HomeHome Metamath Proof Explorer
Theorem List (p. 59 of 309)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-30843)
 

Theorem List for Metamath Proof Explorer - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfovcl 5801 Closure law for an operation. (Contributed by NM, 19-Apr-2007.)
 |-  F : ( R  X.  S ) --> C   =>    |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A F B )  e.  C )
 
Theoremeqfnov 5802* Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)
 |-  ( ( F  Fn  ( A  X.  B ) 
 /\  G  Fn  ( C  X.  D ) ) 
 ->  ( F  =  G  <->  ( ( A  X.  B )  =  ( C  X.  D )  /\  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) ) ) )
 
Theoremeqfnov2 5803* Two operators with the same domain are equal iff their values at each point in the domain are equal. (Contributed by Jeff Madsen, 7-Jun-2010.)
 |-  ( ( F  Fn  ( A  X.  B ) 
 /\  G  Fn  ( A  X.  B ) ) 
 ->  ( F  =  G  <->  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) ) )
 
Theoremfnov 5804* Representation of a function in terms of its values. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( F  Fn  ( A  X.  B )  <->  F  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) ) )
 
Theoremmpt22eqb 5805* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5803. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  ( ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )  <->  A. x  e.  A  A. y  e.  B  C  =  D ) )
 
Theoremrnmpt2 5806* The range of an operation given by the "maps to" notation. (Contributed by FL, 20-Jun-2011.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
 
Theoremreldmmpt2 5807* The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  Rel  dom  F
 
Theoremelrnmpt2g 5808* Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( D  e.  V  ->  ( D  e.  ran 
 F 
 <-> 
 E. x  e.  A  E. y  e.  B  D  =  C )
 )
 
Theoremelrnmpt2 5809* Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   &    |-  C  e.  _V   =>    |-  ( D  e.  ran  F  <->  E. x  e.  A  E. y  e.  B  D  =  C )
 
Theoremralrnmpt2 5810* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   &    |-  (
 z  =  C  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  ( A. z  e. 
 ran  F ph  <->  A. x  e.  A  A. y  e.  B  ps ) )
 
Theoremrexrnmpt2 5811* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   &    |-  (
 z  =  C  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  ( E. z  e. 
 ran  F ph  <->  E. x  e.  A  E. y  e.  B  ps ) )
 
Theoremoprabexd 5812* Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  E* z ps )   &    |-  ( ph  ->  F  =  { <.
 <. x ,  y >. ,  z >.  |  (
 ( x  e.  A  /\  y  e.  B )  /\  ps ) }
 )   =>    |-  ( ph  ->  F  e.  _V )
 
Theoremoprabex 5813* Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  e.  A  /\  y  e.  B )  ->  E* z ph )   &    |-  F  =  { <.
 <. x ,  y >. ,  z >.  |  (
 ( x  e.  A  /\  y  e.  B )  /\  ph ) }   =>    |-  F  e.  _V
 
Theoremoprabex3 5814* Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)
 |-  H  e.  _V   &    |-  F  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  ( H  X.  H )  /\  y  e.  ( H  X.  H ) )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. ) 
 /\  z  =  R ) ) }   =>    |-  F  e.  _V
 
Theoremoprabrexex2 5815* Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
 |-  A  e.  _V   &    |-  { <. <. x ,  y >. ,  z >.  |  ph }  e.  _V   =>    |-  {
 <. <. x ,  y >. ,  z >.  |  E. w  e.  A  ph }  e.  _V
 
Theoremovid 5816* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( ( x  e.  R  /\  y  e.  S )  ->  E! z ph )   &    |-  F  =  { <.
 <. x ,  y >. ,  z >.  |  (
 ( x  e.  R  /\  y  e.  S )  /\  ph ) }   =>    |-  ( ( x  e.  R  /\  y  e.  S )  ->  (
 ( x F y )  =  z  <->  ph ) )
 
