMatijasevic is inconsistent on whether the X and Y sequences have domain NN0 or ZZ; we use ZZ consistently to avoid the need for a separate subtraction law. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( ( A Xrm -u N ) = ( A Xrm N ) /\ ( A Yrm -u N ) = -u ( A Yrm N ) ) )
Theorem	rmxyadd 35739	Addition formula for X and Y sequences. See rmxadd 35745 and rmyadd 35749 for most uses. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ M e. ZZ /\ N e. ZZ ) -> ( ( A Xrm ( M + N ) ) = ( ( ( A Xrm M ) x. ( A Xrm N ) ) + ( ( ( A ^ 2 ) - 1 ) x. ( ( A Yrm M ) x. ( A Yrm N ) ) ) ) /\ ( A Yrm ( M + N ) ) = ( ( ( A Yrm M ) x. ( A Xrm N ) ) + ( ( A Xrm M ) x. ( A Yrm N ) ) ) ) )
Theorem	rmxy1 35740	Value of the X and Y sequences at 1. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( A e. ( ZZ>= ` 2 ) -> ( ( A Xrm 1 ) = A /\ ( A Yrm 1 ) = 1 ) )
Theorem	rmxy0 35741	Value of the X and Y sequences at 0. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( A e. ( ZZ>= ` 2 ) -> ( ( A Xrm 0 ) = 1 /\ ( A Yrm 0 ) = 0 ) )
Theorem	rmxneg 35742	Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 35738, rmxyadd 35739, rmxy0 35741, and rmxy1 35740 via qirropth 35726 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.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Xrm -u N ) = ( A Xrm N ) )
Theorem	rmx0 35743	Value of X sequence at 0. Part 1 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( A e. ( ZZ>= ` 2 ) -> ( A Xrm 0 ) = 1 )
Theorem	rmx1 35744	Value of X sequence at 1. Part 2 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( A e. ( ZZ>= ` 2 ) -> ( A Xrm 1 ) = A )
Theorem	rmxadd 35745	Addition formula for X sequence. Equation 2.7 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ M e. ZZ /\ N e. ZZ ) -> ( A Xrm ( M + N ) ) = ( ( ( A Xrm M ) x. ( A Xrm N ) ) + ( ( ( A ^ 2 ) - 1 ) x. ( ( A Yrm M ) x. ( A Yrm N ) ) ) ) )
Theorem	rmyneg 35746	Negation formula for Y sequence (odd function). (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Yrm -u N ) = -u ( A Yrm N ) )
Theorem	rmy0 35747	Value of Y sequence at 0. Part 1 of equation 2.12 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( A e. ( ZZ>= ` 2 ) -> ( A Yrm 0 ) = 0 )
Theorem	rmy1 35748	Value of Y sequence at 1. Part 2 of equation 2.12 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( A e. ( ZZ>= ` 2 ) -> ( A Yrm 1 ) = 1 )
Theorem	rmyadd 35749	Addition formula for Y sequence. Equation 2.8 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ M e. ZZ /\ N e. ZZ ) -> ( A Yrm ( M + N ) ) = ( ( ( A Yrm M ) x. ( A Xrm N ) ) + ( ( A Xrm M ) x. ( A Yrm N ) ) ) )
Theorem	rmxp1 35750	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.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Xrm ( N + 1 ) ) = ( ( ( A Xrm N ) x. A ) + ( ( ( A ^ 2 ) - 1 ) x. ( A Yrm N ) ) ) )
Theorem	rmyp1 35751	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.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Yrm ( N + 1 ) ) = ( ( ( A Yrm N ) x. A ) + ( A Xrm N ) ) )
Theorem	rmxm1 35752	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.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Xrm ( N - 1 ) ) = ( ( A x. ( A Xrm N ) ) - ( ( ( A ^ 2 ) - 1 ) x. ( A Yrm N ) ) ) )
Theorem	rmym1 35753	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.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Yrm ( N - 1 ) ) = ( ( ( A Yrm N ) x. A ) - ( A Xrm N ) ) )
Theorem	rmxluc 35754	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.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Xrm ( N + 1 ) ) = ( ( ( 2 x. A ) x. ( A Xrm N ) ) - ( A Xrm ( N - 1 ) ) ) )
Theorem	rmyluc 35755	The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 35747 and rmy1 35748. Part 3 of equation 2.12 of [JonesMatijasevic] p. 695. JonesMatijasevic uses this theorem to redefine the X and Y sequences to have domain ( ZZ X. ZZ ), 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.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Yrm ( N + 1 ) ) = ( ( 2 x. ( ( A Yrm N ) x. A ) ) - ( A Yrm ( N - 1 ) ) ) )
Theorem	rmyluc2 35756	Lucas sequence property of Y with better output ordering. (Contributed by Stefan O'Rear, 16-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Yrm ( N + 1 ) ) = ( ( ( 2 x. A ) x. ( A Yrm N ) ) - ( A Yrm ( N - 1 ) ) ) )
Theorem	rmxdbl 35757	"Double-angle formula" for X-values. Equation 2.13 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Xrm ( 2 x. N ) ) = ( ( 2 x. ( ( A Xrm N ) ^ 2 ) ) - 1 ) )
Theorem	rmydbl 35758	"Double-angle formula" for Y-values. Equation 2.14 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Yrm ( 2 x. N ) ) = ( ( 2 x. ( A Xrm N ) ) x. ( A Yrm N ) ) )
21.23.28  Ordering and induction lemmas for the integers
Theorem	monotuz 35759*	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.)
