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Theorem List for Metamath Proof Explorer - 35501-35600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhashgcdlem 35501* A correspondence between elements of specific GCD and relative primes in a smaller ring. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  A  =  { y  e.  (
 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }   &    |-  B  =  { z  e.  (
 0..^ M )  |  ( z  gcd  M )  =  N }   &    |-  F  =  ( x  e.  A  |->  ( x  x.  N ) )   =>    |-  ( ( M  e.  NN  /\  N  e.  NN  /\  N  ||  M )  ->  F : A -1-1-onto-> B )
 
Theoremhashgcdeq 35502* Number of initial positive integers with specified divisors. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  (
 ( M  e.  NN  /\  N  e.  NN )  ->  ( # `  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }
 )  =  if ( N  ||  M ,  ( phi `  ( M  /  N ) ) ,  0 ) )
 
Theoremphisum 35503* The divisor sum identity of the totient function. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  ( N  e.  NN  ->  sum_
 d  e.  { x  e.  NN  |  x  ||  N }  ( phi `  d )  =  N )
 
Theoremproot1hash 35504 If an integral domain has a primitive  N-th root of unity, it has exactly  ( phi `  N ) of them. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  G  =  ( (mulGrp `  R )s  (Unit `  R ) )   &    |-  O  =  ( od `  G )   =>    |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N }
 ) )  ->  ( # `
  ( `' O " { N } )
 )  =  ( phi `  N ) )
 
Theoremproot1ex 35505 The complex field has primitive  N-th roots of unity for all  N. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  G  =  ( (mulGrp ` fld )s  ( CC  \  {
 0 } ) )   &    |-  O  =  ( od `  G )   =>    |-  ( N  e.  NN  ->  ( -u 1  ^c 
 ( 2  /  N ) )  e.  ( `' O " { N } ) )
 
21.23.50  Cyclotomic polynomials
 
Syntaxccytp 35506 Syntax for the sequence of cyclotomic polynomials.
 class CytP
 
Definitiondf-cytp 35507* The Nth cyclotomic polynomial is the polynomial which has as its zeros precisely the primitive Nth roots of unity. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- CytP  =  ( n  e.  NN  |->  ( (mulGrp `  (Poly1 ` fld ) )  gsumg  ( r  e.  ( `' ( od `  (
 (mulGrp ` fld )s  ( CC  \  {
 0 } ) ) ) " { n } )  |->  ( (var1 ` fld ) ( -g `  (Poly1 ` fld )
 ) ( (algSc `  (Poly1 ` fld ) ) `  r
 ) ) ) ) )
 
Theoremisdomn3 35508 Nonzero elements form a multiplicative submonoid of any domain. (Contributed by Stefan O'Rear, 11-Sep-2015.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  U  =  (mulGrp `  R )   =>    |-  ( R  e. Domn  <->  ( R  e.  Ring  /\  ( B  \  {  .0.  } )  e.  (SubMnd `  U ) ) )
 
Theoremmon1pid 35509 Monicity and degree of the unit polynomial. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  .1.  =  ( 1r `  P )   &    |-  M  =  (Monic1p `  R )   &    |-  D  =  ( deg1  `  R )   =>    |-  ( R  e. NzRing  ->  (  .1.  e.  M  /\  ( D `  .1.  )  =  0 ) )
 
Theoremmon1psubm 35510 Monic polynomials are a multiplicative submonoid. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  M  =  (Monic1p `
  R )   &    |-  U  =  (mulGrp `  P )   =>    |-  ( R  e. NzRing  ->  M  e.  (SubMnd `  U ) )
 
Theoremdeg1mhm 35511 Homomorphic property of the polynomial degree. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  D  =  ( deg1  `  R )   &    |-  B  =  ( Base `  P )   &    |-  P  =  (Poly1 `  R )   &    |-  .0.  =  ( 0g `  P )   &    |-  Y  =  ( (mulGrp `  P )s  ( B  \  {  .0.  } ) )   &    |-  N  =  (flds  NN0 )   =>    |-  ( R  e. Domn  ->  ( D  |`  ( B  \  {  .0.  } )
 )  e.  ( Y MndHom  N ) )
 