Theoremovidig 5817* The value of an operation class abstraction. Compare ovidi 5818. The condition  ( x  e.  R  /\  y  e.  S ) is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |- 
 E* z ph   &    |-  F  =  { <.
 <. x ,  y >. ,  z >.  |  ph }   =>    |-  ( ph  ->  ( x F y )  =  z )
 
Theoremovidi 5818* The value of an operation class abstraction (weak version). (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  ( ( x  e.  R  /\  y  e.  S )  ->  E* z ph )   &    |-  F  =  { <.
 <. x ,  y >. ,  z >.  |  (
 ( x  e.  R  /\  y  e.  S )  /\  ph ) }   =>    |-  ( ( x  e.  R  /\  y  e.  S )  ->  ( ph  ->  ( x F y )  =  z ) )
 
Theoremov 5819* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  C  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 z  =  C  ->  ( ch  <->  th ) )   &    |-  (
 ( x  e.  R  /\  y  e.  S )  ->  E! z ph )   &    |-  F  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  R  /\  y  e.  S )  /\  ph ) }   =>    |-  ( ( A  e.  R  /\  B  e.  S )  ->  (
 ( A F B )  =  C  <->  th ) )
 
Theoremovigg 5820* The value of an operation class abstraction. Compare ovig 5821. The condition  ( x  e.  R  /\  y  e.  S ) is been removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph 
 <->  ps ) )   &    |-  E* z ph   &    |-  F  =  { <.
 <. x ,  y >. ,  z >.  |  ph }   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ps 
 ->  ( A F B )  =  C )
 )
 
Theoremovig 5821* The value of an operation class abstraction (weak version). (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 14-Sep-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph 
 <->  ps ) )   &    |-  (
 ( x  e.  R  /\  y  e.  S )  ->  E* z ph )   &    |-  F  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }   =>    |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  D )  ->  ( ps 
 ->  ( A F B )  =  C )
 )
 
Theoremovmpt4g 5822* Value of a function given by the "maps to" notation. (This is the operation analog of fvmpt2 5460.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( ( x  e.  A  /\  y  e.  B  /\  C  e.  V )  ->  ( x F y )  =  C )
 
Theoremovmpt2s 5823* Value of a function given by the "maps to" notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )   =>    |-  ( ( A  e.  C  /\  B  e.  D  /\  [_ A  /  x ]_ [_ B  /  y ]_ R  e.  V )  ->  ( A F B )  = 
 [_ A  /  x ]_
 [_ B  /  y ]_ R )
 
Theoremov2gf 5824* The value of an operation class abstraction. A version of ovmpt2g 5834 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/_ y B   &    |-  F/_ x G   &    |-  F/_ y S   &    |-  ( x  =  A  ->  R  =  G )   &    |-  (
 y  =  B  ->  G  =  S )   &    |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )   =>    |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  H ) 
 ->  ( A F B )  =  S )
 
Theoremovmpt2dxf 5825* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )   &    |-  (
 ( ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )   &    |-  (
 ( ph  /\  x  =  A )  ->  D  =  L )   &    |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  B  e.  L )   &    |-  ( ph  ->  S  e.  X )   &    |-  F/ x ph   &    |-  F/ y ph   &    |-  F/_ y A   &    |-  F/_ x B   &    |-  F/_ x S   &    |-  F/_ y S   =>    |-  ( ph  ->  ( A F B )  =  S )
 
Theoremovmpt2dx 5826* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )   &    |-  (
 ( ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )   &    |-  (
 ( ph  /\  x  =  A )  ->  D  =  L )   &    |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  B  e.  L )   &    |-  ( ph  ->  S  e.  X )   =>    |-  ( ph  ->  ( A F B )  =  S )
 
Theoremovmpt2d 5827* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )   &    |-  (
 ( ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )   &    |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  S  e.  X )   =>    |-  ( ph  ->  ( A F B )  =  S )
 
Theoremovmpt2x 5828* The value of an operation class abstraction. Variant of ovmpt2ga 5829 which does not require  D and  x to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )   &    |-  ( x  =  A  ->  D  =  L )   &    |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )   =>    |-  ( ( A  e.  C  /\  B  e.  L  /\  S  e.  H ) 
 ->  ( A F B )  =  S )
 
