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Theorem List for Metamath Proof Explorer - 6201-6300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcaovcomg 6201* Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   =>    |-  ( ( ph  /\  ( A  e.  S  /\  B  e.  S )
 )  ->  ( A F B )  =  ( B F A ) )
 
Theoremcaovcomd 6202* Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   =>    |-  ( ph  ->  ( A F B )  =  ( B F A ) )
 
Theoremcaovcom 6203* Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x F y )  =  ( y F x )   =>    |-  ( A F B )  =  ( B F A )
 
Theoremcaovassg 6204* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   =>    |-  ( ( ph  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )
 )  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
 
Theoremcaovassd 6205* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   =>    |-  ( ph  ->  (
 ( A F B ) F C )  =  ( A F ( B F C ) ) )
 
Theoremcaovass 6206* Convert an operation associative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  (
 ( x F y ) F z )  =  ( x F ( y F z ) )   =>    |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
 
Theoremcaovcang 6207* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x F y )  =  ( x F z )  <->  y  =  z
 ) )   =>    |-  ( ( ph  /\  ( A  e.  T  /\  B  e.  S  /\  C  e.  S )
 )  ->  ( ( A F B )  =  ( A F C ) 
 <->  B  =  C ) )
 
Theoremcaovcand 6208* Convert an operation cancellation law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x F y )  =  ( x F z )  <->  y  =  z
 ) )   &    |-  ( ph  ->  A  e.  T )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   =>    |-  ( ph  ->  (
 ( A F B )  =  ( A F C )  <->  B  =  C ) )
 
Theoremcaovcanrd 6209* Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x F y )  =  ( x F z )  <->  y  =  z
 ) )   &    |-  ( ph  ->  A  e.  T )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ph  ->  A  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x F y )  =  ( y F x ) )   =>    |-  ( ph  ->  ( ( B F A )  =  ( C F A )  <->  B  =  C ) )
 
Theoremcaovcan 6210* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.)
 |-  C  e.  _V   &    |-  (
 ( x  e.  S  /\  y  e.  S )  ->  ( ( x F y )  =  ( x F z )  ->  y  =  z ) )   =>    |-  ( ( A  e.  S  /\  B  e.  S )  ->  (
 ( A F B )  =  ( A F C )  ->  B  =  C ) )
 
Theoremcaovordig 6211* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x R y  ->  ( z F x ) R ( z F y ) ) )   =>    |-  ( ( ph  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )
 )  ->  ( A R B  ->  ( C F A ) R ( C F B ) ) )
 
Theoremcaovordid 6212* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x R y  ->  ( z F x ) R ( z F y ) ) )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   =>    |-  ( ph  ->  ( A R B  ->  ( C F A ) R ( C F B ) ) )
 
Theoremcaovordg 6213* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x R y  <->  ( z F x ) R ( z F y ) ) )   =>    |-  ( ( ph  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )
 )  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )
 
Theoremcaovordd 6214* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x R y  <->  ( z F x ) R ( z F y ) ) )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   =>    |-  ( ph  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )
 
Theoremcaovord2d 6215* Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x R y  <->  ( z F x ) R ( z F y ) ) )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x F y )  =  ( y F x ) )   =>    |-  ( ph  ->  ( A R B  <->  ( A F C ) R ( B F C ) ) )
 
Theoremcaovord3d 6216* Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x R y  <->  ( z F x ) R ( z F y ) ) )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ph  ->  D  e.  S )   =>    |-  ( ph  ->  ( ( A F B )  =  ( C F D )  ->  ( A R C  <->  D R B ) ) )
 
Theoremcaovord 6217* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( z  e.  S  ->  ( x R y  <->  ( z F x ) R ( z F y ) ) )   =>    |-  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )
 
Theoremcaovord2 6218* Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( z  e.  S  ->  ( x R y  <->  ( z F x ) R ( z F y ) ) )   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   =>    |-  ( C  e.  S  ->  ( A R B  <->  ( A F C ) R ( B F C ) ) )
 