|- ( ( ph /\ y e. H ) -> F < G )   &    |- ( ( ph /\ x e. H ) -> C e. RR )   &    |- H = ( ZZ>= ` I )   &    |- ( x = ( y + 1 ) -> C = G )   &    |- ( x = y -> C = F )   &    |- ( x = A -> C = D )   &    |- ( x = B -> C = E )   =>    |- ( ( ph /\ ( A e. H /\ B e. H ) ) -> ( A < B <-> D < E ) )
Theorem	monotoddzzfi 35760*	A function which is odd and monotonic on NN0 is monotonic on ZZ. This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.)
|- ( ( ph /\ x e. ZZ ) -> ( F ` x ) e. RR )   &    |- ( ( ph /\ x e. ZZ ) -> ( F ` -u x ) = -u ( F ` x ) )   &    |- ( ( ph /\ x e. NN0 /\ y e. NN0 ) -> ( x < y -> ( F ` x ) < ( F ` y ) ) )   =>    |- ( ( ph /\ A e. ZZ /\ B e. ZZ ) -> ( A < B <-> ( F ` A ) < ( F ` B ) ) )
Theorem	monotoddzz 35761*	A function (given implicitly) which is odd and monotonic on NN0 is monotonic on ZZ. This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.)
|- ( ( ph /\ x e. NN0 /\ y e. NN0 ) -> ( x < y -> E < F ) )   &    |- ( ( ph /\ x e. ZZ ) -> E e. RR )   &    |- ( ( ph /\ y e. ZZ ) -> G = -u F )   &    |- ( x = A -> E = C )   &    |- ( x = B -> E = D )   &    |- ( x = y -> E = F )   &    |- ( x = -u y -> E = G )   =>    |- ( ( ph /\ A e. ZZ /\ B e. ZZ ) -> ( A < B <-> C < D ) )
Theorem	oddcomabszz 35762*	An odd function which takes nonnegative values on nonnegative arguments commutes with abs. (Contributed by Stefan O'Rear, 26-Sep-2014.)
|- ( ( ph /\ x e. ZZ ) -> A e. RR )   &    |- ( ( ph /\ x e. ZZ /\ 0 <_ x ) -> 0 <_ A )   &    |- ( ( ph /\ y e. ZZ ) -> C = -u B )   &    |- ( x = y -> A = B )   &    |- ( x = -u y -> A = C )   &    |- ( x = D -> A = E )   &    |- ( x = ( abs ` D ) -> A = F )   =>    |- ( ( ph /\ D e. ZZ ) -> ( abs ` E ) = F )
Theorem	2nn0ind 35763*	Induction on nonnegative integers with two base cases, for use with Lucas-type sequences. (Contributed by Stefan O'Rear, 1-Oct-2014.)
|- ps   &    |- ch   &    |- ( y e. NN -> ( ( th /\ ta ) -> et ) )   &    |- ( x = 0 -> ( ph <-> ps ) )   &    |- ( x = 1 -> ( ph <-> ch ) )   &    |- ( x = ( y - 1 ) -> ( ph <-> th ) )   &    |- ( x = y -> ( ph <-> ta ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> et ) )   &    |- ( x = A -> ( ph <-> rh ) )   =>    |- ( A e. NN0 -> rh )
Theorem	zindbi 35764*	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.)
|- ( y e. ZZ -> ( ps <-> ch ) )   &    |- ( x = y -> ( ph <-> ps ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> ch ) )   &    |- ( x = 0 -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   =>    |- ( A e. ZZ -> ( th <-> ta ) )
Theorem	expmordi 35765	Mantissa ordering relationship for exponentiation. (Contributed by Stefan O'Rear, 16-Oct-2014.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) /\ N e. NN ) -> ( A ^ N ) < ( B ^ N ) )
Theorem	rpexpmord 35766	Mantissa ordering relationship for exponentiation of positive reals. (Contributed by Stefan O'Rear, 16-Oct-2014.)
