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Theorem List for Metamath Proof Explorer - 35201-35300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrmxneg 35201 Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 35197, rmxyadd 35198, rmxy0 35200, and rmxy1 35199 via qirropth 35185 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 ) )
 
Theoremrmx0 35202 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 )
 
Theoremrmx1 35203 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 )
 
Theoremrmxadd 35204 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 ) ) ) ) )
 
Theoremrmyneg 35205 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 ) )
 
Theoremrmy0 35206 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 )
 
Theoremrmy1 35207 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 )
 
Theoremrmyadd 35208 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 ) ) ) )
 
Theoremrmxp1 35209 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 ) ) ) )
 
Theoremrmyp1 35210 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 ) ) )
 
Theoremrmxm1 35211 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 ) ) ) )
 
Theoremrmym1 35212 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 ) ) )
 
Theoremrmxluc 35213 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 ) ) ) )
 
Theoremrmyluc 35214 The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 35206 and rmy1 35207. 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 ) ) ) )
 
Theoremrmyluc2 35215 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 ) ) ) )
 
Theoremrmxdbl 35216 "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 ) )
 
Theoremrmydbl 35217 "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
 
Theoremmonotuz 35218* 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 ) )
 
Theoremmonotoddzzfi 35219* 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 ) ) )
 
Theoremmonotoddzz 35220* 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 ) )
 
Theoremoddcomabszz 35221* 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 )
 
Theorem2nn0ind 35222* 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 )
 
Theoremzindbi 35223* 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 ) )
 
Theoremexpmordi 35224 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 ) )
 
Theoremrpexpmord 35225 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
 
Theoremrmxypos 35226 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 ) ) )
 
Theoremltrmynn0 35227 The Y-sequence is strictly monotonic on  NN0. Strengthened by ltrmy 35231. (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 ) ) )
 
Theoremltrmxnn0 35228 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 ) ) )
 
Theoremlermxnn0 35229 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 ) ) )
 
Theoremrmxnn 35230 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 )
 
Theoremltrmy 35231 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 ) ) )
 
Theoremrmyeq0 35232 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 ) )
 
Theoremrmyeq 35233 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 ) ) )
 
Theoremlermy 35234 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 ) ) )
 
Theoremrmynn 35235 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 )
 
Theoremrmynn0 35236 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 )
 
Theoremrmyabs 35237 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 ) ) )
 
Theoremjm2.24nn 35238 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 ) )
 
Theoremjm2.17a 35239 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 ) ) )
 
Theoremjm2.17b 35240 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 ) )
 
Theoremjm2.17c 35241 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 )
 ) )
 
Theoremjm2.24 35242 Lemma 2.24 of [JonesMatijasevic] p. 697 extended to  ZZ. Could be eliminated with a more careful proof of jm2.26lem3 35285. (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 ) )
 
Theoremrmygeid 35243 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
 
Theoremcongtr 35244 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 ) )
 
Theoremcongadd 35245 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 ) ) )
 
Theoremcongmul 35246 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 ) ) )
 
Theoremcongsym 35247 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 ) )
 
Theoremcongneg 35248 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 ) )
 
Theoremcongsub 35249 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 ) ) )
 
Theoremcongid 35250 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 ) )
 
Theoremmzpcong 35251* 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 ) ) )
 
Theoremcongrep 35252* 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 ) )
 
Theoremcongabseq 35253 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
 
Theoremacongid 35254 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 ) ) )
 
Theoremacongsym 35255 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 ) ) )
 
Theoremacongneg2 35256 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 ) ) )
 
Theoremacongtr 35257 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 ) ) )
 
Theoremacongeq12d 35258 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 ) ) ) )
 
Theoremacongrep 35259* 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 ) ) )
 
Theoremfzmaxdif 35260 Bound on the difference between two integers constrained to two possibly overlapping finite ranges. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( ( C  e.  ZZ  /\  A  e.  ( B ... C ) ) 
 /\  ( F  e.  ZZ  /\  D  e.  ( E ... F ) ) 
 /\  ( C  -  E )  <_  ( F  -  B ) ) 
 ->  ( abs `  ( A  -  D ) ) 
 <_  ( F  -  B ) )
 
Theoremfzneg 35261 Reflection of a finite range of integers about 0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  e.  ( B ... C )  <->  -u A  e.  ( -u C ... -u B ) ) )
 
