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Theorem List for Metamath Proof Explorer - 27001-27100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrmspecpos 27001 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+ )
 
Theoremrmxycomplete 27002* The X and Y sequences taken together enumerate all solutions to the corresponding Pell equation in the right half-plane. (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 ) ) ) )
 
Theoremrmxynorm 27003 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 )
 
Theoremrmbaserp 27004 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+ )
 
Theoremrmxyneg 27005 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 ) ) )
 
Theoremrmxyadd 27006 Addition formula for X and Y sequences. See rmxadd 27012 and rmyadd 27016 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 ) ) ) ) )
 
Theoremrmxy1 27007 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 ) )
 
Theoremrmxy0 27008 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 ) )
 
Theoremrmxneg 27009 Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 27005, rmxyadd 27006, rmxy0 27008, and rmxy1 27007 via qirropth 26993 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 27010 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 27011 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 27012 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 27013 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 27014 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 27015 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 27016 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 27017 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 27018 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 27019 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 27020 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 27021 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 27022 The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 27014 and rmy1 27015. 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 27023 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 27024 "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 27025 "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 ) ) )
 
18.17.30  Ordering and induction lemmas for the integers
 
Theoremmonotuz 27026* A function defined on a set of upper 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 27027* 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 27028* 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 27029* 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 27030* Induction on natural numbers 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 27031* 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 27032 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 27033 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 ) ) )
 
18.17.31  X and Y sequences 2: Order properties
 
Theoremrmxypos 27034 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 27035 The Y-sequence is strictly monotonic on  NN0. Strengthened by ltrmy 27039. (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 27036 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 27037 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 27038 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 27039 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 27040 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 27041 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 27042 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 27043 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 27044 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 27045 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 27046 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 27047 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 27048 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 27049 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 27050 Lemma 2.24 of [JonesMatijasevic] p. 697 extended to  ZZ. Could be eliminated with a more careful proof of jm2.26lem3 27094. (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 27051 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 ) )
 
18.17.32  Congruential equations
 
Theoremcongtr 27052 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 27053 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 27054 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 27055 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 27056 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 27057 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 27058 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 27059* 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 27060* 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 27061 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 ) )
 
18.17.33  Alternating congruential equations
 
Theoremacongid 27062 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 27063 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 27064 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 27065 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 27066 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 27067* 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 27068 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 27069 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 27070 Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 27095 (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 27071 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 ) ) )
 
18.17.34  Additional theorems on integer divisibility
 
Theorembezoutr 27072 Partial converse to bezout 12721. 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 27073 Converse of bezout 12721 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 27074 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 ) ) )
 
Theoremzabscl 27075 The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)
 |-  ( A  e.  ZZ  ->  ( abs `  A )  e.  ZZ )
 
Theoremnn0sqcl 27076 The square of a natural number is a natural number. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  ( A  e.  NN0  ->  ( A ^ 2 )  e. 
 NN0 )
 
Theoremdvdsleabs2 27077 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 27078 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 27079 Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 24-Sep-2014.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  ( M  ||  N  <->  ( N  mod  ( abs `  M )
 )  =  0 ) )
 
Theoremdivalgmodcl 27080 divalgmod 12605 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 ) ) ) )
 
18.17.35  X and Y sequences 3: Divisibility properties
 
Theoremjm2.18 27081 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 27082 Lemma for jm2.19 27086. 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 27083 Lemma for jm2.19 27086. (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 27084 Lemma for jm2.19 27086. (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 27085 Lemma for jm2.19 27086. Extend to ZZ by symmetry. TODO: use zindbi 27031. (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 27086 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 27087 Lemma for jm2.20nn 27090. 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 27088* Lemma for jm2.20nn 27090. 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 27089 Lemma for jm2.20nn 27090. 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 27090 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 27091 Lemma for jm2.26 27095. (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 27092 Lemma for jm2.26 27095. 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 27093 Lemma for jm2.26 27095. 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 27094 Lemma for jm2.26 27095. Use acongrep 27067 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 27095 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 27096 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 27097 Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 27096 if Yrm is redefined as described in rmyluc 27022. (Contributed by Stefan O'Rear, 1-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN0 )  ->  ( A  -  1
 )  ||  ( ( A Yrm 
 N )  -  N ) )
 
18.17.36  X and Y sequences 4: Diophantine representability of Y
 
Theoremjm2.27a 27098 Lemma for jm2.27 27101. 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 27099 Lemma for jm2.27 27101. 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 27100 Lemma for jm2.27 27101. 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 ) ) ) ) ) )
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