Theoremcaovord3 6219* Ordering law. (Contributed by NM, 29-Feb-1996.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( z  e.  S  ->  ( x R y  <->  ( z F x ) R ( z F y ) ) )   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   &    |-  D  e.  _V   =>    |-  (
 ( ( B  e.  S  /\  C  e.  S )  /\  ( A F B )  =  ( C F D ) ) 
 ->  ( A R C  <->  D R B ) )
 
Theoremcaovdig 6220* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)
 |-  ( ( ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) ) )   =>    |-  ( ( ph  /\  ( A  e.  K  /\  B  e.  S  /\  C  e.  S )
 )  ->  ( A G ( B F C ) )  =  ( ( A G B ) H ( A G C ) ) )
 
Theoremcaovdid 6221* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) ) )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   =>    |-  ( ph  ->  ( A G ( B F C ) )  =  ( ( A G B ) H ( A G C ) ) )
 
Theoremcaovdir2d 6222* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) ) )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x G y )  =  ( y G x ) )   =>    |-  ( ph  ->  (
 ( A F B ) G C )  =  ( ( A G C ) F ( B G C ) ) )
 
Theoremcaovdirg 6223* Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  K )
 )  ->  ( ( x F y ) G z )  =  ( ( x G z ) H ( y G z ) ) )   =>    |-  ( ( ph  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  K )
 )  ->  ( ( A F B ) G C )  =  ( ( A G C ) H ( B G C ) ) )
 
Theoremcaovdird 6224* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  K )
 )  ->  ( ( x F y ) G z )  =  ( ( x G z ) H ( y G z ) ) )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  K )   =>    |-  ( ph  ->  (
 ( A F B ) G C )  =  ( ( A G C ) H ( B G C ) ) )
 
Theoremcaovdi 6225* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) )   =>    |-  ( A G ( B F C ) )  =  ( ( A G B ) F ( A G C ) )
 
Theoremcaov32d 6226* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   =>    |-  ( ph  ->  (
 ( A F B ) F C )  =  ( ( A F C ) F B ) )
 
Theoremcaov12d 6227* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   =>    |-  ( ph  ->  ( A F ( B F C ) )  =  ( B F ( A F C ) ) )
 
Theoremcaov31d 6228* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   =>    |-  ( ph  ->  (
 ( A F B ) F C )  =  ( ( C F B ) F A ) )
 
Theoremcaov13d 6229* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   =>    |-  ( ph  ->  ( A F ( B F C ) )  =  ( C F ( B F A ) ) )
 
Theoremcaov4d 6230* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   &    |-  ( ph  ->  D  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   =>    |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( B F D ) ) )
 
Theoremcaov411d 6231* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   &    |-  ( ph  ->  D  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   =>    |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( C F B ) F ( A F D ) ) )
 
Theoremcaov42d 6232* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   &    |-  ( ph  ->  D  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   =>    |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( D F B ) ) )
 
Theoremcaov32 6233* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   &    |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )   =>    |-  ( ( A F B ) F C )  =  ( ( A F C ) F B )
 
Theoremcaov12 6234* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   &    |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )   =>    |-  ( A F ( B F C ) )  =  ( B F ( A F C ) )
 
Theoremcaov31 6235* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   &    |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )   =>    |-  ( ( A F B ) F C )  =  ( ( C F B ) F A )
 
Theoremcaov13 6236* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   &    |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )   =>    |-  ( A F ( B F C ) )  =  ( C F ( B F A ) )
 
Theoremcaov4 6237* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   &    |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )   &    |-  D  e.  _V   =>    |-  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( B F D ) )
 
Theoremcaov411 6238* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   &    |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )   &    |-  D  e.  _V   =>    |-  ( ( A F B ) F ( C F D ) )  =  ( ( C F B ) F ( A F D ) )
 
Theoremcaov42 6239* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   &    |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )   &    |-  D  e.  _V   =>    |-  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( D F B ) )
 
