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Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-eqp 24001* Define an equivalence relation on 
ZZ-indexed sequences of integers such that two sequences are equivalent iff the difference is equivalent to zero, and a sequence is equivalent to zero iff the sum  sum_ k  <_  n f ( k ) ( p ^
k ) is a multiple of  p ^ (
n  +  1 ) for every  n. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- ~Qp  =  ( p  e.  Prime  |->  { <. f ,  g >.  |  ( { f ,  g }  C_  ( ZZ  ^m  ZZ )  /\  A. n  e.  ZZ  sum_ k  e.  ( ZZ>=
 `  -u n ) ( ( ( f `  -u k )  -  (
 g `  -u k ) )  /  ( p ^ ( k  +  ( n  +  1
 ) ) ) )  e.  ZZ ) }
 )
 
Definitiondf-rqp 24002* There is a unique element of  ( ZZ  ^m  (
0 ... ( p  - 
1 ) ) ) ~Qp -equivalent to any element of 
( ZZ  ^m  ZZ ), if the sequences are zero for sufficiently large negative values; this function selects that element. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- /Qp  =  ( p  e.  Prime  |->  (~Qp  i^i  [_
 { f  e.  ( ZZ  ^m  ZZ )  | 
 E. x  e.  ran  ZZ>= ( `' f " ( ZZ  \  { 0 } )
 )  C_  x }  /  y ]_ ( y  X.  ( y  i^i  ( ZZ  ^m  (
 0 ... ( p  -  1 ) ) ) ) ) ) )
 
Definitiondf-qp 24003* Define the  p-adic completion of the rational numbers, as a normed field structure with a total order (that is not compatible with the operations). (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- Qp  =  ( p  e.  Prime  |->  [_
 { h  e.  ( ZZ  ^m  ( 0 ... ( p  -  1
 ) ) )  | 
 E. x  e.  ran  ZZ>= ( `' h " ( ZZ  \  { 0 } )
 )  C_  x }  /  b ]_ ( ( { <. ( Base `  ndx ) ,  b >. , 
 <. ( +g  `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( (/Qp `  p ) `  (
 f  o F  +  g ) ) )
 >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( (/Qp `  p ) `  ( n  e.  ZZ  |->  sum_ k  e.  ZZ  ( ( f `
  k )  x.  ( g `  ( n  -  k ) ) ) ) ) )
 >. }  u.  { <. ( le `  ndx ) ,  { <. f ,  g >.  |  ( { f ,  g }  C_  b  /\  sum_ k  e.  ZZ  ( ( f `  -u k )  x.  (
 ( p  +  1 ) ^ -u k
 ) )  <  sum_ k  e.  ZZ  ( ( g `
  -u k )  x.  ( ( p  +  1 ) ^ -u k
 ) ) ) } >. } ) toNrmGrp  ( f  e.  b  |->  if (
 f  =  ( ZZ 
 X.  { 0 } ) ,  0 ,  ( p ^ -u sup ( ( `' f " ( ZZ  \  { 0 } )
 ) ,  RR ,  `'  <  ) ) ) ) ) )
 
Definitiondf-zp 24004 Define the  p-adic integers, as a subset of the  p-adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- Zp  =  (ZRing  o. Qp )
 
Definitiondf-qpa 24005* Define the completion of the  p-adic rationals. Here we simply define it as the splitting field of a dense sequence of polynomials (using as the  n-th set the collection of polynomials with degree less than  n and with coefficients  <  ( p ^
n )). Krasner's lemma will then show that all monic polynomials have splitting fields isomorphic to a sufficiently close Eisenstein polynomial from the list, and unramified extensions are generated by the polynomial  x ^ (
p ^ n )  -  x, which is in the list. Thus, every finite extension of Qp is a subfield of this field extension, so it is algebraically closed. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- _Qp  =  ( p  e.  Prime  |->  [_ (Qp `  p )  /  r ]_ ( r polySplitLim  ( n  e.  NN  |->  { f  e.  (Poly1 `  r )  |  (
 ( r deg1  f )  <_  n  /\  A. d  e. 
 ran  (coe1 `  f ) ( `' d " ( ZZ  \  { 0 } )
 )  C_  ( 0 ... n ) ) }
 ) ) )
 