Theoremovmpt2ga 5829* Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )   &    |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )   =>    |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  H ) 
 ->  ( A F B )  =  S )
 
Theoremovmpt2a 5830* Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )   &    |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )   &    |-  S  e.  _V   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A F B )  =  S )
 
Theoremovmpt2df 5831* Alternate deduction version of ovmpt2 5835, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  A  e.  C )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  B  e.  D )   &    |-  ( ( ph  /\  ( x  =  A  /\  y  =  B )
 )  ->  R  e.  V )   &    |-  ( ( ph  /\  ( x  =  A  /\  y  =  B ) )  ->  ( ( A F B )  =  R  ->  ps )
 )   &    |-  F/_ x F   &    |-  F/ x ps   &    |-  F/_ y F   &    |- 
 F/ y ps   =>    |-  ( ph  ->  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  ->  ps ) )
 
Theoremovmpt2dv 5832* Alternate deduction version of ovmpt2 5835, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  A  e.  C )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  B  e.  D )   &    |-  ( ( ph  /\  ( x  =  A  /\  y  =  B )
 )  ->  R  e.  V )   &    |-  ( ( ph  /\  ( x  =  A  /\  y  =  B ) )  ->  ( ( A F B )  =  R  ->  ps )
 )   =>    |-  ( ph  ->  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  ->  ps ) )
 
Theoremovmpt2dv2 5833* Alternate deduction version of ovmpt2 5835, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  A  e.  C )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  B  e.  D )   &    |-  ( ( ph  /\  ( x  =  A  /\  y  =  B )
 )  ->  R  e.  V )   &    |-  ( ( ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )   =>    |-  ( ph  ->  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  ->  ( A F B )  =  S ) )
 
Theoremovmpt2g 5834* Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( x  =  A  ->  R  =  G )   &    |-  ( y  =  B  ->  G  =  S )   &    |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )   =>    |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  H )  ->  ( A F B )  =  S )
 
Theoremovmpt2 5835* Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( x  =  A  ->  R  =  G )   &    |-  ( y  =  B  ->  G  =  S )   &    |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )   &    |-  S  e.  _V   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A F B )  =  S )
 
Theoremov3 5836* The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)
 |-  S  e.  _V   &    |-  (
 ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D )
 )  ->  R  =  S )   &    |-  F  =  { <.
 <. x ,  y >. ,  z >.  |  (
 ( x  e.  ( H  X.  H )  /\  y  e.  ( H  X.  H ) )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. ) 
 /\  z  =  R ) ) }   =>    |-  ( ( ( A  e.  H  /\  B  e.  H )  /\  ( C  e.  H  /\  D  e.  H ) )  ->  ( <. A ,  B >. F <. C ,  D >. )  =  S )
 
Theoremov6g 5837* The value of an operation class abstraction. Special case. (Contributed by NM, 13-Nov-2006.)
 |-  ( <. x ,  y >.  =  <. A ,  B >.  ->  R  =  S )   &    |-  F  =  { <. <. x ,  y >. ,  z >.  |  ( <. x ,  y >.  e.  C  /\  z  =  R ) }   =>    |-  ( ( ( A  e.  G  /\  B  e.  H  /\  <. A ,  B >.  e.  C )  /\  S  e.  J )  ->  ( A F B )  =  S )
 
Theoremovg 5838* The value of an operation class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 z  =  C  ->  ( ch  <->  th ) )   &    |-  (
 ( ta  /\  ( x  e.  R  /\  y  e.  S )
 )  ->  E! z ph )   &    |-  F  =  { <.
 <. x ,  y >. ,  z >.  |  (
 ( x  e.  R  /\  y  e.  S )  /\  ph ) }   =>    |-  ( ( ta 
 /\  ( A  e.  R  /\  B  e.  S  /\  C  e.  D ) )  ->  ( ( A F B )  =  C  <->  th ) )
 
