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Theorem List for Metamath Proof Explorer - 26501-26600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremjm2.20nn 26501 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 26502 Lemma for jm2.26 26506. (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 26503 Lemma for jm2.26 26506. 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 26504 Lemma for jm2.26 26506. 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 26505 Lemma for jm2.26 26506. Use acongrep 26478 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 26506 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 26507 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 26508 Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 26507 if Yrm is redefined as described in rmyluc 26433. (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 26509 Lemma for jm2.27 26512. 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 26510 Lemma for jm2.27 26512. 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 26511 Lemma for jm2.27 26512. 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 26512* 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 26509 and jm2.27c 26511. 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 26513* Lemma for rmydioph 26518. 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 26514 Lemma for rmydioph 26518. 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 26515 Lemma for rmydioph 26518. 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 26516 Lemma for rmydioph 26518. 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 26517 Lemma for rmydioph 26518. Used with sselii 3177 to infer membership of midpoints of range; jm2.27dlem2 26514 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  B  =  ( A  +  1 )   &    |-  ( 1 ...
 B )  C_  (
 1 ... C )   =>    |-  ( 1 ...
 A )  C_  (
 1 ... C )
 
Theoremrmydioph 26518 jm2.27 26512 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 )
 
18.17.37  X and Y sequences 5: Diophantine representability of X, ^, _C
 
Theoremrmxdiophlem 26519* 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 26520 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 26521 Lemma for jm3.1 26524. (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 26522 Lemma for jm3.1 26524. (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 26523 Lemma for jm3.1 26524. (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 26524 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 26525* Lemma for expdioph 26527. 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 26526 Lemma for expdioph 26527. 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 26527 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 )
 
18.17.38  Uncategorized stuff not associated with a major project
 
Theoremsetindtr 26528* 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 7414; 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 26529* Epsilon induction scheme without Infinity. See comments at setindtr 26528. (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 26530* Lemma for dford3 26532. (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 26531* Lemma for dford3 26532. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  (
 ( Tr  x  /\  A. y  e.  x  Tr  y )  ->  x  e. 
 On )
 
Theoremdford3 26532* 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 26533* dford3 26532 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 26534 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 26535* Lemma for rpnnen3 26536. (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 26536 Dedekind cut injection of  RR into  ~P QQ. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  RR  ~<_  ~P QQ
 
18.17.39  More equivalents of the Axiom of Choice
 
Theoremaxac10 26537 Characterization of choice similar to dffin1-5 8009. (Contributed by Stefan O'Rear, 6-Jan-2015.)
 |-  (  ~~  " On )  =  _V
 
Theoremharinf 26538 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 26539* 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 26540 Tarski's theorem about choice: infxpidm 8179 is equivalent to ax-ac 8080. (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 26541* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 6964, 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 26542* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 6964, 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 26543* Function value of the pw2f1o2 26542 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 26544* Membership in a mapped set under the pw2f1o2 26542 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 26545 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 26546 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 26547 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 26548 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 26549* 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 26550* 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 )
 
Theoreminisegn0 26551 Non-emptyness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  ( A  e.  ran  F  <->  ( `' F " { A } )  =/= 
 (/) )
 
Theoremdnnumch1 26552* Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 7652 (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 26553* 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 26554* 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 26555* 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 26556* 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 26557* Lemma for fnwe2 26561. 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 26558* Lemma for fnwe2 26561. 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 26559* Lemma for fnwe2 26561. 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 26560* Lemma for fnwe2 26561. 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 26561* 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 6192 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 26562* Lemma for dfac11 26571. 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 26563* Lemma for dfac11 26571. 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 26564* Lemma for dfac11 26571. 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 26565* Lemma for dfac11 26571. Limit case. Patch together well-orderings constructed so far using fnwe2 26561 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 26566* Lemma for dfac11 26571. 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 26567* Lemma for dfac11 26571. 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 26568* Lemma for dfac11 26571. 
( 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 ) )
 
Theoremsupeq123d 26569 Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  ( ph  ->  A  =  D )   &    |-  ( ph  ->  B  =  E )   &    |-  ( ph  ->  C  =  F )   =>    |-  ( ph  ->  sup ( A ,  B ,  C )  =  sup ( D ,  E ,  F ) )
 
Theoremaomclem8 26570* Lemma for dfac11 26571. 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 26571* The right hand side of this theorem (compare with ac4 8097), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 7301, 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 26572* 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 26573 Lemma for kelac2 26574 and dfac21 26575: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( S  e.  V  ->  (
 topGen `  { S ,  { ~P U. S } } )  e.  Comp )
 
Theoremkelac2 26574* 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 26575 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 ) )
 
18.17.40  Finitely generated left modules
 
Syntaxclfig 26576 Extend class notation with the class of finitely generated left modules.
 class LFinGen
 
Definitiondf-lfig 26577 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 26578* 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 26579* 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 26580* 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 ) )
 
Theoremislssfgi 26581 Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  N  =  ( LSpan `  W )   &    |-  V  =  ( Base `  W )   &    |-  X  =  ( Ws  ( N `  B ) )   =>    |-  ( ( W  e.  LMod  /\  B  C_  V  /\  B  e.  Fin )  ->  X  e. LFinGen )
 