Theoremcaovdir 6240* Reverse distributive law. (Contributed by NM, 26-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x G y )  =  ( y G x )   &    |-  ( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) )   =>    |-  ( ( A F B ) G C )  =  ( ( A G C ) F ( B G C ) )
 
Theoremcaovdilem 6241* Lemma used by real number construction. (Contributed by NM, 26-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x G y )  =  ( y G x )   &    |-  ( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) )   &    |-  D  e.  _V   &    |-  H  e.  _V   &    |-  (
 ( x G y ) G z )  =  ( x G ( y G z ) )   =>    |-  ( ( ( A G C ) F ( B G D ) ) G H )  =  ( ( A G ( C G H ) ) F ( B G ( D G H ) ) )
 
Theoremcaovlem2 6242* Lemma used in real number construction. (Contributed by NM, 26-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x G y )  =  ( y G x )   &    |-  ( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) )   &    |-  D  e.  _V   &    |-  H  e.  _V   &    |-  (
 ( x G y ) G z )  =  ( x G ( y G z ) )   &    |-  R  e.  _V   &    |-  ( x F y )  =  ( y F x )   &    |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )   =>    |-  ( ( ( ( A G C ) F ( B G D ) ) G H ) F ( ( ( A G D ) F ( B G C ) ) G R ) )  =  ( ( A G ( ( C G H ) F ( D G R ) ) ) F ( B G ( ( C G R ) F ( D G H ) ) ) )
 
Theoremcaovmo 6243* Uniqueness of inverse element in commutative, associative operation with identity. Remark in proof of Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 4-Mar-1996.)
 |-  B  e.  S   &    |-  dom  F  =  ( S  X.  S )   &    |-  -.  (/)  e.  S   &    |-  ( x F y )  =  ( y F x )   &    |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )   &    |-  ( x  e.  S  ->  ( x F B )  =  x )   =>    |-  E* w ( A F w )  =  B
 
Theoremgrprinvlem 6244* Lemma for grprinvd 6245. (Contributed by NM, 9-Aug-2013.)
 |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ph  ->  O  e.  B )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( O  .+  x )  =  x )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  B  ( y  .+  x )  =  O )   &    |-  (
 ( ph  /\  ps )  ->  X  e.  B )   &    |-  ( ( ph  /\  ps )  ->  ( X  .+  X )  =  X )   =>    |-  ( ( ph  /\  ps )  ->  X  =  O )
 
Theoremgrprinvd 6245* Deduce right inverse from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
 |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ph  ->  O  e.  B )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( O  .+  x )  =  x )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  B  ( y  .+  x )  =  O )   &    |-  (
 ( ph  /\  ps )  ->  X  e.  B )   &    |-  ( ( ph  /\  ps )  ->  N  e.  B )   &    |-  ( ( ph  /\  ps )  ->  ( N  .+  X )  =  O )   =>    |-  ( ( ph  /\  ps )  ->  ( X  .+  N )  =  O )
 
Theoremgrpridd 6246* Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
 |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ph  ->  O  e.  B )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( O  .+  x )  =  x )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  B  ( y  .+  x )  =  O )   =>    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( x  .+  O )  =  x )
 
2.4.11  "Maps to" notation
 
Theoremelmpt2cl 6247* If a two-parameter class is not empty, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( X  e.  ( S F T ) 
 ->  ( S  e.  A  /\  T  e.  B ) )
 
Theoremelmpt2cl1 6248* If a two-parameter class is not empty, the first argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( X  e.  ( S F T ) 
 ->  S  e.  A )
 
Theoremelmpt2cl2 6249* If a two-parameter class is not empty, the second argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( X  e.  ( S F T ) 
 ->  T  e.  B )
 
Theoremelovmpt2 6250* Utility lemma for two-parameter classes.

