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Theorem List for Metamath Proof Explorer - 26201-26300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltrmynn0 26201 The Y-sequence is strictly monotonic on  NN0. Strengthened by ltrmy 26205. (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 26202 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 26203 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 26204 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 26205 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 26206 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 26207 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 26208 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 26209 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 26210 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 26211 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 26212 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 26213 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 26214 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 26215 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 26216 Lemma 2.24 of [JonesMatijasevic] p. 697 extended to  ZZ. Could be eliminated with a more careful proof of jm2.26lem3 26260. (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 26217 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 ) )
 
16.15.32  Congruential equations
 
Theoremcongtr 26218 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 26219 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 26220 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 26221 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 26222 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 26223 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 26224 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 26225* 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 26226* 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 26227 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 ) )
 
16.15.33  Alternating congruential equations
 
Theoremacongid 26228 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 26229 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 26230 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 26231 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 26232 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 26233* 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 26234 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 26235 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 26236 Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 26261 (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 26237 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 ) ) )
 
16.15.34  Additional theorems on integer divisibility
 
Theorembezoutr 26238 Partial converse to bezout 12595. 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 26239 Converse of bezout 12595 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 26240 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 26241 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 26242 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 26243 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 26244 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 26245 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 26246 divalgmod 12479 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 ) ) ) )
 
16.15.35  X and Y sequences 3: Divisibility properties
 
Theoremjm2.18 26247 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 26248 Lemma for jm2.19 26252. 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 26249 Lemma for jm2.19 26252. (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 26250 Lemma for jm2.19 26252. (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 26251 Lemma for jm2.19 26252. Extend to ZZ by symmetry. TODO: use zindbi 26197. (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 26252 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 26253 Lemma for jm2.20nn 26256. 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 26254* Lemma for jm2.20nn 26256. 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 26255 Lemma for jm2.20nn 26256. 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 26256 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 26257 Lemma for jm2.26 26261. (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 26258 Lemma for jm2.26 26261. 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 26259 Lemma for jm2.26 26261. 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 26260 Lemma for jm2.26 26261. Use acongrep 26233 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 26261 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 26262 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 26263 Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 26262 if Yrm is redefined as described in rmyluc 26188. (Contributed by Stefan O'Rear, 1-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN0 )  ->  ( A  -  1
 )  ||  ( ( A Yrm 
 N )  -  N ) )
 
16.15.36  X and Y sequences 4: Diophantine representability of Y
 
Theoremjm2.27a 26264 Lemma for jm2.27 26267. 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 26265 Lemma for jm2.27 26267. 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 26266 Lemma for jm2.27 26267. 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 26267* 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 26264 and jm2.27c 26266. 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 26268* Lemma for rmydioph 26273. 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 26269 Lemma for rmydioph 26273. 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 26270 Lemma for rmydioph 26273. 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 26271 Lemma for rmydioph 26273. 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 26272 Lemma for rmydioph 26273. Used with sselii 3100 to infer membership of midpoints of range; jm2.27dlem2 26269 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  B  =  ( A  +  1 )   &    |-  ( 1 ...
 B )  C_  (
 1 ... C )   =>    |-  ( 1 ...
 A )  C_  (
 1 ... C )
 
Theoremrmydioph 26273 jm2.27 26267 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 )
 
16.15.37  X and Y sequences 5: Diophantine representability of X, ^, _C
 
Theoremrmxdiophlem 26274* 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 26275 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 )
 
Theoremjm3.1lem1 26276 Lemma for jm3.1 26279. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  ( ph  ->  A  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  2 )
 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( K Yrm  ( N  +  1 ) )  <_  A )   =>    |-  ( ph  ->  ( K ^ N )  <  A )
 
Theoremjm3.1lem2 26277 Lemma for jm3.1 26279. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  ( ph  ->  A  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  2 )
 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( K Yrm  ( N  +  1 ) )  <_  A )   =>    |-  ( ph  ->  ( K ^ N )  < 
 ( ( ( ( 2  x.  A )  x.  K )  -  ( K ^ 2 ) )  -  1 ) )
 