Definitiondf-cp 24006 Define the metric completion of the algebraic completion of the  p -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- Cp  =  ( cplMetSp  o. _Qp )
 
18.5  Mathbox for Paul Chapman
 
18.5.1  Group homomorphism and isomorphism
 
Theoremghomgrpilem1 24007 Lemma for ghomgrpi 24009. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  G  e.  GrpOp   &    |-  H  e.  GrpOp   &    |-  F  e.  ( G GrpOpHom  H )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   &    |-  W  =  ran  H   &    |-  T  =  (GId `  H )   &    |-  M  =  ( inv `  H )   &    |-  Z  =  ran  F   &    |-  S  =  ( H  |`  ( Z  X.  Z ) )   =>    |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( F `
  A ) H ( F `  B ) )  =  ( F `  ( A G B ) ) )
 
Theoremghomgrpilem2 24008 Lemma for ghomgrpi 24009. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  G  e.  GrpOp   &    |-  H  e.  GrpOp   &    |-  F  e.  ( G GrpOpHom  H )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   &    |-  W  =  ran  H   &    |-  T  =  (GId `  H )   &    |-  M  =  ( inv `  H )   &    |-  Z  =  ran  F   &    |-  S  =  ( H  |`  ( Z  X.  Z ) )   =>    |-  S  e.  ( SubGrpOp `  H )
 
Theoremghomgrpi 24009 The image of a group homomorphism from  G to  H is a subgroup of  H (inference version). (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  G  e.  GrpOp   &    |-  H  e.  GrpOp   &    |-  F  e.  ( G GrpOpHom  H )   &    |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   =>    |-  S  e.  ( SubGrpOp `  H )
 
Theoremghomsn 24010 The endomorphism of the trivial group. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  A  e.  _V   &    |-  G  =  { <.
 <. A ,  A >. ,  A >. }   =>    |-  (  _I  |`  { A } )  e.  ( G GrpOpHom  G )
 
Theoremghomgrplem 24011 Lemma for ghomgrp 24012. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  ( ph  ->  ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 ) )   &    |-  S  =  { <.
 <. z ,  z >. ,  z >. }   &    |-  J  =  (  _I  |`  { z } )   =>    |-  ( ph  ->  ( H  |`  ( ran  F  X.  ran  F ) )  e.  ( SubGrpOp `  H ) )
 
Theoremghomgrp 24012 The image of a group homomorphism from  G to  H is a subgroup of  H. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  ( SubGrpOp `  H ) )
 
Theoremghomfo 24013 A group homomorphism maps onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
 |-  X  =  ran  G   &    |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   &    |-  Z  =  ran  S   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  ->  F : X -onto-> Z )
 
Theoremghomcl 24014 Closure of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.)
 |-  X  =  ran  G   &    |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   &    |-  Z  =  ran  S   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  ->  ( A  e.  X  ->  ( F `  A )  e.  Z ) )
 
Theoremghomgsg 24015 A group homomorphism from  G to  H is also a group homomorphism from  G to its image in  H. (Contributed by Paul Chapman, 3-Mar-2008.)
 |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F  e.  ( G GrpOpHom  S ) )
 
Theoremghomf1olem 24016* Lemma for ghomf1o 24017. (Contributed by Paul Chapman, 3-Mar-2008.)
 |-  X  =  ran  G   &    |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   &    |-  Z  =  ran  S   &    |-  U  =  (GId `  G )   &    |-  T  =  (GId `  H )   &    |-  N  =  ( inv `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  ->  ( F : X -1-1-onto-> Z  <->  A. x  e.  X  ( ( F `  x )  =  T  ->  x  =  U ) ) )
 