Theoremcytpfn 35512 Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- CytP  Fn  NN
 
Theoremcytpval 35513* Substitutions for the Nth cyclotomic polynomial. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  T  =  ( (mulGrp ` fld )s  ( CC  \  {
 0 } ) )   &    |-  O  =  ( od `  T )   &    |-  P  =  (Poly1 ` fld )   &    |-  X  =  (var1 ` fld )   &    |-  Q  =  (mulGrp `  P )   &    |-  .-  =  ( -g `  P )   &    |-  A  =  (algSc `  P )   =>    |-  ( N  e.  NN  ->  (CytP `  N )  =  ( Q  gsumg  ( r  e.  ( `' O " { N } )  |->  ( X 
 .-  ( A `  r ) ) ) ) )
 
21.23.51  Miscellaneous topology
 
Theoremfgraphopab 35514* Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( F : A --> B  ->  F  =  { <. a ,  b >.  |  (
 ( a  e.  A  /\  b  e.  B )  /\  ( F `  a )  =  b
 ) } )
 
Theoremfgraphxp 35515* Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( F : A --> B  ->  F  =  { x  e.  ( A  X.  B )  |  ( F `  ( 1st `  x ) )  =  ( 2nd `  x ) }
 )
 
Theoremhausgraph 35516 The graph of a continuous function into a Hausdorff space is closed. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  (
 ( K  e.  Haus  /\  F  e.  ( J  Cn  K ) ) 
 ->  F  e.  ( Clsd `  ( J  tX  K ) ) )
 
Syntaxctopsep 35517 The class of separable toplogies.
 class TopSep
 
Syntaxctoplnd 35518 The class of Lindelöf toplogies.
 class TopLnd
 
Definitiondf-topsep 35519* A topology is separable iff it has a countable dense subset. (Contributed by Stefan O'Rear, 8-Jan-2015.)
 |- TopSep  =  {
 j  e.  Top  |  E. x  e.  ~P  U. j ( x  ~<_  om 
 /\  ( ( cls `  j ) `  x )  =  U. j ) }
 
Definitiondf-toplnd 35520* A topology is Lindelöf iff every open cover has a countable subcover. (Contributed by Stefan O'Rear, 8-Jan-2015.)
 |- TopLnd  =  { x  e.  Top  |  A. y  e.  ~P  x ( U. x  =  U. y  ->  E. z  e.  ~P  x ( z  ~<_  om 
 /\  U. x  =  U. z ) ) }
 
21.24  Mathbox for Jon Pennant
 
Theoremioounsn 35521 The closure of the upper end of an open real interval. (Contributed by Jon Pennant, 8-Jun-2019.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  (
 ( A (,) B )  u.  { B }
 )  =  ( A (,] B ) )
 
Theoremiocunico 35522 Split an open interval into two pieces at point B, Co-author TA. (Contributed by Jon Pennant, 8-Jun-2019.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  ( ( A (,] B )  u.  ( B [,) C ) )  =  ( A (,) C ) )
 
Theoremiocinico 35523 The intersection of two sets that meet at a point is that point. (Contributed by Jon Pennant, 12-Jun-2019.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  ( ( A (,] B )  i^i  ( B [,) C ) )  =  { B }
 )
 
Theoremiocmbl 35524 An open-below, closed-above real interval is measurable. (Contributed by Jon Pennant, 12-Jun-2019.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  e.  dom  vol )
 
Theoremcnioobibld 35525* A bounded, continuous function on an open bounded interval is integrable. The function must be bounded. For a counterexample, consider  F  =  ( x  e.  ( 0 (,) 1 )  |->  ( 1  /  x ) ). See cniccibl 22537 for closed bounded intervals. (Contributed by Jon Pennant, 31-May-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  dom  F ( abs `  ( F `  y ) )  <_  x )   =>    |-  ( ph  ->  F  e.  L^1 )
 