Theoremovres 5839 The value of a restricted operation. (Contributed by FL, 10-Nov-2006.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A ( F  |`  ( C  X.  D ) ) B )  =  ( A F B ) )
 
Theoremovresd 5840 Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  X )   =>    |-  ( ph  ->  ( A ( D  |`  ( X  X.  X ) ) B )  =  ( A D B ) )
 
Theoremoprssov 5841 The value of a member of the domain of a subclass of an operation. (Contributed by NM, 23-Aug-2007.)
 |-  ( ( ( Fun 
 F  /\  G  Fn  ( C  X.  D ) 
 /\  G  C_  F )  /\  ( A  e.  C  /\  B  e.  D ) )  ->  ( A F B )  =  ( A G B ) )
 
Theoremfovrn 5842 An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.)
 |-  ( ( F :
 ( R  X.  S )
 --> C  /\  A  e.  R  /\  B  e.  S )  ->  ( A F B )  e.  C )
 
Theoremfovrnda 5843 An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  ( ph  ->  F : ( R  X.  S ) --> C )   =>    |-  ( ( ph  /\  ( A  e.  R  /\  B  e.  S )
 )  ->  ( A F B )  e.  C )
 
Theoremfovrnd 5844 An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  ( ph  ->  F : ( R  X.  S ) --> C )   &    |-  ( ph  ->  A  e.  R )   &    |-  ( ph  ->  B  e.  S )   =>    |-  ( ph  ->  ( A F B )  e.  C )
 
Theoremfnrnov 5845* The range of an operation expressed as a collection of the operation's values. (Contributed by NM, 29-Oct-2006.)
 |-  ( F  Fn  ( A  X.  B )  ->  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) }
 )
 
Theoremfoov 5846* An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.)
 |-  ( F : ( A  X.  B )
 -onto-> C  <->  ( F :
 ( A  X.  B )
 --> C  /\  A. z  e.  C  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) ) )
 
Theoremfnovrn 5847 An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.)
 |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  A  /\  D  e.  B ) 
 ->  ( C F D )  e.  ran  F )
 
Theoremovelrn 5848* A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
 |-  ( F  Fn  ( A  X.  B )  ->  ( C  e.  ran  F  <->  E. x  e.  A  E. y  e.  B  C  =  ( x F y ) ) )
 
Theoremfunimassov 5849* Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013.)
 |-  ( ( Fun  F  /\  ( A  X.  B )  C_  dom  F )  ->  ( ( F "
 ( A  X.  B ) )  C_  C  <->  A. x  e.  A  A. y  e.  B  ( x F y )  e.  C ) )
 
Theoremovelimab 5850* Operation value in an image. (Contributed by Mario Carneiro, 23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( F  Fn  A  /\  ( B  X.  C )  C_  A ) 
 ->  ( D  e.  ( F " ( B  X.  C ) )  <->  E. x  e.  B  E. y  e.  C  D  =  ( x F y ) ) )
 
Theoremovconst2 5851 The value of a constant operation. (Contributed by NM, 5-Nov-2006.)
 |-  C  e.  _V   =>    |-  ( ( R  e.  A  /\  S  e.  B )  ->  ( R ( ( A  X.  B )  X.  { C } ) S )  =  C )
 
Theoremab2rexex 5852* Existence of a class abstraction of existentially restricted sets. Variables  x and  y are normally free-variable parameters in the class expression substituted for  C, which can be thought of as  C ( x ,  y ). See comments for abrexex 5615. (Contributed by NM, 20-Sep-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  e.  _V
 
Theoremab2rexex2 5853* Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x,  y, and  z. Compare abrexex2 5632. (Contributed by NM, 20-Sep-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  { z  | 
 ph }  e.  _V   =>    |-  { z  |  E. x  e.  A  E. y  e.  B  ph
 }  e.  _V
 
Theoremoprssdm 5854* Domain of closure of an operation. (Contributed by NM, 24-Aug-1995.)
 |- 
 -.  (/)  e.  S   &    |-  (
 ( x  e.  S  /\  y  e.  S )  ->  ( x F y )  e.  S )   =>    |-  ( S  X.  S )  C_  dom  F
 