Theoremghomf1o 24017* Two ways of saying a group homomorphism is 1-1-onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
 |-  X  =  ran  G   &    |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   &    |-  Z  =  ran  S   &    |-  U  =  (GId `  G )   &    |-  T  =  (GId `  H )   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  ->  ( F : X -1-1-onto-> Z  <->  A. x  e.  X  ( ( F `  x )  =  T  ->  x  =  U ) ) )
 
Theoremelgiso 24018 Membership in the set of group isomorphisms from  G to  H. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  (
 ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G  GrpOpIso  H )  <->  ( F  e.  ( G GrpOpHom  H )  /\  F : ran  G -1-1-onto-> ran  H ) ) )
 
18.5.2  Real and complex numbers (cont.)
 
Theoremclimuzcnv 24019* Utility lemma to convert between  m  <_  k and  k  e.  ( ZZ>= `  m ) in limit theorems. (Contributed by Paul Chapman, 10-Nov-2012.)
 |-  ( m  e.  NN  ->  ( ( k  e.  ( ZZ>=
 `  m )  ->  ph )  <->  ( k  e. 
 NN  ->  ( m  <_  k  ->  ph ) ) ) )
 
Theoremsinccvglem 24020*  ( ( sin `  x )  /  x )  ~~>  1 as (real)  x  ~~>  0. (Contributed by Paul Chapman, 10-Nov-2012.) (Revised by Mario Carneiro, 21-May-2014.)
 |-  ( ph  ->  F : NN --> ( RR  \  { 0 } ) )   &    |-  ( ph  ->  F  ~~>  0 )   &    |-  G  =  ( x  e.  ( RR  \  { 0 } )  |->  ( ( sin `  x )  /  x ) )   &    |-  H  =  ( x  e.  CC  |->  ( 1  -  ( ( x ^ 2 ) 
 /  3 ) ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  (
 ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( abs `  ( F `  k
 ) )  <  1
 )   =>    |-  ( ph  ->  ( G  o.  F )  ~~>  1 )
 
Theoremsinccvg 24021*  ( ( sin `  x )  /  x )  ~~>  1 as (real)  x  ~~>  0. (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)
 |-  (
 ( F : NN --> ( RR  \  { 0 } )  /\  F  ~~>  0 )  ->  ( ( x  e.  ( RR  \  { 0 } )  |->  ( ( sin `  x )  /  x ) )  o.  F )  ~~>  1 )
 
Theoremcircum 24022* The circumference of a circle of radius  R, defined as the limit as  n  ~~>  +oo of the perimeter of an inscribed n-sided isogons, is  ( (
2  x.  pi )  x.  R ). (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)
 |-  A  =  ( ( 2  x.  pi )  /  n )   &    |-  P  =  ( n  e.  NN  |->  ( ( 2  x.  n )  x.  ( R  x.  ( sin `  ( A  /  2 ) ) ) ) )   &    |-  R  e.  RR   =>    |-  P  ~~>  ( ( 2  x.  pi )  x.  R )
 
18.5.3  Miscellaneous theorems
 
Theoremelfzm12 24023 Membership in a curtailed finite sequence of integers. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( N  e.  NN  ->  ( M  e.  ( 1
 ... ( N  -  1 ) )  ->  M  e.  ( 1 ... N ) ) )
 
Theoremnn0seqcvg 24024* A strictly-decreasing nonnegative integer sequence with initial term  N reaches zero by the  N th term. Inference version. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  F : NN0 --> NN0   &    |-  N  =  ( F `
  0 )   &    |-  (
 k  e.  NN0  ->  ( ( F `  (
 k  +  1 ) )  =/=  0  ->  ( F `  ( k  +  1 ) )  <  ( F `  k ) ) )   =>    |-  ( F `  N )  =  0
 
