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Theorem List for Metamath Proof Explorer - 35901-36000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremacongrep 35901* 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 35902 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 35903 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 35904 Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 35928. (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 35905 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 35906 Partial converse to bezout 14589. 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 35907 Converse of bezout 14589 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 35908 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 35909 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 35910 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 35911 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 35912 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 35911 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 35913 divalgmod 14466 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 35914 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 35915 Lemma for jm2.19 35919. 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 35916 Lemma for jm2.19 35919. (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 35917 Lemma for jm2.19 35919. (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 35918 Lemma for jm2.19 35919. Extend to ZZ by symmetry. TODO: use zindbi 35865. (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 35919 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 35920 Lemma for jm2.20nn 35923. 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 35921* Lemma for jm2.20nn 35923. 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 35922 Lemma for jm2.20nn 35923. 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 35923 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 35924 Lemma for jm2.26 35928. (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 35925 Lemma for jm2.26 35928. 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 35926 Lemma for jm2.26 35928. 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 35927 Lemma for jm2.26 35928. Use acongrep 35901 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 35928 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 35929 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 35930 Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 35929 if Yrm is redefined as described in rmyluc 35856. (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 35931 Lemma for jm2.27 35934. 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 35932 Lemma for jm2.27 35934. 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 35933 Lemma for jm2.27 35934. 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 35934* 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 35931 and jm2.27c 35933. 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 35935* Lemma for rmydioph 35940. 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 35936 Lemma for rmydioph 35940. 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 35937 Lemma for rmydioph 35940. 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 35938 Lemma for rmydioph 35940. 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 35939 Lemma for rmydioph 35940. Used with sselii 3415 to infer membership of midpoints of range; jm2.27dlem2 35936 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  B  =  ( A  +  1 )   &    |-  ( 1 ...
 B )  C_  (
 1 ... C )   =>    |-  ( 1 ...
 A )  C_  (
 1 ... C )
 
Theoremrmydioph 35940 jm2.27 35934 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 35941* 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 35942 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 35943 Lemma for jm3.1 35946. (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 35944 Lemma for jm3.1 35946. (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 35945 Lemma for jm3.1 35946. (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 35946 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 35947* Lemma for expdioph 35949. 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 35948 Lemma for expdioph 35949. 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 35949 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 )
 
21.23.36  Uncategorized stuff not associated with a major project
 
Theoremsetindtr 35950* 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 8236; 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 35951* Epsilon induction scheme without Infinity. See comments at setindtr 35950. (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 35952* Lemma for dford3 35954. (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 35953* Lemma for dford3 35954. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  (
 ( Tr  x  /\  A. y  e.  x  Tr  y )  ->  x  e. 
 On )
 
Theoremdford3 35954* 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 35955* dford3 35954 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 35956 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 35957* Lemma for rpnnen3 35958. (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 35958 Dedekind cut injection of  RR into  ~P QQ. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  RR  ~<_  ~P QQ
 
21.23.37  More equivalents of the Axiom of Choice
 
Theoremaxac10 35959 Characterization of choice similar to dffin1-5 8836. (Contributed by Stefan O'Rear, 6-Jan-2015.)
 |-  (  ~~  " On )  =  _V
 
Theoremharinf 35960 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 35961* Deduction for weak dominance by a Cartesian 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 35962 Tarski's theorem about choice: infxpidm 9005 is equivalent to ax-ac 8907. (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 35963* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7697, 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 35964* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7697, 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 35965* Function value of the pw2f1o2 35964 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 35966* Membership in a mapped set under the pw2f1o2 35964 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 35967 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 ) )
 
Theoremfreq12d 35968 Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  ( ph  ->  R  =  S )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( R  Fr  A  <->  S  Fr  B ) )
 
Theoremweeq12d 35969 Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  ( ph  ->  R  =  S )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( R  We  A  <->  S  We  B ) )
 
Theoremlimsuc2 35970 Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  (
 ( Ord  A  /\  A  =  U. A ) 
 ->  ( B  e.  A  <->  suc 
 B  e.  A ) )
 