Theoremjm3.1lem3 26278 Lemma for jm3.1 26279. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  ( ph  ->  A  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  2 )
 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( K Yrm  ( N  +  1 ) )  <_  A )   =>    |-  ( ph  ->  (
 ( ( ( 2  x.  A )  x.  K )  -  ( K ^ 2 ) )  -  1 )  e. 
 NN )
 
Theoremjm3.1 26279 Diophantine expression for exponentiation. Lemma 3.1 of [JonesMatijasevic] p. 698. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  (
 ( ( A  e.  ( ZZ>= `  2 )  /\  K  e.  ( ZZ>= `  2 )  /\  N  e.  NN )  /\  ( K Yrm  ( N  +  1 ) )  <_  A )  ->  ( K ^ N )  =  ( (
 ( A Xrm  N )  -  ( ( A  -  K )  x.  ( A Yrm 
 N ) ) ) 
 mod  ( ( ( ( 2  x.  A )  x.  K )  -  ( K ^ 2 ) )  -  1 ) ) )
 
Theoremexpdiophlem1 26280* Lemma for expdioph 26282. Fully expanded expression for exponential. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  ( C  e.  NN0  ->  (
 ( ( A  e.  ( ZZ>= `  2 )  /\  B  e.  NN )  /\  C  =  ( A ^ B ) )  <->  E. d  e.  NN0  E. e  e.  NN0  E. f  e. 
 NN0  ( ( A  e.  ( ZZ>= `  2
 )  /\  B  e.  NN )  /\  ( ( A  e.  ( ZZ>= `  2 )  /\  d  =  ( A Yrm  ( B  +  1 ) ) ) 
 /\  ( ( d  e.  ( ZZ>= `  2
 )  /\  e  =  ( d Yrm  B ) ) 
 /\  ( ( d  e.  ( ZZ>= `  2
 )  /\  f  =  ( d Xrm  B ) ) 
 /\  ( C  <  ( ( ( ( 2  x.  d )  x.  A )  -  ( A ^ 2 ) )  -  1 )  /\  ( ( ( ( 2  x.  d )  x.  A )  -  ( A ^ 2 ) )  -  1 ) 
 ||  ( ( f  -  ( ( d  -  A )  x.  e ) )  -  C ) ) ) ) ) ) ) )
 
Theoremexpdiophlem2 26281 Lemma for expdioph 26282. Exponentiation on a restricted domain is Diophantine. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  { a  e.  ( NN0  ^m  (
 1 ... 3 ) )  |  ( ( ( a `  1 )  e.  ( ZZ>= `  2
 )  /\  ( a `  2 )  e.  NN )  /\  ( a `  3 )  =  (
 ( a `  1
 ) ^ ( a `
  2 ) ) ) }  e.  (Dioph `  3 )
 
Theoremexpdioph 26282 The exponential function is Diophantine. This result completes and encapsulates our development using Pell equation solution sequences and is sometimes regarded as Matiyasevich's theorem properly. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  { a  e.  ( NN0  ^m  (
 1 ... 3 ) )  |  ( a `  3 )  =  (
 ( a `  1
 ) ^ ( a `
  2 ) ) }  e.  (Dioph `  3 )
 
16.15.38  Uncategorized stuff not associated with a major project
 
Theoremsetindtr 26283* Epsilon induction for sets contained in a transitive set. If we are allowed to assume Infinity, then all sets have a transitive closure and this reduces to setind 7303; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( A. x ( x  C_  A  ->  x  e.  A )  ->  ( E. y
 ( Tr  y  /\  B  e.  y )  ->  B  e.  A ) )
 
Theoremsetindtrs 26284* Epsilon induction scheme without Infinity. See comments at setindtr 26283. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( A. y  e.  x  ps  ->  ph )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  ( ph  <->  ch ) )   =>    |-  ( E. z ( Tr  z  /\  B  e.  z )  ->  ch )
 