Theoremzmodid2 24025 Identity law for modulo restricted to integers. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M  mod  N )  =  M  <->  M  e.  (
 0 ... ( N  -  1 ) ) ) )
 
Theoremmodaddabs 24026 Absorbtion law for modulo. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( ( ( A 
 mod  C )  +  ( B  mod  C ) ) 
 mod  C )  =  ( ( A  +  B )  mod  C ) )
 
Theoremelfzp1b 24027 An integer is a member of a 0-based finite set of sequential integers iff its successor is a member of the corresponding 1-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  (
 ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  (
 0 ... ( N  -  1 ) )  <->  ( K  +  1 )  e.  (
 1 ... N ) ) )
 
Theoremlediv2aALT 24028 Division of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  (
 ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  <_  C ) )  ->  ( A 
 <_  B  ->  ( C  /  B )  <_  ( C  /  A ) ) )
 
Theoremabs2sqlei 24029 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  (
 ( abs `  A )  <_  ( abs `  B ) 
 <->  ( ( abs `  A ) ^ 2 )  <_  ( ( abs `  B ) ^ 2 ) )
 
Theoremabs2sqlti 24030 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  (
 ( abs `  A )  <  ( abs `  B ) 
 <->  ( ( abs `  A ) ^ 2 )  < 
 ( ( abs `  B ) ^ 2 ) )
 
Theoremabs2sqle 24031 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A )  <_  ( abs `  B ) 
 <->  ( ( abs `  A ) ^ 2 )  <_  ( ( abs `  B ) ^ 2 ) ) )
 
Theoremabs2sqlt 24032 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A )  <  ( abs `  B ) 
 <->  ( ( abs `  A ) ^ 2 )  < 
 ( ( abs `  B ) ^ 2 ) ) )
 
Theoremabs2difi 24033 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  (
 ( abs `  A )  -  ( abs `  B ) )  <_  ( abs `  ( A  -  B ) )
 
Theoremabs2difabsi 24034 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( abs `  ( ( abs `  A )  -  ( abs `  B ) ) )  <_  ( abs `  ( A  -  B ) )
 
18.6  Mathbox for Drahflow

This is the mathbox of Jens-Wolfhard Schicke-Uffmann, reachable at drahflow@gmx.de / drahflow.name

 
Theoremsbcung 24035* Distribution of class substitution over union of two classes. (Contributed by Drahflow, 23-Sep-2015.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( C  u.  D )  =  ( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D ) )
 
Theoremsbcuni 24036* Distribution of class substitution over union of two classes, inference version. (Contributed by Drahflow, 23-Sep-2015.)
 |-  A  e.  _V   =>    |-  [_ A  /  x ]_ ( C  u.  D )  =  ( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D )
 
Theoremsbcopg 24037* Distribution of class substitution over ordered pairs. (Contributed by Drahflow, 25-Sep-2015.) (Revised by Mario Carneiro, 29-Oct-2015.)
 |-  ( A  e.  _V  ->  [_ A  /  x ]_ <. C ,  D >.  = 
 <. [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >. )
 
Syntaxcrelexp 24038 Extend class notation to include relation exponentiation.
 class  ^ r
 
Definitiondf-relexp 24039* Definition of repeated composition of a relation with itself, aka relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ^ r  =  ( r  e.  _V ,  n  e.  NN0  |->  (  seq  0 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) ) ,  ( z  e.  _V  |->  (  _I  |`  U. U. r
 ) ) ) `  n ) )
 
Theoremrelexp0 24040 A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( R ^ r 0 )  =  (  _I  |`  U. U. R ) )
 
Theoremrelexpsucr 24041 A reduction for relation exponentiation to the right. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  ( R ^ r ( N  +  1 ) )  =  ( ( R ^ r N )  o.  R ) ) )
 
Theoremrelexp1 24042 A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( R ^ r 1 )  =  R )
 
Theoremrelexpsucl 24043 A reduction for relation exponentiation to the left. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  ( R ^ r ( N  +  1 ) )  =  ( R  o.  ( R ^ r N ) ) ) )
 