Theoremwepwsolem 35971* Transfer an ordering on characteristic functions by isomorphism to the power set. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  T  =  { <. x ,  y >.  |  E. z  e.  A  ( ( z  e.  y  /\  -.  z  e.  x )  /\  A. w  e.  A  ( w R z  ->  ( w  e.  x  <->  w  e.  y ) ) ) }   &    |-  U  =  { <. x ,  y >.  | 
 E. z  e.  A  ( ( x `  z )  _E  (
 y `  z )  /\  A. w  e.  A  ( w R z  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  F  =  ( a  e.  ( 2o 
 ^m  A )  |->  ( `' a " { 1o } ) )   =>    |-  ( A  e.  _V  ->  F  Isom  U ,  T  ( ( 2o  ^m  A ) ,  ~P A ) )
 
Theoremwepwso 35972* A well-ordering induces a strict ordering on the power set. EDITORIAL: when well-orderings are set like, this can be strengthened to remove  A  e.  V. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  T  =  { <. x ,  y >.  |  E. z  e.  A  ( ( z  e.  y  /\  -.  z  e.  x )  /\  A. w  e.  A  ( w R z  ->  ( w  e.  x  <->  w  e.  y ) ) ) }   =>    |-  ( ( A  e.  V  /\  R  We  A )  ->  T  Or  ~P A )
 
Theoremdnnumch1 35973* Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 8479. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F  = recs ( ( z  e. 
 _V  |->  ( G `  ( A  \  ran  z
 ) ) ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )   =>    |-  ( ph  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
 
Theoremdnnumch2 35974* Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F  = recs ( ( z  e. 
 _V  |->  ( G `  ( A  \  ran  z
 ) ) ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )   =>    |-  ( ph  ->  A 
 C_  ran  F )
 
Theoremdnnumch3lem 35975* Value of the ordinal injection function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F  = recs ( ( z  e. 
 _V  |->  ( G `  ( A  \  ran  z
 ) ) ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )   =>    |-  ( ( ph  /\  w  e.  A ) 
 ->  ( ( x  e.  A  |->  |^| ( `' F " { x } )
 ) `  w )  =  |^| ( `' F " { w } )
 )
 
Theoremdnnumch3 35976* Define an injection from a set into the ordinals using a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F  = recs ( ( z  e. 
 _V  |->  ( G `  ( A  \  ran  z
 ) ) ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  |^| ( `' F " { x } ) ) : A -1-1-> On )
 
Theoremdnwech 35977* Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F  = recs ( ( z  e. 
 _V  |->  ( G `  ( A  \  ran  z
 ) ) ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )   &    |-  H  =  { <. v ,  w >.  |  |^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) }   =>    |-  ( ph  ->  H  We  A )
 
Theoremfnwe2val 35978* Lemma for fnwe2 35982. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  (
 z  =  ( F `
  x )  ->  S  =  U )   &    |-  T  =  { <. x ,  y >.  |  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x U y ) ) }   =>    |-  (
 a T b  <->  ( ( F `
  a ) R ( F `  b
 )  \/  ( ( F `  a )  =  ( F `  b )  /\  a [_ ( F `  a ) 
 /  z ]_ S b ) ) )
 
Theoremfnwe2lem1 35979* Lemma for fnwe2 35982. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  (
 z  =  ( F `
  x )  ->  S  =  U )   &    |-  T  =  { <. x ,  y >.  |  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x U y ) ) }   &    |-  (
 ( ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y
 )  =  ( F `
  x ) }
 )   =>    |-  ( ( ph  /\  a  e.  A )  ->  [_ ( F `  a )  /  z ]_ S  We  {
 y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
 
Theoremfnwe2lem2 35980* Lemma for fnwe2 35982. An element which is in a minimal fiber and minimal within its fiber is minimal globally; thus  T is well-founded. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  (
 z  =  ( F `
  x )  ->  S  =  U )   &    |-  T  =  { <. x ,  y >.  |  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x U y ) ) }   &    |-  (
 ( ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y
 )  =  ( F `
  x ) }
 )   &    |-  ( ph  ->  ( F  |`  A ) : A --> B )   &    |-  ( ph  ->  R  We  B )   &    |-  ( ph  ->  a  C_  A )   &    |-  ( ph  ->  a  =/=  (/) )   =>    |-  ( ph  ->  E. b  e.  a  A. c  e.  a  -.  c T b )
 