Theoremitgpowd 35526* The integral of a monomial on a closed bounded interval of the real line. Co-authors TA and MC. (Contributed by Jon Pennant, 31-May-2019.) (Revised by Thierry Arnoux, 14-Jun-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  S. ( A [,] B ) ( x ^ N )  _d x  =  ( ( ( B ^
 ( N  +  1 ) )  -  ( A ^ ( N  +  1 ) ) ) 
 /  ( N  +  1 ) ) )
 
Theoremarearect 35527 The area of a rectangle whose sides are parallel to the coordinate axes in  ( RR  X.  RR ) is its width multiplied by its height. (Contributed by Jon Pennant, 19-Mar-2019.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   &    |-  D  e.  RR   &    |-  A  <_  B   &    |-  C  <_  D   &    |-  S  =  ( ( A [,] B )  X.  ( C [,] D ) )   =>    |-  (area `  S )  =  ( ( B  -  A )  x.  ( D  -  C ) )
 
Theoremareaquad 35528* The area of a quadrilateral with two sides which are parallel to the y-axis in  ( RR  X.  RR ) is its width multiplied by the average height of its higher edge minus the average height of its lower edge. Co-author TA. (Contributed by Jon Pennant, 31-May-2019.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   &    |-  D  e.  RR   &    |-  E  e.  RR   &    |-  F  e.  RR   &    |-  A  <  B   &    |-  C  <_  E   &    |-  D  <_  F   &    |-  U  =  ( C  +  ( ( ( x  -  A )  /  ( B  -  A ) )  x.  ( D  -  C ) ) )   &    |-  V  =  ( E  +  (
 ( ( x  -  A )  /  ( B  -  A ) )  x.  ( F  -  E ) ) )   &    |-  S  =  { <. x ,  y >.  |  ( x  e.  ( A [,] B )  /\  y  e.  ( U [,] V ) ) }   =>    |-  (area `  S )  =  ( (
 ( ( F  +  E )  /  2
 )  -  ( ( D  +  C ) 
 /  2 ) )  x.  ( B  -  A ) )
 
21.25  Mathbox for Richard Penner
 
21.25.1  Short Studies
 
21.25.1.1  Additional work on conditional logical operator
 
Theoremifpan123g 35529 Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  ta )  /\ if- ( ps ,  th ,  et ) )  <->  ( ( ( -.  ph  \/  ch )  /\  ( ph  \/  ta ) )  /\  ( ( -.  ps  \/  th )  /\  ( ps  \/  et ) ) ) )
 
Theoremifpan23 35530 Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.)
 |-  (
 (if- ( ph ,  ps ,  ch )  /\ if- (
 ph ,  th ,  ta ) )  <-> if- ( ph ,  ( ps  /\  th ) ,  ( ch  /\  ta ) ) )
 
Theoremifpdfor2 35531 Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  \/  ps )  <-> if- (
 ph ,  ph ,  ps ) )
 
Theoremifporcor 35532 Corollary of commutation of or. (Contributed by RP, 20-Apr-2020.)
 |-  (if- ( ph ,  ph ,  ps )  <-> if- ( ps ,  ps ,  ph ) )
 
Theoremifpdfan2 35533 Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  (
 ( ph  /\  ps )  <-> if- (
 ph ,  ps ,  ph ) )
 
Theoremifpancor 35534 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  ps ,  ph )  <-> if- ( ps ,  ph ,  ps ) )
 
Theoremifpdfor 35535 Define or in terms of conditional logic operator and true. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  \/  ps )  <-> if- (
 ph , T.  ,  ps ) )
 
Theoremifpdfan 35536 Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  /\  ps )  <-> if- (
 ph ,  ps , F.  ) )
 
Theoremifpbi2 35537 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
 |-  (
 ( ph  <->  ps )  ->  (if- ( ch ,  ph ,  th )  <-> if- ( ch ,  ps ,  th ) ) )
 