Theoremrelexpcnv 24044 Distributivity of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  `' ( R ^ r N )  =  ( `' R ^ r N ) ) )
 
TheoremrelexpexOLD 24045 Obsolete; use ovex 5899 instead - NM 5-Apr-2016. The exponentiation of a relation exists. (Contributed by Drahflow, 12-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( R ^ r N )  e.  _V
 
Theoremrelexprel 24046 The exponentiation of a relation is a relation. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  Rel  ( R ^ r N ) ) )
 
Theoremrelexpdm 24047 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  dom  ( R ^ r N ) 
 C_  U. U. R ) )
 
Theoremrelexprn 24048 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  ran  ( R ^ r N ) 
 C_  U. U. R ) )
 
Theoremrelexpfld 24049 The field of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  U. U. ( R ^ r N )  C_  U. U. R ) )
 
Theoremrelexpadd 24050 Relation composition becomes addition under exponentiation. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (
 ( N  e.  NN0  /\  M  e.  NN0 )  ->  ( ( R ^
 r N )  o.  ( R ^ r M ) )  =  ( R ^ r
 ( N  +  M ) ) ) )
 
Theoremrelexpindlem 24051* Principle of transitive induction, finite and non-class version. The first three hypotheses give various existences, the next three give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( et  ->  Rel  R )   &    |-  ( et  ->  R  e.  _V )   &    |-  ( et  ->  S  e.  _V )   &    |-  ( i  =  S  ->  ( ph  <->  ch ) )   &    |-  ( i  =  x  ->  ( ph  <->  ps ) )   &    |-  ( i  =  j  ->  ( ph  <->  th ) )   &    |-  ( et  ->  ch )   &    |-  ( et  ->  ( j R x  ->  ( th  ->  ps )
 ) )   =>    |-  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^
 r n ) x 
 ->  ps ) ) )
 
Theoremrelexpind 24052* Principle of transitive induction, finite version. The first three hypotheses give various existences, the next three give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( et  ->  Rel  R )   &    |-  ( et  ->  R  e.  _V )   &    |-  ( et  ->  S  e.  _V )   &    |-  ( et  ->  X  e.  _V )   &    |-  (
 i  =  S  ->  (
 ph 
 <->  ch ) )   &    |-  (
 i  =  x  ->  ( ph  <->  ps ) )   &    |-  (
 i  =  j  ->  ( ph  <->  th ) )   &    |-  ( x  =  X  ->  ( ps  <->  ta ) )   &    |-  ( et  ->  ch )   &    |-  ( et  ->  ( j R x  ->  ( th  ->  ps )
 ) )   =>    |-  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^
 r n ) X 
 ->  ta ) ) )
 
Syntaxcrtrcl 24053 Extend class notation with recursively defined reflexive, transitive closure.
 class  t *rec
 
Definitiondf-rtrclrec 24054* The reflexive, transitive closure of a relation constructed as the union of all finite exponentiations. (Contributed by Drahflow, 12-Nov-2015.)
 |-  t *rec  =  ( r  e. 
 _V  |->  U_ n  e.  NN0  ( r ^ r n ) )
 
Theoremdfrtrclrec2 24055* If two elements are connected by a reflexive, transitive closure, then they are connected via  n instances the relation, for some  n. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( A ( t *rec `  R ) B  <->  E. n  e.  NN0  A ( R ^ r n ) B ) )
 
Theoremrtrclreclem.refl 24056 The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (  _I  |`  U. U. R )  C_  ( t *rec `  R ) )
 
Theoremrtrclreclem.subset 24057 The reflexive, transitive closure is indeed a closure. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  R  C_  ( t *rec `  R ) )
 
Theoremrtrclreclem.trans 24058 The reflexive, transitive closure is indeed transitive. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (
 ( t *rec `  R )  o.  (
 t *rec `  R ) )  C_  ( t *rec `  R )
 )
 