Theoremfnwe2lem3 35981* Lemma for fnwe2 35982. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  (
 z  =  ( F `
  x )  ->  S  =  U )   &    |-  T  =  { <. x ,  y >.  |  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x U y ) ) }   &    |-  (
 ( ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y
 )  =  ( F `
  x ) }
 )   &    |-  ( ph  ->  ( F  |`  A ) : A --> B )   &    |-  ( ph  ->  R  We  B )   &    |-  ( ph  ->  a  e.  A )   &    |-  ( ph  ->  b  e.  A )   =>    |-  ( ph  ->  ( a T b  \/  a  =  b  \/  b T a ) )
 
Theoremfnwe2 35982* A well-ordering can be constructed on a partitioned set by patching together well-orderings on each partition using a well-ordering on the partitions themselves. Similar to fnwe 6931 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  (
 z  =  ( F `
  x )  ->  S  =  U )   &    |-  T  =  { <. x ,  y >.  |  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x U y ) ) }   &    |-  (
 ( ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y
 )  =  ( F `
  x ) }
 )   &    |-  ( ph  ->  ( F  |`  A ) : A --> B )   &    |-  ( ph  ->  R  We  B )   =>    |-  ( ph  ->  T  We  A )
 
Theoremaomclem1 35983* Lemma for dfac11 35991. This is the beginning of the proof that multiple choice is equivalent to choice. Our goal is to construct, by transfinite recursion, a well-ordering of  ( R1 `  A ). In what follows,  A is the index of the rank we wish to well-order,  z is the collection of well-orderings constructed so far,  dom  z is the set of ordinal indexes of constructed ranks i.e. the next rank to construct, and  y is a postulated multiple-choice function.

Successor case 1, define a simple ordering from the well-ordered predecessor. (Contributed by Stefan O'Rear, 18-Jan-2015.)

 |-  B  =  { <. a ,  b >.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a ) 
 /\  A. d  e.  ( R1 `  U. dom  z
 ) ( d ( z `  U. dom  z ) c  ->  ( d  e.  a  <->  d  e.  b ) ) ) }   &    |-  ( ph  ->  dom  z  e.  On )   &    |-  ( ph  ->  dom  z  =  suc  U. dom  z )   &    |-  ( ph  ->  A. a  e. 
 dom  z ( z `
  a )  We  ( R1 `  a
 ) )   =>    |-  ( ph  ->  B  Or  ( R1 `  dom  z ) )
 
Theoremaomclem2 35984* Lemma for dfac11 35991. Successor case 2, a choice function for subsets of  ( R1 `  dom  z ). (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  B  =  { <. a ,  b >.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a ) 
 /\  A. d  e.  ( R1 `  U. dom  z
 ) ( d ( z `  U. dom  z ) c  ->  ( d  e.  a  <->  d  e.  b ) ) ) }   &    |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z ) ,  B ) )   &    |-  ( ph  ->  dom  z  e. 
 On )   &    |-  ( ph  ->  dom  z  =  suc  U. dom  z )   &    |-  ( ph  ->  A. a  e.  dom  z
 ( z `  a
 )  We  ( R1
 `  a ) )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  dom  z  C_  A )   &    |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
 y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
 } ) ) )   =>    |-  ( ph  ->  A. a  e. 
 ~P  ( R1 `  dom  z ) ( a  =/=  (/)  ->  ( C `  a )  e.  a
 ) )
 
Theoremaomclem3 35985* Lemma for dfac11 35991. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  B  =  { <. a ,  b >.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a ) 
 /\  A. d  e.  ( R1 `  U. dom  z
 ) ( d ( z `  U. dom  z ) c  ->  ( d  e.  a  <->  d  e.  b ) ) ) }   &    |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z ) ,  B ) )   &    |-  D  = recs ( (
 a  e.  _V  |->  ( C `  ( ( R1 `  dom  z
 )  \  ran  a ) ) ) )   &    |-  E  =  { <. a ,  b >.  |  |^| ( `' D " { a } )  e.  |^| ( `' D " { b } ) }   &    |-  ( ph  ->  dom  z  e.  On )   &    |-  ( ph  ->  dom  z  =  suc  U. dom  z )   &    |-  ( ph  ->  A. a  e.  dom  z
 ( z `  a
 )  We  ( R1
 `  a ) )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  dom  z  C_  A )   &    |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
 y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
 } ) ) )   =>    |-  ( ph  ->  E  We  ( R1 `  dom  z
 ) )
 