Theoremifpbi3 35538 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
 |-  (
 ( ph  <->  ps )  ->  (if- ( ch ,  th ,  ph )  <-> if- ( ch ,  th ,  ps ) ) )
 
Theoremifpim1 35539 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  ->  ps )  <-> if- ( -.  ph , T.  ,  ps ) )
 
Theoremifpnot 35540 Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  ( -.  ph  <-> if- ( ph , F.  , T.  ) )
 
Theoremifpid2 35541 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  ( ph 
 <-> if- ( ph , T.  , F.  ) )
 
Theoremifpim2 35542 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  ->  ps )  <-> if- ( ps , T.  ,  -.  ph ) )
 
Theoremifpbi23 35543 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th ) )  ->  (if- ( ta ,  ph ,  ch )  <-> if- ( ta ,  ps ,  th ) ) )
 
Theoremifpdfbi 35544 Define biimplication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  <->  ps )  <-> if- ( ph ,  ps ,  -.  ps ) )
 
Theoremifpbiidcor 35545 Restatement of biid 236. (Contributed by RP, 25-Apr-2020.)
 |- if- ( ph ,  ph ,  -.  ph )
 
Theoremifpbicor 35546 Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  ps ,  -.  ps )  <-> if- ( ps ,  ph ,  -.  ph ) )
 
Theoremifpxorcor 35547 Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  -.  ps ,  ps )  <-> if- ( ps ,  -.  ph ,  ph ) )
 
Theoremifpbi1 35548 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
 |-  (
 ( ph  <->  ps )  ->  (if- ( ph ,  ch ,  th )  <-> if- ( ps ,  ch ,  th ) ) )
 
Theoremifpnot23 35549 Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.)
 |-  ( -. if- ( ph ,  ps ,  ch )  <-> if- ( ph ,  -.  ps ,  -.  ch )
 )
 
Theoremifpnotnotb 35550 Factor conditional logic operator over negation in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  -.  ps ,  -.  ch )  <->  -. if- ( ph ,  ps ,  ch ) )
 
Theoremifpnorcor 35551 Corollary of commutation of nor. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  -.  ph ,  -.  ps )  <-> if- ( ps ,  -.  ps ,  -.  ph )
 )
 
Theoremifpnancor 35552 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  -.  ps ,  -.  ph )  <-> if- ( ps ,  -.  ph ,  -.  ps )
 )
 
Theoremifpnot23b 35553 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -. if- ( ph ,  -.  ps ,  ch )  <-> if- ( ph ,  ps ,  -.  ch ) )
 
Theoremifpbiidcor2 35554 Restatement of biid 236. (Contributed by RP, 25-Apr-2020.)
 |-  -. if- (
 ph ,  -.  ph ,  ph )
 
Theoremifpnot23c 35555 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -. if- ( ph ,  ps ,  -.  ch )  <-> if- ( ph ,  -.  ps ,  ch ) )
 
Theoremifpnot23d 35556 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -. if- ( ph ,  -.  ps ,  -.  ch )  <-> if- (
 ph ,  ps ,  ch ) )
 
Theoremifpdfnan 35557 Define nand as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  -/\  ps )  <-> if- (
 ph ,  -.  ps , T.  ) )
 
Theoremifpdfxor 35558 Define xor as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  \/_  ps )  <-> if- (
 ph ,  -.  ps ,  ps ) )
 
Theoremifpbi12 35559 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th ) )  ->  (if- ( ph ,  ch ,  ta )  <-> if- ( ps ,  th ,  ta ) ) )
 
Theoremifpbi13 35560 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th ) )  ->  (if- ( ph ,  ta ,  ch )  <-> if- ( ps ,  ta ,  th ) ) )
 
Theoremifpbi123 35561 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th )  /\  ( ta 
 <->  et ) )  ->  (if- ( ph ,  ch ,  ta )  <-> if- ( ps ,  th ,  et ) ) )
 