Theoremfglmod 26582 Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  ( M  e. LFinGen  ->  M  e.  LMod
 )
 
Theoremlsmfgcl 26583 The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  U  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  D  =  ( Ws  A )   &    |-  E  =  ( Ws  B )   &    |-  F  =  ( Ws  ( A  .(+)  B ) )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   &    |-  ( ph  ->  D  e. LFinGen )   &    |-  ( ph  ->  E  e. LFinGen )   =>    |-  ( ph  ->  F  e. LFinGen )
 
18.17.41  Noetherian left modules I
 
Syntaxclnm 26584 Extend class notation with the class of Noetherian left modules.
 class LNoeM
 
Definitiondf-lnm 26585* A left-module is Noetherian iff it is hereditarily finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |- LNoeM  =  { w  e.  LMod  |  A. i  e.  ( LSubSp `  w ) ( ws  i )  e. LFinGen }
 
Theoremislnm 26586* Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  S  =  ( LSubSp `  M )   =>    |-  ( M  e. LNoeM  <->  ( M  e.  LMod  /\  A. i  e.  S  ( Ms  i )  e. LFinGen )
 )
 
Theoremislnm2 26587* Property of being a Noetherian left module with finite generation expanded in terms of spans. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  B  =  ( Base `  M )   &    |-  S  =  ( LSubSp `  M )   &    |-  N  =  ( LSpan `  M )   =>    |-  ( M  e. LNoeM  <->  ( M  e.  LMod  /\  A. i  e.  S  E. g  e.  ( ~P B  i^i  Fin )
 i  =  ( N `
  g ) ) )
 
Theoremlnmlmod 26588 A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  ( M  e. LNoeM  ->  M  e.  LMod
 )
 
Theoremlnmlssfg 26589 A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  S  =  ( LSubSp `  M )   &    |-  R  =  ( Ms  U )   =>    |-  ( ( M  e. LNoeM  /\  U  e.  S ) 
 ->  R  e. LFinGen )
 
Theoremlnmlsslnm 26590 All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  S  =  ( LSubSp `  M )   &    |-  R  =  ( Ms  U )   =>    |-  ( ( M  e. LNoeM  /\  U  e.  S ) 
 ->  R  e. LNoeM )
 
Theoremlnmfg 26591 A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  ( M  e. LNoeM  ->  M  e. LFinGen )
 
Theoremkercvrlsm 26592 The domain of a linear function is the subspace sum of the kernel and any subspace which covers the range. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  U  =  ( LSubSp `  S )   &    |-  .(+)  =  (
 LSSum `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  K  =  ( `' F " {  .0.  } )   &    |-  B  =  (
 Base `  S )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   &    |-  ( ph  ->  D  e.  U )   &    |-  ( ph  ->  ( F " D )  = 
 ran  F )   =>    |-  ( ph  ->  ( K  .(+)  D )  =  B )
 
Theoremlmhmfgima 26593 A homomorphism maps finitely generated submodules to finitely generated submodules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( Ts  ( F " A ) )   &    |-  X  =  ( Ss  A )   &    |-  U  =  (
 LSubSp `  S )   &    |-  ( ph  ->  X  e. LFinGen )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   =>    |-  ( ph  ->  Y  e. LFinGen )
 
Theoremlnmepi 26594 Epimorphic images of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  B  =  ( Base `  T )   =>    |-  (
 ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  T  e. LNoeM )
 
Theoremlmhmfgsplit 26595 If the kernel and range of a homomorphism of left modules are finitely generated, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  .0.  =  ( 0g `  T )   &    |-  K  =  ( `' F " {  .0.  } )   &    |-  U  =  ( Ss  K )   &    |-  V  =  ( Ts 
 ran  F )   =>    |-  ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen ) 
 ->  S  e. LFinGen )
 
Theoremlmhmlnmsplit 26596 If the kernel and range of a homomorphism of left modules are Noetherian, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 12-Jun-2015.)
 |-  .0.  =  ( 0g `  T )   &    |-  K  =  ( `' F " {  .0.  } )   &    |-  U  =  ( Ss  K )   &    |-  V  =  ( Ts 
 ran  F )   =>    |-  ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM ) 
 ->  S  e. LNoeM )
 
Theoremlnmlmic 26597 Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( R  ~=ph𝑚 
 S  ->  ( R  e. LNoeM  <->  S  e. LNoeM ) )
 
18.17.42  Addenda for structure powers
 
Theorempwssplit0 26598* Splitting for structure powers, part 0: restriction is a function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  U )   &    |-  Z  =  ( W  ^s  V )   &    |-  B  =  ( Base `  Y )   &    |-  C  =  ( Base `  Z )   &    |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )   =>    |-  ( ( W  e.  T  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
 
Theorempwssplit1 26599* Splitting for structure powers, part 1: restriction is an onto function. The only actual monoid law we need here is that the base set is nonempty. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  U )   &    |-  Z  =  ( W  ^s  V )   &    |-  B  =  ( Base `  Y )   &    |-  C  =  ( Base `  Z )   &    |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )   =>    |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  F : B -onto-> C )
 
Theorempwssplit2 26600* Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  U )   &    |-  Z  =  ( W  ^s  V )   &    |-  B  =  ( Base `  Y )   &    |-  C  =  ( Base `  Z )   &    |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )   =>    |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
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