Theoremacongeq 35262 Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 35286 (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( A  e.  NN  /\  B  e.  ( 0
 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( B  =  C  <->  ( ( 2  x.  A )  ||  ( B  -  C )  \/  (
 2  x.  A ) 
 ||  ( B  -  -u C ) ) ) )
 
Theoremdvdsacongtr 35263 Alternating congruence passes from a base to a dividing base. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( D  ||  A  /\  ( A  ||  ( B  -  C )  \/  A  ||  ( B  -  -u C ) ) ) )  ->  ( D  ||  ( B  -  C )  \/  D  ||  ( B  -  -u C ) ) )
 
21.23.32  Additional theorems on integer divisibility
 
Theorembezoutr 35264 Partial converse to bezout 14387. Existence of a linear combination does not set the GCD, but it does upper bound it. (Contributed by Stefan O'Rear, 23-Sep-2014.)
 |-  (
 ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  ->  ( A 
 gcd  B )  ||  (
 ( A  x.  X )  +  ( B  x.  Y ) ) )
 
Theorembezoutr1 35265 Converse of bezout 14387 for the gcd = 1 case, sufficient condition for relative primality. (Contributed by Stefan O'Rear, 23-Sep-2014.)
 |-  (
 ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  ->  ( ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1  ->  ( A  gcd  B )  =  1 ) )
 
Theoremcoprmdvdsb 35266 Multiplication by a coprime number does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)
 |-  (
 ( K  e.  ZZ  /\  N  e.  ZZ  /\  ( M  e.  ZZ  /\  ( K  gcd  M )  =  1 )
 )  ->  ( K  ||  N  <->  K  ||  ( M  x.  N ) ) )
 
Theoremdvdsleabs2 35267 Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( M  ||  N  ->  ( abs `  M )  <_  ( abs `  N ) ) )
 
Theoremmodabsdifz 35268 Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 26-Sep-2014.)
 |-  (
 ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( ( N  -  ( N  mod  ( abs `  M ) ) ) 
 /  M )  e. 
 ZZ )
 
Theoremdvdsabsmod0 35269 Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 24-Sep-2014.) (Proof shortened by OpenAI, 3-Jul-2020.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  ( M  ||  N  <->  ( N  mod  ( abs `  M )
 )  =  0 ) )
 
Theoremdvdsabsmod0OLD 35270 Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 24-Sep-2014.) Obsolete version of dvdsabsmod0 35269 as of 3-Jul-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  ( M  ||  N  <->  ( N  mod  ( abs `  M )
 )  =  0 ) )
 
Theoremdivalgmodcl 35271 divalgmod 14271 using a class variable. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  (
 ( N  e.  ZZ  /\  D  e.  NN  /\  R  e.  NN0 )  ->  ( R  =  ( N  mod  D )  <->  ( R  <  D 
 /\  D  ||  ( N  -  R ) ) ) )
 
21.23.33  X and Y sequences 3: Divisibility properties
 
Theoremjm2.18 35272 Theorem 2.18 of [JonesMatijasevic] p. 696. Direct relationship of the exponential function to X and Y sequences. (Contributed by Stefan O'Rear, 14-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  K  e.  NN0  /\  N  e.  NN0 )  ->  (
 ( ( ( 2  x.  A )  x.  K )  -  ( K ^ 2 ) )  -  1 )  ||  ( ( ( A Xrm  N )  -  ( ( A  -  K )  x.  ( A Yrm  N ) ) )  -  ( K ^ N ) ) )
 
Theoremjm2.19lem1 35273 Lemma for jm2.19 35277. X and Y values are coprime. (Contributed by Stefan O'Rear, 23-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  M  e.  ZZ )  ->  ( ( A Xrm  M ) 
 gcd  ( A Yrm  M ) )  =  1 )
 
Theoremjm2.19lem2 35274 Lemma for jm2.19 35277. (Contributed by Stefan O'Rear, 23-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
 ( A Yrm  M )  ||  ( A Yrm  N )  <->  ( A Yrm  M ) 
 ||  ( A Yrm  ( N  +  M ) ) ) )
 
Theoremjm2.19lem3 35275 Lemma for jm2.19 35277. (Contributed by Stefan O'Rear, 26-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ )  /\  I  e.  NN0 )  ->  ( ( A Yrm  M ) 
 ||  ( A Yrm  N )  <-> 
 ( A Yrm  M )  ||  ( A Yrm  ( N  +  ( I  x.  M ) ) ) ) )
 