Theoremaomclem4 35986* Lemma for dfac11 35991. Limit case. Patch together well-orderings constructed so far using fnwe2 35982 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  F  =  { <. a ,  b >.  |  ( ( rank `  a )  _E  ( rank `  b )  \/  ( ( rank `  a
 )  =  ( rank `  b )  /\  a
 ( z `  suc  ( rank `  a )
 ) b ) ) }   &    |-  ( ph  ->  dom  z  e.  On )   &    |-  ( ph  ->  dom  z  =  U.
 dom  z )   &    |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )   =>    |-  ( ph  ->  F  We  ( R1 `  dom  z
 ) )
 
Theoremaomclem5 35987* Lemma for dfac11 35991. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  B  =  { <. a ,  b >.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a ) 
 /\  A. d  e.  ( R1 `  U. dom  z
 ) ( d ( z `  U. dom  z ) c  ->  ( d  e.  a  <->  d  e.  b ) ) ) }   &    |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z ) ,  B ) )   &    |-  D  = recs ( (
 a  e.  _V  |->  ( C `  ( ( R1 `  dom  z
 )  \  ran  a ) ) ) )   &    |-  E  =  { <. a ,  b >.  |  |^| ( `' D " { a } )  e.  |^| ( `' D " { b } ) }   &    |-  F  =  { <. a ,  b >.  |  ( ( rank `  a )  _E  ( rank `  b )  \/  ( ( rank `  a
 )  =  ( rank `  b )  /\  a
 ( z `  suc  ( rank `  a )
 ) b ) ) }   &    |-  G  =  ( if ( dom  z  =  U. dom  z ,  F ,  E )  i^i  ( ( R1
 `  dom  z )  X.  ( R1 `  dom  z ) ) )   &    |-  ( ph  ->  dom  z  e. 
 On )   &    |-  ( ph  ->  A. a  e.  dom  z
 ( z `  a
 )  We  ( R1
 `  a ) )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  dom  z  C_  A )   &    |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
 y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
 } ) ) )   =>    |-  ( ph  ->  G  We  ( R1 `  dom  z
 ) )
 
Theoremaomclem6 35988* Lemma for dfac11 35991. Transfinite induction, close over  z. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  B  =  { <. a ,  b >.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a ) 
 /\  A. d  e.  ( R1 `  U. dom  z
 ) ( d ( z `  U. dom  z ) c  ->  ( d  e.  a  <->  d  e.  b ) ) ) }   &    |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z ) ,  B ) )   &    |-  D  = recs ( (
 a  e.  _V  |->  ( C `  ( ( R1 `  dom  z
 )  \  ran  a ) ) ) )   &    |-  E  =  { <. a ,  b >.  |  |^| ( `' D " { a } )  e.  |^| ( `' D " { b } ) }   &    |-  F  =  { <. a ,  b >.  |  ( ( rank `  a )  _E  ( rank `  b )  \/  ( ( rank `  a
 )  =  ( rank `  b )  /\  a
 ( z `  suc  ( rank `  a )
 ) b ) ) }   &    |-  G  =  ( if ( dom  z  =  U. dom  z ,  F ,  E )  i^i  ( ( R1
 `  dom  z )  X.  ( R1 `  dom  z ) ) )   &    |-  H  = recs ( (
 z  e.  _V  |->  G ) )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
 y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
 } ) ) )   =>    |-  ( ph  ->  ( H `  A )  We  ( R1 `  A ) )
 
Theoremaomclem7 35989* Lemma for dfac11 35991. 
( R1 `  A
) is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  B  =  { <. a ,  b >.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a ) 
 /\  A. d  e.  ( R1 `  U. dom  z
 ) ( d ( z `  U. dom  z ) c  ->  ( d  e.  a  <->  d  e.  b ) ) ) }   &    |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z ) ,  B ) )   &    |-  D  = recs ( (
 a  e.  _V  |->  ( C `  ( ( R1 `  dom  z
 )  \  ran  a ) ) ) )   &    |-  E  =  { <. a ,  b >.  |  |^| ( `' D " { a } )  e.  |^| ( `' D " { b } ) }   &    |-  F  =  { <. a ,  b >.  |  ( ( rank `  a )  _E  ( rank `  b )  \/  ( ( rank `  a
 )  =  ( rank `  b )  /\  a
 ( z `  suc  ( rank `  a )
 ) b ) ) }   &    |-  G  =  ( if ( dom  z  =  U. dom  z ,  F ,  E )  i^i  ( ( R1
 `  dom  z )  X.  ( R1 `  dom  z ) ) )   &    |-  H  = recs ( (
 z  e.  _V  |->  G ) )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
 y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
 } ) ) )   =>    |-  ( ph  ->  E. b  b  We  ( R1 `  A ) )
 
Theoremaomclem8 35990* Lemma for dfac11 35991. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  ( y `  a )  e.  (
 ( ~P a  i^i 
 Fin )  \  { (/)
 } ) ) )   =>    |-  ( ph  ->  E. b  b  We  ( R1 `  A ) )
 
Theoremdfac11 35991* The right-hand side of this theorem (compare with ac4 8923), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 8125, despite not mentioning the cumulative hierarchy in any way as most consequences of regularity do.