|- ( ( N e. NN /\ A e. RR+ /\ B e. RR+ ) -> ( A < B <-> ( A ^ N ) < ( B ^ N ) ) )
21.23.29  X and Y sequences 2: Order properties
Theorem	rmxypos 35767	For all nonnegative indices, X is positive and Y is nonnegative. (Contributed by Stefan O'Rear, 24-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( 0 < ( A Xrm N ) /\ 0 <_ ( A Yrm N ) ) )
Theorem	ltrmynn0 35768	The Y-sequence is strictly monotonic on NN0. Strengthened by ltrmy 35772. (Contributed by Stefan O'Rear, 24-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ M e. NN0 /\ N e. NN0 ) -> ( M < N <-> ( A Yrm M ) < ( A Yrm N ) ) )
Theorem	ltrmxnn0 35769	The X-sequence is strictly monotonic on NN0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ M e. NN0 /\ N e. NN0 ) -> ( M < N <-> ( A Xrm M ) < ( A Xrm N ) ) )
Theorem	lermxnn0 35770	The X-sequence is monotonic on NN0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> ( A Xrm M ) <_ ( A Xrm N ) ) )
Theorem	rmxnn 35771	The X-sequence is defined to range over NN0 but never actually takes the value 0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Xrm N ) e. NN )
Theorem	ltrmy 35772	The Y-sequence is strictly monotonic over ZZ. (Contributed by Stefan O'Rear, 25-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( A Yrm M ) < ( A Yrm N ) ) )
Theorem	rmyeq0 35773	Y is zero only at zero. (Contributed by Stefan O'Rear, 26-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( N = 0 <-> ( A Yrm N ) = 0 ) )
Theorem	rmyeq 35774	Y is one-to-one. (Contributed by Stefan O'Rear, 3-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ M e. ZZ /\ N e. ZZ ) -> ( M = N <-> ( A Yrm M ) = ( A Yrm N ) ) )
Theorem	lermy 35775	Y is monotonic (non-strict). (Contributed by Stefan O'Rear, 3-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( A Yrm M ) <_ ( A Yrm N ) ) )
Theorem	rmynn 35776	Yrm is positive for positive arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. NN ) -> ( A Yrm N ) e. NN )
Theorem	rmynn0 35777	Yrm is nonnegative for nonnegative arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( A Yrm N ) e. NN0 )
Theorem	rmyabs 35778	Yrm commutes with abs. (Contributed by Stefan O'Rear, 26-Sep-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ZZ ) -> ( abs ` ( A Yrm B ) ) = ( A Yrm ( abs ` B ) ) )
Theorem	jm2.24nn 35779	X(n) is strictly greater than Y(n) + Y(n-1). Lemma 2.24 of [JonesMatijasevic] p. 697 restricted to NN. (Contributed by Stefan O'Rear, 3-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. NN ) -> ( ( A Yrm ( N - 1 ) ) + ( A Yrm N ) ) < ( A Xrm N ) )
Theorem	jm2.17a 35780	First half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 14-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( ( ( 2 x. A ) - 1 ) ^ N ) <_ ( A Yrm ( N + 1 ) ) )
Theorem	jm2.17b 35781	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.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( A Yrm ( N + 1 ) ) <_ ( ( 2 x. A ) ^ N ) )
Theorem	jm2.17c 35782	Second half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 15-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. NN ) -> ( A Yrm ( ( N + 1 ) + 1 ) ) < ( ( 2 x. A ) ^ ( N + 1 ) ) )
Theorem	jm2.24 35783	Lemma 2.24 of [JonesMatijasevic] p. 697 extended to ZZ. Could be eliminated with a more careful proof of jm2.26lem3 35826. (Contributed by Stefan O'Rear, 3-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( ( A Yrm ( N - 1 ) ) + ( A Yrm N ) ) < ( A Xrm N ) )
Theorem	rmygeid 35784	Y(n) increases faster than n. Used implicitly without proof or comment in lemma 2.27 of [JonesMatijasevic] p. 697. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> N <_ ( A Yrm N ) )
21.23.30  Congruential equations
Theorem	congtr 35785	A wff of the form A || ( B - C ) is interpreted as a congruential equation. This is similar to ( B mod A ) = ( C mod A ), but is defined such that behavior is regular for zero and negative values of A. To use this concept effectively, we need to show that congruential equations behave similarly to normal equations; first a transitivity law. Idea for the future: If there was a congruential equation symbol, it could incorporate type constraints, so that most of these would not need them. (Contributed by Stefan O'Rear, 1-Oct-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. ZZ /\ D e. ZZ ) /\ ( A || ( B - C ) /\ A || ( C - D ) ) ) -> A || ( B - D ) )