Theoremjm2.19lem4 35276 Lemma for jm2.19 35277. Extend to ZZ by symmetry. TODO: use zindbi 35223. (Contributed by Stefan O'Rear, 26-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ )  /\  I  e.  ZZ )  ->  ( ( A Yrm  M )  ||  ( A Yrm  N ) 
 <->  ( A Yrm  M )  ||  ( A Yrm  ( N  +  ( I  x.  M ) ) ) ) )
 
Theoremjm2.19 35277 Lemma 2.19 of [JonesMatijasevic] p. 696. Transfer divisibility constraints between Y-values and their indices. (Contributed by Stefan O'Rear, 24-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( A Yrm  M ) 
 ||  ( A Yrm  N ) ) )
 
Theoremjm2.21 35278 Lemma for jm2.20nn 35281. Express X and Y values as a binomial. (Contributed by Stefan O'Rear, 26-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ  /\  J  e.  ZZ )  ->  (
 ( A Xrm  ( N  x.  J ) )  +  ( ( sqr `  (
 ( A ^ 2
 )  -  1 ) )  x.  ( A Yrm  ( N  x.  J ) ) ) )  =  ( ( ( A Xrm  N )  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( A Yrm  N ) ) ) ^ J ) )
 
Theoremjm2.22 35279* Lemma for jm2.20nn 35281. Applying binomial theorem and taking irrational part. (Contributed by Stefan O'Rear, 26-Sep-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ  /\  J  e.  NN0 )  ->  ( A Yrm 
 ( N  x.  J ) )  =  sum_ i  e.  { x  e.  ( 0 ... J )  |  -.  2  ||  x }  ( ( J  _C  i )  x.  ( ( ( A Xrm  N ) ^ ( J  -  i ) )  x.  ( ( ( A Yrm  N ) ^ i
 )  x.  ( ( ( A ^ 2
 )  -  1 ) ^ ( ( i  -  1 )  / 
 2 ) ) ) ) ) )
 
Theoremjm2.23 35280 Lemma for jm2.20nn 35281. Truncate binomial expansion p-adicly. (Contributed by Stefan O'Rear, 26-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ  /\  J  e.  NN )  ->  (
 ( A Yrm  N ) ^
 3 )  ||  (
 ( A Yrm  ( N  x.  J ) )  -  ( J  x.  (
 ( ( A Xrm  N ) ^ ( J  -  1 ) )  x.  ( A Yrm  N ) ) ) ) )
 
Theoremjm2.20nn 35281 Lemma 2.20 of [JonesMatijasevic] p. 696, the "first step down lemma". (Contributed by Stefan O'Rear, 27-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  M  e.  NN  /\  N  e.  NN )  ->  (
 ( ( A Yrm  N ) ^ 2 )  ||  ( A Yrm  M )  <->  ( N  x.  ( A Yrm  N ) ) 
 ||  M ) )
 
Theoremjm2.25lem1 35282 Lemma for jm2.26 35286. (Contributed by Stefan O'Rear, 2-Oct-2014.)
 |-  (
 ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D ) ) )  ->  ( ( A  ||  ( D  -  B )  \/  A  ||  ( D  -  -u B ) )  <->  ( A  ||  ( C  -  B )  \/  A  ||  ( C  -  -u B ) ) ) )
 
Theoremjm2.25 35283 Lemma for jm2.26 35286. Remainders mod X(2n) are negaperiodic mod 2n. (Contributed by Stefan O'Rear, 2-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ )  /\  I  e.  ZZ )  ->  ( ( A Xrm  N )  ||  ( ( A Yrm 
 ( M  +  ( I  x.  ( 2  x.  N ) ) ) )  -  ( A Yrm  M ) )  \/  ( A Xrm 
 N )  ||  (
 ( A Yrm  ( M  +  ( I  x.  (
 2  x.  N ) ) ) )  -  -u ( A Yrm  M ) ) ) )
 
Theoremjm2.26a 35284 Lemma for jm2.26 35286. Reverse direction is required to prove forward direction, so do it separatly. Induction on difference between K and M, together with the addition formula fact that adding 2N only inverts sign. (Contributed by Stefan O'Rear, 2-Oct-2014.)
 |-  (
 ( ( A  e.  ( ZZ>= `  2 )  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  M  e.  ZZ )
 )  ->  ( (
 ( 2  x.  N )  ||  ( K  -  M )  \/  (
 2  x.  N ) 
 ||  ( K  -  -u M ) )  ->  ( ( A Xrm  N ) 
 ||  ( ( A Yrm  K )  -  ( A Yrm  M ) )  \/  ( A Xrm 
 N )  ||  (
 ( A Yrm  K )  -  -u ( A Yrm  M ) ) ) ) )
 