Theoremifpidg 35562 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( th  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( ( ( ph  /\  ps )  ->  th )  /\  (
 ( ph  /\  th )  ->  ps ) )  /\  ( ( ch  ->  (
 ph  \/  th )
 )  /\  ( th  ->  ( ph  \/  ch ) ) ) ) )
 
Theoremifpid3g 35563 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ch  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( (
 ph  /\  ps )  ->  ch )  /\  (
 ( ph  /\  ch )  ->  ps ) ) )
 
Theoremifpid2g 35564 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ps  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( ps 
 ->  ( ph  \/  ch ) )  /\  ( ch 
 ->  ( ph  \/  ps ) ) ) )
 
Theoremifpid1g 35565 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( ch 
 ->  ph )  /\  ( ph  ->  ps ) ) )
 
Theoremifpim23g 35566 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  (
 ( ( ph  ->  ps )  <-> if- ( ch ,  ps ,  -.  ph ) )  <->  ( ( (
 ph  /\  ps )  ->  ch )  /\  ( ch  ->  ( ph  \/  ps ) ) ) )
 
Theoremifpim3 35567 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  (
 ( ph  ->  ps )  <-> if- (
 ph ,  ps ,  -.  ph ) )
 
Theoremifpnim1 35568 Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -.  ( ph  ->  ps )  <-> if- (
 ph ,  -.  ps ,  ph ) )
 
Theoremifpim4 35569 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  (
 ( ph  ->  ps )  <-> if- ( ps ,  ps ,  -.  ph ) )
 
Theoremifpnim2 35570 Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -.  ( ph  ->  ps )  <-> if- ( ps ,  -.  ps ,  ph ) )
 
Theoremifpim123g 35571 Implication of conditional logical operators. The right hand side is basically conjunctive normal form which is useful in proofs. (Contributed by RP, 16-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  ta )  -> if- ( ps ,  th ,  et ) )  <->  ( ( ( ( ph  ->  -.  ps )  \/  ( ch  ->  th ) )  /\  (
 ( ps  ->  ph )  \/  ( ta  ->  th )
 ) )  /\  (
 ( ( ph  ->  ps )  \/  ( ch 
 ->  et ) )  /\  ( ( -.  ps  -> 
 ph )  \/  ( ta  ->  et ) ) ) ) )
 
Theoremifpim1g 35572 Implication of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  th )  -> if- ( ps ,  ch ,  th ) )  <->  ( ( ( ps  ->  ph )  \/  ( th  ->  ch )
 )  /\  ( ( ph  ->  ps )  \/  ( ch  ->  th ) ) ) )
 
Theoremifp1bi 35573 Substitute the first element of conditional logical operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  th )  <-> if- ( ps ,  ch ,  th ) )  <->  ( ( ( ( ph  ->  ps )  \/  ( ch  ->  th )
 )  /\  ( ( ph  ->  ps )  \/  ( th  ->  ch ) ) ) 
 /\  ( ( ( ps  ->  ph )  \/  ( ch  ->  th )
 )  /\  ( ( ps  ->  ph )  \/  ( th  ->  ch ) ) ) ) )
 
Theoremifpbi1b 35574 When the first variable is irrelevant, it can be replaced. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  ch ,  ch )  <-> if- ( ps ,  ch ,  ch ) )
 
Theoremifpimimb 35575 Factor conditional logic operator over implication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 ->  ch ) ,  ( th  ->  ta ) )  <->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta ) ) )
 
Theoremifpororb 35576 Factor conditional logic operator over disjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 \/  ch ) ,  ( th  \/  ta ) )  <-> 
 (if- ( ph ,  ps ,  th )  \/ if-
 ( ph ,  ch ,  ta ) ) )
 
Theoremifpananb 35577 Factor conditional logic operator over conjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 /\  ch ) ,  ( th  /\  ta ) )  <-> 
 (if- ( ph ,  ps ,  th )  /\ if- (
 ph ,  ch ,  ta ) ) )
 