Theoremrtrclreclem.min 24059* The reflexive, transitive closure of  R is the smallest reflexive, transitive relation which contains  R and the identity. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  A. s
 ( ( (  _I  |`  ( dom  R  u.  ran 
 R ) )  C_  s  /\  R  C_  s  /\  ( s  o.  s
 )  C_  s )  ->  ( t *rec `  R )  C_  s ) )
 
Theoremdfrtrcl2 24060 The two definitions  t * and  t
*rec of the reflexive, transitive closure coincide if  R is indeed a relation. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (
 t * `  R )  =  ( t *rec `  R ) )
 
Theoremrtrclind 24061* Principle of transitive induction. The first four hypotheses give various existences, the next four give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( et  ->  Rel  R )   &    |-  ( et  ->  R  e.  _V )   &    |-  ( et  ->  S  e.  _V )   &    |-  ( et  ->  X  e.  _V )   &    |-  (
 i  =  S  ->  (
 ph 
 <->  ch ) )   &    |-  (
 i  =  x  ->  ( ph  <->  ps ) )   &    |-  (
 i  =  j  ->  ( ph  <->  th ) )   &    |-  ( x  =  X  ->  ( ps  <->  ta ) )   &    |-  ( et  ->  ch )   &    |-  ( et  ->  ( j R x  ->  ( th  ->  ps )
 ) )   =>    |-  ( et  ->  ( S ( t * `
  R ) X 
 ->  ta ) )
 
18.7  Mathbox for Scott Fenton
 
18.7.1  ZFC Axioms in primitive form
 
Theoremaxextprim 24062 ax-ext 2277 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
 |-  -.  A. x  -.  ( ( x  e.  y  ->  x  e.  z )  ->  ( ( x  e.  z  ->  x  e.  y )  ->  y  =  z ) )
 
Theoremaxrepprim 24063 ax-rep 4147 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
 |-  -.  A. x  -.  ( -. 
 A. y  -.  A. z ( ph  ->  z  =  y )  ->  A. z  -.  (
 ( A. y  z  e.  x  ->  -.  A. x ( A. z  x  e.  y  ->  -.  A. y ph ) )  ->  -.  ( -.  A. x ( A. z  x  e.  y  ->  -.  A. y ph )  ->  A. y  z  e.  x ) ) )
 
Theoremaxunprim 24064 ax-un 4528 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
 |-  -.  A. x  -.  A. y
 ( -.  A. x ( y  e.  x  ->  -.  x  e.  z
 )  ->  y  e.  x )
 
Theoremaxpowprim 24065 ax-pow 4204 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
 |-  ( A. x  -.  A. y
 ( A. x ( -. 
 A. z  -.  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x ) 
 ->  x  =  y
 )
 
Theoremaxregprim 24066 ax-reg 7322 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
 |-  ( x  e.  y  ->  -. 
 A. x ( x  e.  y  ->  -.  A. z ( z  e.  x  ->  -.  z  e.  y ) ) )
 
Theoremaxinfprim 24067 ax-inf 7355 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 13-Oct-2010.)
 |-  -.  A. x  -.  ( y  e.  z  ->  -.  (
 y  e.  x  ->  -.  A. y ( y  e.  x  ->  -.  A. z ( y  e.  z  ->  -.  z  e.  x ) ) ) )
 
Theoremaxacprim 24068 ax-ac 8101 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 26-Oct-2010.)
 |-  -.  A. x  -.  A. y A. z ( A. x  -.  ( y  e.  z  ->  -.  z  e.  w )  ->  -.  A. w  -.  A. y  -.  ( ( -.  A. w ( y  e.  z  ->  ( z  e.  w  ->  ( y  e.  w  ->  -.  w  e.  x ) ) )  ->  y  =  w )  ->  -.  ( y  =  w  ->  -.  A. w ( y  e.  z  ->  ( z  e.  w  ->  ( y  e.  w  ->  -.  w  e.  x ) ) ) ) ) )
 