Theoremndmovg 5855 The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.)
 |-  ( ( dom  F  =  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S ) )  ->  ( A F B )  =  (/) )
 
Theoremndmov 5856 The value of an operation outside its domain. (Contributed by NM, 24-Aug-1995.)
 |- 
 dom  F  =  ( S  X.  S )   =>    |-  ( -.  ( A  e.  S  /\  B  e.  S )  ->  ( A F B )  =  (/) )
 
Theoremndmovcl 5857 The closure of an operation outside its domain, when the domain includes the empty set. This technical lemma can make the operation more convenient to work in some cases. It is is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by NM, 24-Sep-2004.)
 |- 
 dom  F  =  ( S  X.  S )   &    |-  (
 ( A  e.  S  /\  B  e.  S ) 
 ->  ( A F B )  e.  S )   &    |-  (/)  e.  S   =>    |-  ( A F B )  e.  S
 
Theoremndmovrcl 5858 Reverse closure law, when an operation's domain doesn't contain the empty set. (Contributed by NM, 3-Feb-1996.)
 |- 
 dom  F  =  ( S  X.  S )   &    |-  -.  (/) 
 e.  S   =>    |-  ( ( A F B )  e.  S  ->  ( A  e.  S  /\  B  e.  S ) )
 
Theoremndmovcom 5859 Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.)
 |- 
 dom  F  =  ( S  X.  S )   =>    |-  ( -.  ( A  e.  S  /\  B  e.  S )  ->  ( A F B )  =  ( B F A ) )
 
Theoremndmovass 5860 Any operation is associative outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
 |- 
 dom  F  =  ( S  X.  S )   &    |-  -.  (/) 
 e.  S   =>    |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
 
Theoremndmovdistr 5861 Any operation is distributive outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
 |- 
 dom  F  =  ( S  X.  S )   &    |-  -.  (/) 
 e.  S   &    |-  dom  G  =  ( S  X.  S )   =>    |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) 
 ->  ( A G ( B F C ) )  =  ( ( A G B ) F ( A G C ) ) )
 
Theoremndmovord 5862 Elimination of redundant antecedents in an ordering law. (Contributed by NM, 7-Mar-1996.)
 |- 
 dom  F  =  ( S  X.  S )   &    |-  R  C_  ( S  X.  S )   &    |- 
 -.  (/)  e.  S   &    |-  (
 ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )   =>    |-  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )
 
Theoremndmovordi 5863 Elimination of redundant antecedent in an ordering law. (Contributed by NM, 25-Jun-1998.)
 |- 
 dom  F  =  ( S  X.  S )   &    |-  R  C_  ( S  X.  S )   &    |- 
 -.  (/)  e.  S   &    |-  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )   =>    |-  ( ( C F A ) R ( C F B ) 
 ->  A R B )
 
Theoremcaovclg 5864* Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)
 |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  D )
 )  ->  ( x F y )  e.  E )   =>    |-  ( ( ph  /\  ( A  e.  C  /\  B  e.  D )
 )  ->  ( A F B )  e.  E )
 
Theoremcaovcld 5865* Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  D )
 )  ->  ( x F y )  e.  E )   &    |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  B  e.  D )   =>    |-  ( ph  ->  ( A F B )  e.  E )
 
Theoremcaovcl 5866* Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x F y )  e.  S )   =>    |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A F B )  e.  S )
 
Theoremcaovcomg 5867* 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 5868* 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 5869* 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 5870* 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 5871* 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 5872* 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 5873* 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 5874* 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 5875* 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 5876* 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 5877* 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 5878* 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 5879* 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 5880* 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 5881* 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 5882* 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 5883* 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 5884* 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 5885* 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 5886* 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 5887* 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 5888* 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 5889* 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 5890* 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 5891* 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 5892* 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 5893* 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 5894* 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 5895* 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 5896* 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 5897* 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 5898* 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 5899* 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 5900* 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 ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30843
  Copyright terms: Public domain < Previous  Next >