Theoremifpnannanb 35578 Factor conditional logic operator over nand in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps  -/\  ch ) ,  ( th  -/\  ta ) )  <-> 
 (if- ( ph ,  ps ,  th )  -/\ if- (
 ph ,  ch ,  ta ) ) )
 
Theoremifpor123g 35579 Disjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  ta )  \/ if-
 ( ps ,  th ,  et ) )  <->  ( ( ( ( ph  ->  -.  ps )  \/  ( ch  \/  th ) )  /\  (
 ( ps  ->  ph )  \/  ( ta  \/  th ) ) )  /\  ( ( ( ph  ->  ps )  \/  ( ch  \/  et ) ) 
 /\  ( ( -. 
 ps  ->  ph )  \/  ( ta  \/  et ) ) ) ) )
 
Theoremifpimim 35580 Consequnce of implication. (Contributed by RP, 17-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 ->  ch ) ,  ( th  ->  ta ) )  ->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta )
 ) )
 
Theoremifpbibib 35581 Factor conditional logic operator over biimplication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps  <->  ch ) ,  ( th  <->  ta ) )  <->  (if- ( ph ,  ps ,  th )  <-> if- ( ph ,  ch ,  ta ) ) )
 
Theoremifpxorxorb 35582 Factor conditional logic operator over xor in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 \/_  ch ) ,  ( th  \/_  ta ) )  <-> 
 (if- ( ph ,  ps ,  th )  \/_ if- (
 ph ,  ch ,  ta ) ) )
 
21.25.1.2  Sophisms
 
Theoremrp-fakeimass 35583 A special case where implication appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
 |-  (
 ( ph  \/  ch )  <->  ( ( ( ph  ->  ps )  ->  ch )  <->  (
 ph  ->  ( ps  ->  ch ) ) ) )
 
Theoremrp-fakeanorass 35584 A special case where a mixture of and and or appears to conform to a mixed associative law. (Contributed by Richard Penner, 26-Feb-2020.)
 |-  (
 ( ch  ->  ph )  <->  ( ( ( ph  /\  ps )  \/  ch )  <->  ( ph  /\  ( ps  \/  ch ) ) ) )
 
Theoremrp-fakeoranass 35585 A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
 |-  (
 ( ph  ->  ch )  <->  ( ( ( ph  \/  ps )  /\  ch )  <->  (
 ph  \/  ( ps  /\ 
 ch ) ) ) )
 
Theoremrp-fakenanass 35586 A special case where nand appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
 |-  (
 ( ph  <->  ch )  <->  ( ( (
 ph  -/\  ps )  -/\  ch )  <->  ( ph  -/\  ( ps  -/\  ch ) ) ) )
 
Theoremrp-fakeinunass 35587 A special case where a mixture of intersection and union appears to conform to a mixed associative law. (Contributed by Richard Penner, 26-Feb-2020.)
 |-  ( C  C_  A  <->  ( ( A  i^i  B )  u.  C )  =  ( A  i^i  ( B  u.  C ) ) )
 
Theoremrp-fakeuninass 35588 A special case where a mixture of union and intersection appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
 |-  ( A  C_  C  <->  ( ( A  u.  B )  i^i 
 C )  =  ( A  u.  ( B  i^i  C ) ) )
 
21.25.1.3  Finite Sets

Membership in the class of finite sets can be expressed in many ways.

 
Theoremrp-isfinite5 35589* A set is said to be finite if it can be put in one-to-one correspondence with all the natural numbers between 1 and some  n  e.  NN0. (Contributed by Richard Penner, 3-Mar-2020.)
 |-  ( A  e.  Fin  <->  E. n  e.  NN0  ( 1 ... n )  ~~  A )
 
Theoremrp-isfinite6 35590* A set is said to be finite if it is either empty or it can be put in one-to-one correspondence with all the natural numbers between 1 and some  n  e.  NN. (Contributed by Richard Penner, 10-Mar-2020.)
 |-  ( A  e.  Fin  <->  ( A  =  (/) 
 \/  E. n  e.  NN  ( 1 ... n )  ~~  A ) )
 