18.7.2  Untangled classes
 
Theoremuntelirr 24069* We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 24219). Using this concept, we can avoid a lot of the uses of the Axiom of Regularity. Here, we prove a series of properties of untanged classes. First, we prove that an untangled class is not a member of itself. (Contributed by Scott Fenton, 28-Feb-2011.)
 |-  ( A. x  e.  A  -.  x  e.  x  ->  -.  A  e.  A )
 
Theoremuntuni 24070* The union of a class is untangled iff all its members are untangled. (Contributed by Scott Fenton, 28-Feb-2011.)
 |-  ( A. x  e.  U. A  -.  x  e.  x  <->  A. y  e.  A  A. x  e.  y  -.  x  e.  x )
 
Theoremuntsucf 24071* If a class is untangled, then so is its successor. (Contributed by Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ y A   =>    |-  ( A. x  e.  A  -.  x  e.  x  ->  A. y  e. 
 suc  A  -.  y  e.  y )
 
Theoremunt0 24072 The null set is untangled. (Contributed by Scott Fenton, 10-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  A. x  e.  (/)  -.  x  e.  x
 
Theoremuntint 24073* If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
 |-  ( E. x  e.  A  A. y  e.  x  -.  y  e.  y  ->  A. y  e.  |^| A  -.  y  e.  y
 )
 
Theoremefrunt 24074* If  A is well-founded by  _E, then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
 |-  (  _E  Fr  A  ->  A. x  e.  A  -.  x  e.  x )
 
Theoremuntangtr 24075* A transitive class is untangled iff its elements are. (Contributed by Scott Fenton, 7-Mar-2011.)
 |-  ( Tr  A  ->  ( A. x  e.  A  -.  x  e.  x  <->  A. x  e.  A  A. y  e.  x  -.  y  e.  y )
 )
 
18.7.3  Extra propositional calculus theorems
 
Theorem3orel1 24076 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.)
 |-  ( -.  ph  ->  ( ( ph  \/  ps  \/  ch )  ->  ( ps  \/  ch ) ) )
 
Theorem3orel2 24077 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( -.  ps  ->  ( ( ph  \/  ps  \/  ch )  ->  ( ph  \/  ch ) ) )
 
Theorem3orel3 24078 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.)
 |-  ( -.  ch  ->  ( ( ph  \/  ps  \/  ch )  ->  ( ph  \/  ps ) ) )
 
Theorem3pm3.2ni 24079 Triple negated disjuntion introduction. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  -.  ph   &    |-  -. 
 ps   &    |- 
 -.  ch   =>    |- 
 -.  ( ph  \/  ps 
 \/  ch )
 
Theorem3jaodd 24080 Double deduction form of 3jaoi 1245. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  et )
 ) )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  et ) ) )   &    |-  ( ph  ->  ( ps  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ( ch 
 \/  th  \/  ta )  ->  et ) ) )
 
Theorem3orit 24081 Closed form of 3ori 1242, (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  (
 ( ph  \/  ps  \/  ch )  <->  ( ( -.  ph  /\  -.  ps )  ->  ch ) )
 
Theorem3mix1d 24082 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  \/  ch  \/  th ) )
 
Theorem3mix2d 24083 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  ps  \/  th ) )
 
Theorem3mix3d 24084 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  th  \/  ps ) )
 
Theorembiimpexp 24085 A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.)
 |-  (
 ( ( ph  <->  ps )  ->  ch )  <->  ( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  ch )
 ) )
 
Theorem3orel13 24086 Elimination of two disjuncts in a triple disjunction. (Contributed by Scott Fenton, 9-Jun-2011.)
 |-  (
 ( -.  ph  /\  -.  ch )  ->  ( ( ph  \/  ps  \/  ch )  ->  ps ) )
 
