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Theorem List for Metamath Proof Explorer - 23401-23500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremghomsn 23401 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 23402 Lemma for ghomgrp 23403. (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 23403 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 23404 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 23405 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 23406 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 23407* Lemma for ghomf1o 23408. (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 23408* 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 23409 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 23410* 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 23411*  ( ( 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 23412*  ( ( 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 23413* 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 23414 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 23415* 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 23416 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 23417 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 23418 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 23419 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 23420 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 23421 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 23422 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 23423 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 23424 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 23425 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 23426* 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 23427* 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 23428* 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 23429 Extend class notation to include relation exponentiation.
 class  ^ r
 
Definitiondf-relexp 23430* 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 23431 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 23432 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 23433 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 23434 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 23435 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 23436 Obsolete; use ovex 5844 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 23437 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 23438 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 23439 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 23440 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 23441 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 23442* 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 23443* 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 23444 Extend class notation with recursively defined reflexive, transitive closure.
 class  t *rec
 
Definitiondf-rtrclrec 23445* 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 23446* 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 23447 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 23448 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 23449 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 23450* 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 23451 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 23452* 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 23453 ax-ext 2264 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 23454 ax-rep 4131 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 23455 ax-un 4510 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 23456 ax-pow 4186 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 23457 ax-reg 7301 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 23458 ax-inf 7334 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 23459 ax-ac 8080 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 23460* We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 23551). 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 23461* 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 23462* 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 23463 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 23464* 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 23465* 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 23466* 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 23467 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 23468 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 23469 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 23470 Triple negated disjuntion introduction. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  -.  ph   &    |-  -. 
 ps   &    |- 
 -.  ch   =>    |- 
 -.  ( ph  \/  ps 
 \/  ch )
 
Theorem3jaodd 23471 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 23472 Closed form of 3ori 1242, (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  (
 ( ph  \/  ps  \/  ch )  <->  ( ( -.  ph  /\  -.  ps )  ->  ch ) )
 
Theorem3mix1d 23473 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  \/  ch  \/  th ) )
 
Theorem3mix2d 23474 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  ps  \/  th ) )
 
Theorem3mix3d 23475 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  th  \/  ps ) )
 
Theorembiimpexp 23476 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 23477 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 23478 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 23479 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 23480* 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 23481 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { <. A ,  B >. ,  C ,  D } B
 
Theorembrtpid2 23482 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { C ,  <. A ,  B >. ,  D } B
 
Theorembrtpid3 23483 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { C ,  D ,  <. A ,  B >. } B
 
18.7.5  Properties of reals and complexes
 
Theoremsqdivzi 23484 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 23485 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 23486 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 23487* The Dedekind cut theorem. This theorem, which may be used to replace ax-pre-sup 8810 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 23488* 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 23489 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 23490 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 ) ) )
 
Theoremmulge0b 23491 A condition for multiplication to be non-negative. (Contributed by Scott Fenton, 25-Jun-2013.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  ( A  x.  B )  <->  ( ( A 
 <_  0  /\  B  <_  0 )  \/  ( 0 
 <_  A  /\  0  <_  B ) ) ) )
 
Theoremmulle0b 23492 A condition for multiplication to be non-positive. (Contributed by Scott Fenton, 25-Jun-2013.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  <_  0  <->  ( ( A 
 <_  0  /\  0  <_  B )  \/  (
 0  <_  A  /\  B  <_  0 ) ) ) )
 
Theoremmulsuble0b 23493 A condition for multiplication of subtraction to be non-positive. (Contributed by Scott Fenton, 25-Jun-2013.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( ( A  -  B )  x.  ( C  -  B ) )  <_  0  <->  ( ( A 
 <_  B  /\  B  <_  C )  \/  ( C 
 <_  B  /\  B  <_  A ) ) ) )
 
Theoremrelin01 23494 An interval law for less than or equal. (Contributed by Scott Fenton, 27-Jun-2013.)
 |-  ( A  e.  RR  ->  ( A  <_  0  \/  ( 0  <_  A  /\  A  <_  1 )  \/  1  <_  A ) )
 
Theoremsubdivcomb1 23495 Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( (
 ( C  x.  A )  -  B )  /  C )  =  ( A  -  ( B  /  C ) ) )
 
Theoremsubdivcomb2 23496 Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  -  ( C  x.  B ) )  /  C )  =  (
 ( A  /  C )  -  B ) )
 
Theoremsubeqrev 23497 Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.)
 |-  (
 ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  -  B )  =  ( C  -  D )  <->  ( B  -  A )  =  ( D  -  C ) ) )
 
Theoremfznatpl1 23498 Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)
 |-  (
 ( N  e.  NN  /\  I  e.  ( 1
 ... ( N  -  1 ) ) ) 
 ->  ( I  +  1 )  e.  ( 1
 ... N ) )
 
Theoremsupfz 23499 The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  sup (
 ( M ... N ) ,  ZZ ,  <  )  =  N )
 
Theoreminffz 23500 The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  sup (
 ( M ... N ) ,  ZZ ,  `'  <  )  =  M )
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