Theoremjm2.26lem3 35285 Lemma for jm2.26 35286. Use acongrep 35259 to find K', M' ~ K, M in [ 0,N ]. Thus Y(K') ~ Y(M') and both are small; K' = M' on pain of contradicting 2.24, so K ~ M (Contributed by Stefan O'Rear, 3-Oct-2014.)
 |-  (
 ( ( A  e.  ( ZZ>= `  2 )  /\  N  e.  NN )  /\  ( K  e.  (
 0 ... N )  /\  M  e.  ( 0 ... N ) )  /\  ( ( A Xrm  N ) 
 ||  ( ( A Yrm  K )  -  ( A Yrm  M ) )  \/  ( A Xrm 
 N )  ||  (
 ( A Yrm  K )  -  -u ( A Yrm  M ) ) ) )  ->  K  =  M )
 
Theoremjm2.26 35286 Lemma 2.26 of [JonesMatijasevic] p. 697, the "second step down lemma". (Contributed by Stefan O'Rear, 2-Oct-2014.)
 |-  (
 ( ( A  e.  ( ZZ>= `  2 )  /\  N  e.  NN )  /\  ( K  e.  ZZ  /\  M  e.  ZZ )
 )  ->  ( (
 ( A Xrm  N )  ||  ( ( A Yrm  K )  -  ( A Yrm  M ) )  \/  ( A Xrm  N )  ||  ( ( A Yrm 
 K )  -  -u ( A Yrm 
 M ) ) )  <-> 
 ( ( 2  x.  N )  ||  ( K  -  M )  \/  ( 2  x.  N )  ||  ( K  -  -u M ) ) ) )
 
Theoremjm2.15nn0 35287 Lemma 2.15 of [JonesMatijasevic] p. 695. Yrm is a polynomial for fixed N, so has the expected congruence property. (Contributed by Stefan O'Rear, 1-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  B  e.  ( ZZ>= `  2 )  /\  N  e.  NN0 )  ->  ( A  -  B )  ||  (
 ( A Yrm  N )  -  ( B Yrm  N ) ) )
 
Theoremjm2.16nn0 35288 Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 35287 if Yrm is redefined as described in rmyluc 35214. (Contributed by Stefan O'Rear, 1-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN0 )  ->  ( A  -  1
 )  ||  ( ( A Yrm 
 N )  -  N ) )
 
21.23.34  X and Y sequences 4: Diophantine representability of Y
 
Theoremjm2.27a 35289 Lemma for jm2.27 35292. Reverse direction after existential quantifiers are expanded. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( ph  ->  A  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  C  e.  NN )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  E  e.  NN0 )   &    |-  ( ph  ->  F  e.  NN0 )   &    |-  ( ph  ->  G  e.  NN0 )   &    |-  ( ph  ->  H  e.  NN0 )   &    |-  ( ph  ->  I  e.  NN0 )   &    |-  ( ph  ->  J  e.  NN0 )   &    |-  ( ph  ->  ( ( D ^ 2
 )  -  ( ( ( A ^ 2
 )  -  1 )  x.  ( C ^
 2 ) ) )  =  1 )   &    |-  ( ph  ->  ( ( F ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( E ^ 2 ) ) )  =  1 )   &    |-  ( ph  ->  G  e.  ( ZZ>= `  2 )
 )   &    |-  ( ph  ->  (
 ( I ^ 2
 )  -  ( ( ( G ^ 2
 )  -  1 )  x.  ( H ^
 2 ) ) )  =  1 )   &    |-  ( ph  ->  E  =  ( ( J  +  1 )  x.  ( 2  x.  ( C ^
 2 ) ) ) )   &    |-  ( ph  ->  F 
 ||  ( G  -  A ) )   &    |-  ( ph  ->  ( 2  x.  C )  ||  ( G  -  1 ) )   &    |-  ( ph  ->  F  ||  ( H  -  C ) )   &    |-  ( ph  ->  ( 2  x.  C )  ||  ( H  -  B ) )   &    |-  ( ph  ->  B  <_  C )   &    |-  ( ph  ->  P  e.  ZZ )   &    |-  ( ph  ->  D  =  ( A Xrm  P ) )   &    |-  ( ph  ->  C  =  ( A Yrm  P ) )   &    |-  ( ph  ->  Q  e.  ZZ )   &    |-  ( ph  ->  F  =  ( A Xrm  Q ) )   &    |-  ( ph  ->  E  =  ( A Yrm  Q ) )   &    |-  ( ph  ->  R  e.  ZZ )   &    |-  ( ph  ->  I  =  ( G Xrm  R ) )   &    |-  ( ph  ->  H  =  ( G Yrm  R ) )   =>    |-  ( ph  ->  C  =  ( A Yrm  B ) )
 