This is definition (MC) of [Schechter] p. 141. EDITORIAL: the proof is not original with me of course but I lost my reference sometime after writing it.

A multiple choice function allows any total order to be extended to a choice function, which in turn defines a well-ordering. Since a well ordering on a set defines a simple ordering of the power set, this allows the trivial well-ordering of the empty set to be transfinitely bootstrapped up the cumulative hierarchy to any desired level. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Stefan O'Rear, 1-Jun-2015.)

 |-  (CHOICE  <->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  (
 f `  z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
 } ) ) )
 
Theoremkelac1 35992* Kelley's choice, basic form: if a collection of sets can be cast as closed sets in the factors of a topology, and there is a definable element in each topology (which need not be in the closed set - if it were this would be trivial), then compactness (via finite intersection) guarantees that the final product is nonempty. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  (
 ( ph  /\  x  e.  I )  ->  S  =/= 
 (/) )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  J  e.  Top )   &    |-  (
 ( ph  /\  x  e.  I )  ->  C  e.  ( Clsd `  J )
 )   &    |-  ( ( ph  /\  x  e.  I )  ->  B : S -1-1-onto-> C )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  U  e.  U. J )   &    |-  ( ph  ->  ( Xt_ `  ( x  e.  I  |->  J ) )  e.  Comp )   =>    |-  ( ph  ->  X_ x  e.  I  S  =/=  (/) )
 
Theoremkelac2lem 35993 Lemma for kelac2 35994 and dfac21 35995: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( S  e.  V  ->  (
 topGen `  { S ,  { ~P U. S } } )  e.  Comp )
 
Theoremkelac2 35994* Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  (
 ( ph  /\  x  e.  I )  ->  S  e.  V )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  S  =/=  (/) )   &    |-  ( ph  ->  ( Xt_ `  ( x  e.  I  |->  (
 topGen `  { S ,  { ~P U. S } } ) ) )  e.  Comp )   =>    |-  ( ph  ->  X_ x  e.  I  S  =/=  (/) )
 
Theoremdfac21 35995 Tychonoff's theorem is a choice equivalent. Definition AC21 of Schechter p. 461. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 27-Aug-2015.)
 |-  (CHOICE  <->  A. f ( f : dom  f --> Comp  ->  (
 Xt_ `  f )  e.  Comp ) )
 
21.23.38  Finitely generated left modules
 
Syntaxclfig 35996 Extend class notation with the class of finitely generated left modules.
 class LFinGen
 
Definitiondf-lfig 35997 Define the class of finitely generated left modules. Finite generation of subspaces can be intepreted using ↾s. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |- LFinGen  =  { w  e.  LMod  |  (
 Base `  w )  e.  ( ( LSpan `  w ) " ( ~P ( Base `  w )  i^i 
 Fin ) ) }
 
Theoremislmodfg 35998* Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  B  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( W  e.  LMod  ->  ( W  e. LFinGen  <->  E. b  e.  ~P  B ( b  e. 
 Fin  /\  ( N `  b )  =  B ) ) )
 
Theoremislssfg 35999* Property of a finitely generated left (sub-)module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  (
 ( W  e.  LMod  /\  U  e.  S ) 
 ->  ( X  e. LFinGen  <->  E. b  e.  ~P  U ( b  e. 
 Fin  /\  ( N `  b )  =  U ) ) )
 
Theoremislssfg2 36000* Property of a finitely generated left (sub-)module, with a relaxed constraint on the spanning vectors. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  B  =  ( Base `  W )   =>    |-  (
 ( W  e.  LMod  /\  U  e.  S ) 
 ->  ( X  e. LFinGen  <->  E. b  e.  ( ~P B  i^i  Fin )
 ( N `  b
 )  =  U ) )
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