21.25.1.4  Infinite Sets
 
Theorempwelg 35591* The powerclass is an element of a class closed under union and powerclass operations iff the element is a member of that class. (Contributed by RP, 21-Mar-2020.)
 |-  ( A. x  e.  B  ( U. x  e.  B  /\  ~P x  e.  B )  ->  ( A  e.  B 
 <->  ~P A  e.  B ) )
 
Theorempwinfig 35592* The powerclass of an infinite set is an infinite set, and vice-versa. Here  B is a class which is closed under both the union and the powerclass operations and which may have infinite sets as members. (Contributed by RP, 21-Mar-2020.)
 |-  ( A. x  e.  B  ( U. x  e.  B  /\  ~P x  e.  B )  ->  ( A  e.  ( B  \  Fin )  <->  ~P A  e.  ( B 
 \  Fin ) ) )
 
Theorempwinfi2 35593 The powerclass of an infinite set is an infinite set, and vice-versa. Here  U is a weak universe. (Contributed by RP, 21-Mar-2020.)
 |-  ( U  e. WUni  ->  ( A  e.  ( U  \  Fin )  <->  ~P A  e.  ( U  \  Fin ) ) )
 
Theorempwinfi3 35594 The powerclass of an infinite set is an infinite set, and vice-versa. Here  T is a transitive Tarski universe. (Contributed by RP, 21-Mar-2020.)
 |-  (
 ( T  e.  Tarski  /\ 
 Tr  T )  ->  ( A  e.  ( T  \  Fin )  <->  ~P A  e.  ( T  \  Fin ) ) )
 
Theorempwinfi 35595 The powerclass of an infinite set is an infinite set, and vice-versa. (Contributed by RP, 21-Mar-2020.)
 |-  ( A  e.  ( _V  \ 
 Fin )  <->  ~P A  e.  ( _V  \  Fin ) )
 
21.25.1.5  Finite intersection property

While there is not yet a definition, the finite intersection property of a class is introduced by fiint 7830 where two textbook definitions are shown to be equivalent.

This property is seen often with ordinal numbers (onin 5440, ordelinel 5507 ), chains of sets ordered by the proper subset relation (sorpssin 6569), various sets in the field of topology (inopn 19698, incld 19834, innei 19917, ... ) and "universal" classes like weak universes (wunin 9120, tskin 9166) and the class of all sets (inex1g 4536) .

 
Theoremfipjust 35596* A definition of the finite intersection property of a class based on closure under pair-wise intersection of its elements is independent of the dummy variables. (Contributed by Richard Penner, 1-Jan-2020.)
 |-  ( A. u  e.  A  A. v  e.  A  ( u  i^i  v )  e.  A  <->  A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A )
 
Theoremcllem0 35597* The class of all sets with property  ph ( z ) is closed under the binary operation on sets defined in  R ( x ,  y ). (Contributed by Richard Penner, 3-Jan-2020.)
 |-  V  =  { z  |  ph }   &    |-  R  e.  U   &    |-  ( z  =  R  ->  ( ph  <->  ps ) )   &    |-  ( z  =  x  ->  ( ph  <->  ch ) )   &    |-  ( z  =  y  ->  ( ph  <->  th ) )   &    |-  ( ( ch 
 /\  th )  ->  ps )   =>    |-  A. x  e.  V  A. y  e.  V  R  e.  V
 
Theoremsuperficl 35598* The class of all supersets of a class has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  B  C_  z }   =>    |- 
 A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A
 
Theoremsuperuncl 35599* The class of all supersets of a class is closed under binary union. (Contributed by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  B  C_  z }   =>    |- 
 A. x  e.  A  A. y  e.  A  ( x  u.  y )  e.  A
 
Theoremssficl 35600* The class of all subsets of a class has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  z 
 C_  B }   =>    |-  A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A
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