Theoremjm2.27b 35290 Lemma for jm2.27 35292. Expand existential quantifiers for reverse direction. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( ph  ->  A  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  C  e.  NN )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  E  e.  NN0 )   &    |-  ( ph  ->  F  e.  NN0 )   &    |-  ( ph  ->  G  e.  NN0 )   &    |-  ( ph  ->  H  e.  NN0 )   &    |-  ( ph  ->  I  e.  NN0 )   &    |-  ( ph  ->  J  e.  NN0 )   &    |-  ( ph  ->  ( ( D ^ 2
 )  -  ( ( ( A ^ 2
 )  -  1 )  x.  ( C ^
 2 ) ) )  =  1 )   &    |-  ( ph  ->  ( ( F ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( E ^ 2 ) ) )  =  1 )   &    |-  ( ph  ->  G  e.  ( ZZ>= `  2 )
 )   &    |-  ( ph  ->  (
 ( I ^ 2
 )  -  ( ( ( G ^ 2
 )  -  1 )  x.  ( H ^
 2 ) ) )  =  1 )   &    |-  ( ph  ->  E  =  ( ( J  +  1 )  x.  ( 2  x.  ( C ^
 2 ) ) ) )   &    |-  ( ph  ->  F 
 ||  ( G  -  A ) )   &    |-  ( ph  ->  ( 2  x.  C )  ||  ( G  -  1 ) )   &    |-  ( ph  ->  F  ||  ( H  -  C ) )   &    |-  ( ph  ->  ( 2  x.  C )  ||  ( H  -  B ) )   &    |-  ( ph  ->  B  <_  C )   =>    |-  ( ph  ->  C  =  ( A Yrm  B ) )
 
Theoremjm2.27c 35291 Lemma for jm2.27 35292. Forward direction with substitutions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( ph  ->  A  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  C  e.  NN )   &    |-  ( ph  ->  C  =  ( A Yrm  B ) )   &    |-  D  =  ( A Xrm  B )   &    |-  Q  =  ( B  x.  ( A Yrm 
 B ) )   &    |-  E  =  ( A Yrm  ( 2  x.  Q ) )   &    |-  F  =  ( A Xrm  ( 2  x.  Q ) )   &    |-  G  =  ( A  +  (
 ( F ^ 2
 )  x.  ( ( F ^ 2 )  -  A ) ) )   &    |-  H  =  ( G Yrm  B )   &    |-  I  =  ( G Xrm  B )   &    |-  J  =  ( ( E  /  (
 2  x.  ( C ^ 2 ) ) )  -  1 )   =>    |-  ( ph  ->  ( (
 ( D  e.  NN0  /\  E  e.  NN0  /\  F  e.  NN0 )  /\  ( G  e.  NN0  /\  H  e.  NN0  /\  I  e.  NN0 ) )  /\  ( J  e.  NN0  /\  (
 ( ( ( ( D ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( C ^ 2
 ) ) )  =  1  /\  ( ( F ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( E ^ 2
 ) ) )  =  1  /\  G  e.  ( ZZ>= `  2 )
 )  /\  ( (
 ( I ^ 2
 )  -  ( ( ( G ^ 2
 )  -  1 )  x.  ( H ^
 2 ) ) )  =  1  /\  E  =  ( ( J  +  1 )  x.  (
 2  x.  ( C ^ 2 ) ) )  /\  F  ||  ( G  -  A ) ) )  /\  ( ( ( 2  x.  C )  ||  ( G  -  1
 )  /\  F  ||  ( H  -  C ) ) 
 /\  ( ( 2  x.  C )  ||  ( H  -  B )  /\  B  <_  C ) ) ) ) ) )
 