Theoremdford3lem1 26285* Lemma for dford3 26287. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  (
 ( Tr  N  /\  A. y  e.  N  Tr  y )  ->  A. b  e.  N  ( Tr  b  /\  A. y  e.  b  Tr  y ) )
 
Theoremdford3lem2 26286* Lemma for dford3 26287. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  (
 ( Tr  x  /\  A. y  e.  x  Tr  y )  ->  x  e. 
 On )
 
Theoremdford3 26287* Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( Ord  N  <->  ( Tr  N  /\  A. x  e.  N  Tr  x ) )
 
Theoremdford4 26288* dford3 26287 expressed in primitives to demonstrate shortness. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( Ord  N  <->  A. a A. b A. c ( ( a  e.  N  /\  b  e.  a )  ->  (
 b  e.  N  /\  ( c  e.  b  ->  c  e.  a ) ) ) )
 
Theoremwopprc 26289 Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  (
 ( A  e.  _V  /\  B  e.  _V )  <->  -. 
 1o  e.  { { { A } ,  (/) } ,  { { B } } } )
 
Theoremrpnnen3lem 26290* Lemma for rpnnen3 26291. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  (
 ( ( a  e. 
 RR  /\  b  e.  RR )  /\  a  < 
 b )  ->  { c  e.  QQ  |  c  < 
 a }  =/=  {
 c  e.  QQ  |  c  <  b } )
 
Theoremrpnnen3 26291 Dedekind cut injection of  RR into  ~P QQ. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  RR  ~<_  ~P QQ
 
16.15.39  More equivalents of the Axiom of Choice
 
Theoremaxac10 26292 Characterization of choice similar to dffin1-5 7898. (Contributed by Stefan O'Rear, 6-Jan-2015.)
 |-  (  ~~  " On )  =  _V
 
Theoremharinf 26293 The Hartogs number of an infinite set is at least  om. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
 |-  (
 ( S  e.  V  /\  -.  S  e.  Fin )  ->  om  C_  (har `  S ) )
 
Theoremwdom2d2 26294* Deduction for weak dominance by a cross product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   &    |-  (
 ( ph  /\  x  e.  A )  ->  E. y  e.  B  E. z  e.  C  x  =  X )   =>    |-  ( ph  ->  A  ~<_*  ( B  X.  C ) )
 
Theoremttac 26295 Tarski's theorem about choice: infxpidm 8066 is equivalent to ax-ac 7969. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.)
 |-  (CHOICE  <->  A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c ) )
 
Theorempw2f1ocnv 26296* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 6854, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 9-Jul-2015.)
 |-  F  =  ( x  e.  ( 2o  ^m  A )  |->  ( `' x " { 1o } ) )   =>    |-  ( A  e.  V  ->  ( F : ( 2o  ^m  A ) -1-1-onto-> ~P A  /\  `' F  =  ( y  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  y ,  1o ,  (/) ) ) ) ) )
 
Theorempw2f1o2 26297* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 6854, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F  =  ( x  e.  ( 2o  ^m  A )  |->  ( `' x " { 1o } ) )   =>    |-  ( A  e.  V  ->  F : ( 2o 
 ^m  A ) -1-1-onto-> ~P A )
 
Theorempw2f1o2val 26298* Function value of the pw2f1o2 26297 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  F  =  ( x  e.  ( 2o  ^m  A )  |->  ( `' x " { 1o } ) )   =>    |-  ( X  e.  ( 2o  ^m  A )  ->  ( F `  X )  =  ( `' X " { 1o } )
 )
 
Theorempw2f1o2val2 26299* Membership in a mapped set under the pw2f1o2 26297 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  F  =  ( x  e.  ( 2o  ^m  A )  |->  ( `' x " { 1o } ) )   =>    |-  ( ( X  e.  ( 2o  ^m  A ) 
 /\  Y  e.  A )  ->  ( Y  e.  ( F `  X )  <-> 
 ( X `  Y )  =  1o )
 )
 
Theoremsoeq12d 26300 Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  ( ph  ->  R  =  S )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( R  Or  A  <->  S  Or  B ) )
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