EDITORIAL: can simplify isghm 14961, islmhm 16058. (Contributed by Stefan O'Rear, 21-Jan-2015.)

 |-  D  =  ( a  e.  A ,  b  e.  B  |->  C )   &    |-  C  e.  _V   &    |-  ( ( a  =  X  /\  b  =  Y )  ->  C  =  E )   =>    |-  ( F  e.  ( X D Y )  <->  ( X  e.  A  /\  Y  e.  B  /\  F  e.  E ) )
 
Theoremrelmptopab 6251* Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 7-Aug-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
 |-  F  =  ( x  e.  A  |->  { <. y ,  z >.  |  ph } )   =>    |- 
 Rel  ( F `  B )
 
Theoremf1ocnvd 6252* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  F  =  ( x  e.  A  |->  C )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  W )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  D  e.  X )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D )
 ) )   =>    |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  ( y  e.  B  |->  D ) ) )
 
Theoremf1od 6253* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
 |-  F  =  ( x  e.  A  |->  C )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  W )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  D  e.  X )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D )
 ) )   =>    |-  ( ph  ->  F : A -1-1-onto-> B )
 
Theoremf1ocnv2d 6254* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  F  =  ( x  e.  A  |->  C )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  B )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  D  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  ( x  =  D  <->  y  =  C ) )   =>    |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  ( y  e.  B  |->  D ) ) )
 
Theoremf1o2d 6255* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
 |-  F  =  ( x  e.  A  |->  C )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  B )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  D  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  ( x  =  D  <->  y  =  C ) )   =>    |-  ( ph  ->  F : A -1-1-onto-> B )
 
TheoremxpexgALT 6256 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4948 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B )  e.  _V )
 
Theoremf1opw2 6257* A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6258 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.)
 |-  ( ph  ->  F : A -1-1-onto-> B )   &    |-  ( ph  ->  ( `' F " a )  e.  _V )   &    |-  ( ph  ->  ( F "
 b )  e.  _V )   =>    |-  ( ph  ->  (
 b  e.  ~P A  |->  ( F " b ) ) : ~P A -1-1-onto-> ~P B )
 
Theoremf1opw 6258* A one-to-one mapping induces a one-to-one mapping on power sets. (Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  ( F : A -1-1-onto-> B  ->  ( b  e.  ~P A  |->  ( F "
 b ) ) : ~P A -1-1-onto-> ~P B )
 
Theoremsuppss2 6259* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 22-Mar-2015.)
 |-  ( ( ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )   =>    |-  ( ph  ->  ( `' ( k  e.  A  |->  B ) " ( _V  \  { Z }
 ) )  C_  W )
 
Theoremsuppssfv 6260* Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  ( ph  ->  ( `' ( x  e.  D  |->  A ) " ( _V  \  { Y }
 ) )  C_  L )   &    |-  ( ph  ->  ( F `  Y )  =  Z )   &    |-  ( ( ph  /\  x  e.  D ) 
 ->  A  e.  V )   =>    |-  ( ph  ->  ( `' ( x  e.  D  |->  ( F `  A ) ) " ( _V  \  { Z } )
 )  C_  L )
 
Theoremsuppssov1 6261* Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  ( ph  ->  ( `' ( x  e.  D  |->  A ) " ( _V  \  { Y }
 ) )  C_  L )   &    |-  ( ( ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )   &    |-  ( ( ph  /\  x  e.  D ) 
 ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  D )  ->  B  e.  R )   =>    |-  ( ph  ->  ( `' ( x  e.  D  |->  ( A O B ) ) " ( _V  \  { Z } )
 )  C_  L )
 
2.4.12  Function operation
 
Syntaxcof 6262 Extend class notation to include mapping of an operation to a function operation.
 class  o F R
 
Syntaxcofr 6263 Extend class notation to include mapping of a binary relation to a function relation.
 class  o R R
 
Definitiondf-of 6264* Define the function operation map. The definition is designed so that if  R is a binary operation, then  o F R is the analogous operation on functions which corresponds to applying  R pointwise to the values of the functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  o F R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g
 )  |->  ( ( f `
  x ) R ( g `  x ) ) ) )
 