Theoremjm2.27 35292* Lemma 2.27 of [JonesMatijasevic] p. 697; rmY is a diophantine relation. 0 was excluded from the range of B and the lower limit of G was imposed because the source proof does not seem to work otherwise; quite possible I'm just missing something. The source proof uses both i and I; i has been changed to j to avoid collision. This theorem is basically nothing but substitution instances, all the work is done in jm2.27a 35289 and jm2.27c 35291. Once Diophantine relations have been defined, the content of the theorem is "rmY is Diophantine" (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  =  ( A Yrm  B ) 
 <-> 
 E. d  e.  NN0  E. e  e.  NN0  E. f  e.  NN0  E. g  e. 
 NN0  E. h  e.  NN0  E. i  e.  NN0  E. j  e.  NN0  ( ( ( ( ( d ^
 2 )  -  (
 ( ( A ^
 2 )  -  1
 )  x.  ( C ^ 2 ) ) )  =  1  /\  ( ( f ^
 2 )  -  (
 ( ( A ^
 2 )  -  1
 )  x.  ( e ^ 2 ) ) )  =  1  /\  g  e.  ( ZZ>= `  2 ) )  /\  ( ( ( i ^ 2 )  -  ( ( ( g ^ 2 )  -  1 )  x.  ( h ^ 2 ) ) )  =  1  /\  e  =  ( (
 j  +  1 )  x.  ( 2  x.  ( C ^ 2
 ) ) )  /\  f  ||  ( g  -  A ) ) ) 
 /\  ( ( ( 2  x.  C ) 
 ||  ( g  -  1 )  /\  f  ||  ( h  -  C ) )  /\  ( ( 2  x.  C ) 
 ||  ( h  -  B )  /\  B  <_  C ) ) ) ) )
 
Theoremjm2.27dlem1 35293* Lemma for rmydioph 35298. Subsitution of a tuple restriction into a projection that doesn't care. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  ( 1 ... B )   =>    |-  ( a  =  ( b  |`  ( 1 ... B ) )  ->  ( a `  A )  =  ( b `  A ) )
 
Theoremjm2.27dlem2 35294 Lemma for rmydioph 35298. This theorem is used along with the next three to efficiently infer steps like 
7  e.  ( 1 ... 10 ). (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  ( 1 ... B )   &    |-  C  =  ( B  +  1 )   &    |-  B  e.  NN   =>    |-  A  e.  ( 1
 ... C )
 
Theoremjm2.27dlem3 35295 Lemma for rmydioph 35298. Infer membership of the endpoint of a range. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  NN   =>    |-  A  e.  ( 1
 ... A )
 
Theoremjm2.27dlem4 35296 Lemma for rmydioph 35298. Infer  NN-hood of large numbers. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  NN   &    |-  B  =  ( A  +  1 )   =>    |-  B  e.  NN
 
Theoremjm2.27dlem5 35297 Lemma for rmydioph 35298. Used with sselii 3438 to infer membership of midpoints of range; jm2.27dlem2 35294 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  B  =  ( A  +  1 )   &    |-  ( 1 ...
 B )  C_  (
 1 ... C )   =>    |-  ( 1 ...
 A )  C_  (
 1 ... C )
 
Theoremrmydioph 35298 jm2.27 35292 restated in terms of Diophantine sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  { a  e.  ( NN0  ^m  (
 1 ... 3 ) )  |  ( ( a `
  1 )  e.  ( ZZ>= `  2 )  /\  ( a `  3
 )  =  ( ( a `  1 ) Yrm  ( a `  2 ) ) ) }  e.  (Dioph `  3 )
 
21.23.35  X and Y sequences 5: Diophantine representability of X, ^, _C
 
Theoremrmxdiophlem 35299* X can be expressed in terms of Y, so it is also Diophantine. (Contributed by Stefan O'Rear, 15-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN0  /\  X  e.  NN0 )  ->  ( X  =  ( A Xrm  N ) 
 <-> 
 E. y  e.  NN0  ( y  =  ( A Yrm 
 N )  /\  (
 ( X ^ 2
 )  -  ( ( ( A ^ 2
 )  -  1 )  x.  ( y ^
 2 ) ) )  =  1 ) ) )
 
Theoremrmxdioph 35300 X is a Diophantine function. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  { a  e.  ( NN0  ^m  (
 1 ... 3 ) )  |  ( ( a `
  1 )  e.  ( ZZ>= `  2 )  /\  ( a `  3
 )  =  ( ( a `  1 ) Xrm  ( a `  2 ) ) ) }  e.  (Dioph `  3 )
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