Theorem	congadd 35786	If two pairs of numbers are componentwise congruent, so are their sums. (Contributed by Stefan O'Rear, 1-Oct-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( D e. ZZ /\ E e. ZZ ) /\ ( A || ( B - C ) /\ A || ( D - E ) ) ) -> A || ( ( B + D ) - ( C + E ) ) )
Theorem	congmul 35787	If two pairs of numbers are componentwise congruent, so are their products. (Contributed by Stefan O'Rear, 1-Oct-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( D e. ZZ /\ E e. ZZ ) /\ ( A || ( B - C ) /\ A || ( D - E ) ) ) -> A || ( ( B x. D ) - ( C x. E ) ) )
Theorem	congsym 35788	Congruence mod A is a symmetric/commutative relation. (Contributed by Stefan O'Rear, 1-Oct-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. ZZ /\ A || ( B - C ) ) ) -> A || ( C - B ) )
Theorem	congneg 35789	If two integers are congruent mod A, so are their negatives. (Contributed by Stefan O'Rear, 1-Oct-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. ZZ /\ A || ( B - C ) ) ) -> A || ( -u B - -u C ) )
Theorem	congsub 35790	If two pairs of numbers are componentwise congruent, so are their differences. (Contributed by Stefan O'Rear, 2-Oct-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( D e. ZZ /\ E e. ZZ ) /\ ( A || ( B - C ) /\ A || ( D - E ) ) ) -> A || ( ( B - D ) - ( C - E ) ) )
Theorem	congid 35791	Every integer is congruent to itself mod every base. (Contributed by Stefan O'Rear, 1-Oct-2014.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> A || ( B - B ) )
Theorem	mzpcong 35792*	Polynomials commute with congruences. (Does this characterize them?) (Contributed by Stefan O'Rear, 5-Oct-2014.)
|- ( ( F e. (mzPoly ` V ) /\ ( X e. ( ZZ ^m V ) /\ Y e. ( ZZ ^m V ) ) /\ ( N e. ZZ /\ A. k e. V N || ( ( X ` k ) - ( Y ` k ) ) ) ) -> N || ( ( F ` X ) - ( F ` Y ) ) )
Theorem	congrep 35793*	Every integer is congruent to some number in the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.)
|- ( ( A e. NN /\ N e. ZZ ) -> E. a e. ( 0 ... ( A - 1 ) ) A || ( a - N ) )
Theorem	congabseq 35794	If two integers are congruent, they are either equal or separated by at least the congruence base. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( ( ( A e. NN /\ B e. ZZ /\ C e. ZZ ) /\ A || ( B - C ) ) -> ( ( abs ` ( B - C ) ) < A <-> B = C ) )
21.23.31  Alternating congruential equations
Theorem	acongid 35795	A wff like that in this theorem will be known as an "alternating congruence". A special symbol might be considered if more uses come up. They have many of the same properties as normal congruences, starting with reflexivity. JonesMatijasevic uses "a ≡ ± b (mod c)" for this construction. The disjunction of divisibility constraints seems to adequately capture the concept, but it's rather verbose and somewhat inelegant. Use of an explicit equivalence relation might also work. (Contributed by Stefan O'Rear, 2-Oct-2014.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A || ( B - B ) \/ A || ( B - -u B ) ) )
Theorem	acongsym 35796	Symmetry of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( A || ( B - C ) \/ A || ( B - -u C ) ) ) -> ( A || ( C - B ) \/ A || ( C - -u B ) ) )
Theorem	acongneg2 35797	Negate right side of alternating congruence. Makes essential use of the "alternating" part. (Contributed by Stefan O'Rear, 3-Oct-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( A || ( B - -u C ) \/ A || ( B - -u -u C ) ) ) -> ( A || ( B - C ) \/ A || ( B - -u C ) ) )
Theorem	acongtr 35798	Transitivity of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. ZZ /\ D e. ZZ ) /\ ( ( A || ( B - C ) \/ A || ( B - -u C ) ) /\ ( A || ( C - D ) \/ A || ( C - -u D ) ) ) ) -> ( A || ( B - D ) \/ A || ( B - -u D ) ) )
Theorem	acongeq12d 35799	Substitution deduction for alternating congruence. (Contributed by Stefan O'Rear, 3-Oct-2014.)
|- ( ph -> B = C )   &    |- ( ph -> D = E )   =>    |- ( ph -> ( ( A || ( B - D ) \/ A || ( B - -u D ) ) <-> ( A || ( C - E ) \/ A || ( C - -u E ) ) ) )
Theorem	acongrep 35800*	Every integer is alternating-congruent to some number in the first half of the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.)
|- ( ( A e. NN /\ N e. ZZ ) -> E. a e. ( 0 ... A ) ( ( 2 x. A ) || ( a - N ) \/ ( 2 x. A ) || ( a - -u N ) ) )