Definitiondf-ofr 6265* Define the function relation map. The definition is designed so that if  R is a binary relation, then  o F R is the analogous relation on functions which is true when each element of the left function relates to the corresponding element of the right function. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  o R R  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i 
 dom  g ) ( f `  x ) R ( g `  x ) }
 
Theoremofeq 6266 Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( R  =  S  ->  o F R  =  o F S )
 
Theoremofreq 6267 Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( R  =  S  ->  o R R  =  o R S )
 
Theoremofexg 6268 A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
 |-  ( A  e.  V  ->  (  o F R  |`  A )  e.  _V )
 
Theoremnfof 6269* Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  F/_ x R   =>    |-  F/_ x  o F R
 
Theoremnfofr 6270* Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  F/_ x R   =>    |-  F/_ x  o R R
 
Theoremoffval 6271* Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  S   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  x )  =  C )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( G `  x )  =  D )   =>    |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  S  |->  ( C R D ) ) )
 
Theoremofrfval 6272* Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  S   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  x )  =  C )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( G `  x )  =  D )   =>    |-  ( ph  ->  ( F  o R R G  <->  A. x  e.  S  C R D ) )
 
Theoremofval 6273 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  S   &    |-  (
 ( ph  /\  X  e.  A )  ->  ( F `
  X )  =  C )   &    |-  ( ( ph  /\  X  e.  B ) 
 ->  ( G `  X )  =  D )   =>    |-  (
 ( ph  /\  X  e.  S )  ->  ( ( F  o F R G ) `  X )  =  ( C R D ) )
 
Theoremofrval 6274 Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  S   &    |-  (
 ( ph  /\  X  e.  A )  ->  ( F `
  X )  =  C )   &    |-  ( ( ph  /\  X  e.  B ) 
 ->  ( G `  X )  =  D )   =>    |-  (
 ( ph  /\  F  o R R G  /\  X  e.  S )  ->  C R D )
 
Theoremoffn 6275 The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  S   =>    |-  ( ph  ->  ( F  o F R G )  Fn  S )
 
Theoremfnfvof 6276 Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Jun-2015.)
 |-  ( ( ( F  Fn  A  /\  G  Fn  A )  /\  ( A  e.  V  /\  X  e.  A )
 )  ->  ( ( F  o F R G ) `  X )  =  ( ( F `  X ) R ( G `  X ) ) )
 
Theoremoffval3 6277* General value of  ( F  o F R G ) with no assumptions on functionality of  F and  G. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( F  o F R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
  x ) R ( G `  x ) ) ) )
 
Theoremoffres 6278 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F  o F R G )  |`  D )  =  ( ( F  |`  D )  o F R ( G  |`  D )
 ) )
 
Theoremoff 6279* The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  T )
 )  ->  ( x R y )  e.  U )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : B
 --> T )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  C   =>    |-  ( ph  ->  ( F  o F R G ) : C --> U )
 
Theoremofres 6280 Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  C   =>    |-  ( ph  ->  ( F  o F R G )  =  (
 ( F  |`  C )  o F R ( G  |`  C )
 ) )
 
Theoremoffval2 6281* The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  X )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )   =>    |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  A  |->  ( B R C ) ) )
 
Theoremofrfval2 6282* The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  X )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )   =>    |-  ( ph  ->  ( F  o R R G 
 <-> 
 A. x  e.  A  B R C ) )
 
Theoremofco 6283 The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  H : D --> C )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( A  i^i  B )  =  C   =>    |-  ( ph  ->  ( ( F  o F R G )  o.  H )  =  ( ( F  o.  H )  o F R ( G  o.  H ) ) )
 
Theoremoffveq 6284* Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  A )   &    |-  ( ph  ->  H  Fn  A )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( F `  x )  =  B )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( G `  x )  =  C )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( B R C )  =  ( H `  x ) )   =>    |-  ( ph  ->  ( F  o F R G )  =  H )
 