18.7.4  Misc. Useful Theorems
 
Theoremnepss 24087 Two classes are inequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.)
 |-  ( A  =/=  B  <->  ( ( A  i^i  B )  C.  A  \/  ( A  i^i  B )  C.  B ) )
 
Theorem3ccased 24088 Triple disjunction form of ccased 913. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ph  ->  ( ( ch 
 /\  et )  ->  ps )
 )   &    |-  ( ph  ->  (
 ( ch  /\  ze )  ->  ps ) )   &    |-  ( ph  ->  ( ( ch 
 /\  si )  ->  ps )
 )   &    |-  ( ph  ->  (
 ( th  /\  et )  ->  ps ) )   &    |-  ( ph  ->  ( ( th  /\ 
 ze )  ->  ps )
 )   &    |-  ( ph  ->  (
 ( th  /\  si )  ->  ps ) )   &    |-  ( ph  ->  ( ( ta 
 /\  et )  ->  ps )
 )   &    |-  ( ph  ->  (
 ( ta  /\  ze )  ->  ps ) )   &    |-  ( ph  ->  ( ( ta 
 /\  si )  ->  ps )
 )   =>    |-  ( ph  ->  (
 ( ( ch  \/  th 
 \/  ta )  /\  ( et  \/  ze  \/  si ) )  ->  ps )
 )
 
Theoremdfso3 24089* Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.)
 |-  ( R  Or  A  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z )  /\  ( x R y  \/  x  =  y  \/  y R x ) ) )
 
Theorembrtpid1 24090 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { <. A ,  B >. ,  C ,  D } B
 
Theorembrtpid2 24091 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { C ,  <. A ,  B >. ,  D } B
 
Theorembrtpid3 24092 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { C ,  D ,  <. A ,  B >. } B
 
Theoremceqsrexv2 24093* Alternate elimitation of a restricted existential quantifier, using implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.)
 |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ( A  e.  B  /\  ps ) )
 
18.7.5  Properties of reals and complexes
 
Theoremsqdivzi 24094 Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B  =/=  0  ->  (
 ( A  /  B ) ^ 2 )  =  ( ( A ^
 2 )  /  ( B ^ 2 ) ) )
 
Theoremdivelunit 24095 A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  e.  ( 0 [,] 1 )  <->  A  <_  B ) )
 
Theorempm2.61iine 24096 Equality version of pm2.61ii 157. (Contributed by Scott Fenton, 13-Jun-2013.)
 |-  (
 ( A  =/=  C  /\  B  =/=  D ) 
 ->  ph )   &    |-  ( A  =  C  ->  ph )   &    |-  ( B  =  D  ->  ph )   =>    |-  ph
 
Theoremdedekind 24097* The Dedekind cut theorem. This theorem, which may be used to replace ax-pre-sup 8831 with appropriate adjustments, states that, if  A completely preceeds  B, then there is some number separating the two of them. (Contributed by Scott Fenton, 13-Jun-2013.)
 |-  (
 ( A  C_  RR  /\  B  C_  RR  /\  A. x  e.  A  A. y  e.  B  x  <  y
 )  ->  E. z  e.  RR  A. x  e.  A  A. y  e.  B  ( x  <_  z  /\  z  <_  y
 ) )
 
Theoremdedekindle 24098* The Dedekind cut theorem, with the hypothesis weakened to only require non-strict less than. (Contributed by Scott Fenton, 2-Jul-2013.)
 |-  (
 ( A  C_  RR  /\  B  C_  RR  /\  A. x  e.  A  A. y  e.  B  x  <_  y
 )  ->  E. z  e.  RR  A. x  e.  A  A. y  e.  B  ( x  <_  z  /\  z  <_  y
 ) )
 
Theoremmulcan1g 24099 A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  =  ( A  x.  C )  <->  ( A  =  0  \/  B  =  C ) ) )
 
Theoremmulcan2g 24100 A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  C )  =  ( B  x.  C )  <->  ( A  =  B  \/  C  =  0 ) ) )
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