Theoremoffveqb 6285* Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  A )   &    |-  ( ph  ->  H  Fn  A )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( F `  x )  =  B )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( G `  x )  =  C )   =>    |-  ( ph  ->  ( H  =  ( F  o F R G )  <->  A. x  e.  A  ( H `  x )  =  ( B R C ) ) )
 
Theoremofc1 6286 Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  F  Fn  A )   &    |-  ( ( ph  /\  X  e.  A )  ->  ( F `  X )  =  C )   =>    |-  ( ( ph  /\  X  e.  A )  ->  (
 ( ( A  X.  { B } )  o F R F ) `
  X )  =  ( B R C ) )
 
Theoremofc2 6287 Right operation by a constant. (Contributed by NM, 7-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  F  Fn  A )   &    |-  ( ( ph  /\  X  e.  A )  ->  ( F `  X )  =  C )   =>    |-  ( ( ph  /\  X  e.  A )  ->  (
 ( F  o F R ( A  X.  { B } ) ) `
  X )  =  ( C R B ) )
 
Theoremofc12 6288 Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   =>    |-  ( ph  ->  (
 ( A  X.  { B } )  o F R ( A  X.  { C } ) )  =  ( A  X.  { ( B R C ) } ) )
 
Theoremcaofref 6289* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ( ph  /\  x  e.  S )  ->  x R x )   =>    |-  ( ph  ->  F  o R R F )
 
Theoremcaofinvl 6290* Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  N : S --> S )   &    |-  ( ph  ->  G  =  ( v  e.  A  |->  ( N `  ( F `
  v ) ) ) )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( ( N `  x ) R x )  =  B )   =>    |-  ( ph  ->  ( G  o F R F )  =  ( A  X.  { B } ) )
 
Theoremcaofid0l 6291* Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( B R x )  =  x )   =>    |-  ( ph  ->  ( ( A  X.  { B }
 )  o F R F )  =  F )
 
Theoremcaofid0r 6292* Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( x R B )  =  x )   =>    |-  ( ph  ->  ( F  o F R ( A  X.  { B } ) )  =  F )
 
Theoremcaofid1 6293* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   &    |-  (
 ( ph  /\  x  e.  S )  ->  ( x R B )  =  C )   =>    |-  ( ph  ->  ( F  o F R ( A  X.  { B } ) )  =  ( A  X.  { C } ) )
 
Theoremcaofid2 6294* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   &    |-  (
 ( ph  /\  x  e.  S )  ->  ( B R x )  =  C )   =>    |-  ( ph  ->  (
 ( A  X.  { B } )  o F R F )  =  ( A  X.  { C } ) )
 
Theoremcaofcom 6295* Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x R y )  =  ( y R x ) )   =>    |-  ( ph  ->  ( F  o F R G )  =  ( G  o F R F ) )
 
Theoremcaofrss 6296* Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x R y  ->  x T y ) )   =>    |-  ( ph  ->  ( F  o R R G  ->  F  o R T G ) )
 
Theoremcaofass 6297* Transfer an associative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x R y ) T z )  =  ( x O ( y P z ) ) )   =>    |-  ( ph  ->  (
 ( F  o F R G )  o F T H )  =  ( F  o F O ( G  o F P H ) ) )
 
Theoremcaoftrn 6298* Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x R y  /\  y T z )  ->  x U z ) )   =>    |-  ( ph  ->  ( ( F  o R R G  /\  G  o R T H )  ->  F  o R U H ) )
 
Theoremcaofdi 6299* Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> K )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x T ( y R z ) )  =  ( ( x T y ) O ( x T z ) ) )   =>    |-  ( ph  ->  ( F  o F T ( G  o F R H ) )  =  ( ( F  o F T G )  o F O ( F  o F T H ) ) )
 
Theoremcaofdir 6300* Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> K )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  K )
 )  ->  ( ( x R y ) T z )  =  ( ( x T z ) O ( y T z ) ) )   =>    |-  ( ph  ->  (
 ( G  o F R H )  o F T F )  =  ( ( G  o F T F )  o F O ( H  o F T F ) ) )
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