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Theorem List for Metamath Proof Explorer - 24101-24200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmulge0b 24101 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 24102 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 24103 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 24104 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 24105 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 24106 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 24107 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 24108 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 24109 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 24110 The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  sup (
 ( M ... N ) ,  ZZ ,  `'  <  )  =  M )
 
Theorembcnm1 24111 The binomial coefficent of  ( N  -  1 ) is  N. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  _C  ( N  -  1 ) )  =  N )
 
Theoremfz0n 24112 The sequence  ( 0 ... ( N  -  1 ) ) is empty iff  N is zero. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( N  e.  NN0  ->  (
 ( 0 ... ( N  -  1 ) )  =  (/)  <->  N  =  0
 ) )
 
Theorem4bc3eq4 24113 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
 |-  (
 4  _C  3 )  =  4
 
Theorem4bc2eq6 24114 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  (
 4  _C  2 )  =  6
 
Theoremhalfthird 24115 Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  (
 ( 1  /  2
 )  -  ( 1 
 /  3 ) )  =  ( 1  / 
 6 )
 
Theorem5recm6rec 24116 One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  (
 ( 1  /  5
 )  -  ( 1 
 /  6 ) )  =  ( 1  / ; 3 0 )
 
Theoremfaclimlem1 24117 Lemma for faclim 24126. Set up substitution rules. (Contributed by Scott Fenton, 26-Nov-2017.)
 |-  ( A  =  B  ->  ( ( ( 1  +  ( 1  /  N ) ) ^ A )  /  ( 1  +  ( A  /  N ) ) )  =  ( ( ( 1  +  ( 1  /  N ) ) ^ B )  /  (
 1  +  ( B 
 /  N ) ) ) )
 
Theoremfaclimlem2 24118 Lemma for faclim 24126. Set up substitution rules. (Contributed by Scott Fenton, 26-Nov-2017.)
 |-  ( N  =  M  ->  ( ( ( 1  +  ( 1  /  N ) ) ^ A )  /  ( 1  +  ( A  /  N ) ) )  =  ( ( ( 1  +  ( 1  /  M ) ) ^ A )  /  (
 1  +  ( A 
 /  M ) ) ) )
 
Theoremfaclimlem3 24119 Lemma for faclim 24126. Base case for induction. (Contributed by Scott Fenton, 26-Nov-2017.)
 |-  seq  1 (  x.  ,  ( n  e.  NN  |->  ( ( ( 1  +  (
 1  /  n )
 ) ^ 0 ) 
 /  ( 1  +  ( 0  /  n ) ) ) ) )  ~~>  1
 
Theoremfaclimlem4 24120 Lemma for faclim 24126. Closure of a particular expression. (Contributed by Scott Fenton, 26-Nov-2017.)
 |-  (
 ( B  e.  NN0  /\  N  e.  NN )  ->  ( ( N  +  1 )  /  ( B  +  ( N  +  1 ) ) )  e.  CC )
 
Theoremfaclimlem5 24121* Lemma for faclim 24126. A convergence statement in the induction. (Contributed by Scott Fenton, 26-Nov-2017.)
 |-  ( B  e.  NN0  ->  ( n  e.  NN  |->  ( ( B  +  1 )  x.  ( ( n  +  1 )  /  ( B  +  ( n  +  1 )
 ) ) ) )  ~~>  ( B  +  1
 ) )
 
Theoremfaclimlem6 24122 Lemma for faclim 24126. Closure of the core expression. (Contributed by Scott Fenton, 26-Nov-2017.)
 |-  (
 ( B  e.  NN0  /\  N  e.  NN )  ->  ( ( ( 1  +  ( 1  /  N ) ) ^ B )  /  (
 1  +  ( B 
 /  N ) ) )  e.  CC )
 
Theoremfaclimlem7 24123* Lemma for faclim 24126. Sequence closure. (Contributed by Scott Fenton, 26-Nov-2017.)
 |-  (
 ( B  e.  NN0  /\  K  e.  NN )  ->  (  seq  1 (  x.  ,  ( n  e.  NN  |->  ( ( ( 1  +  (
 1  /  n )
 ) ^ B ) 
 /  ( 1  +  ( B  /  n ) ) ) ) ) `  K )  e.  CC )
 
Theoremfaclimlem8 24124 Lemma for faclim 24126. Base induction case. (Contributed by Scott Fenton, 29-Nov-2017.)
 |-  ( B  e.  NN0  ->  (
 ( ( 1  +  ( 1  /  1
 ) ) ^ ( B  +  1 )
 )  /  ( 1  +  ( ( B  +  1 )  /  1
 ) ) )  =  ( ( ( ( 1  +  ( 1 
 /  1 ) ) ^ B )  /  ( 1  +  ( B  /  1 ) ) )  x.  ( ( B  +  1 )  x.  ( ( 1  +  1 )  /  ( B  +  (
 1  +  1 ) ) ) ) ) )
 
Theoremfaclimlem9 24125* Lemma for faclim 24126. Final inductive step. (Contributed by Scott Fenton, 29-Nov-2017.)
 |-  (
 ( M  e.  NN  /\  B  e.  NN0 )  ->  ( ( (  seq  1 (  x.  ,  ( n  e.  NN  |->  ( ( ( 1  +  (
 1  /  n )
 ) ^ B ) 
 /  ( 1  +  ( B  /  n ) ) ) ) ) `  M )  x.  ( ( B  +  1 )  x.  ( ( M  +  1 )  /  ( B  +  ( M  +  1 ) ) ) ) )  x.  ( ( ( 1  +  ( 1  /  ( M  +  1
 ) ) ) ^
 ( B  +  1 ) )  /  (
 1  +  ( ( B  +  1 ) 
 /  ( M  +  1 ) ) ) ) )  =  ( ( (  seq  1
 (  x.  ,  ( n  e.  NN  |->  ( ( ( 1  +  (
 1  /  n )
 ) ^ B ) 
 /  ( 1  +  ( B  /  n ) ) ) ) ) `  M )  x.  ( ( ( 1  +  ( 1 
 /  ( M  +  1 ) ) ) ^ B )  /  ( 1  +  ( B  /  ( M  +  1 ) ) ) ) )  x.  (
 ( B  +  1 )  x.  ( ( ( M  +  1 )  +  1 ) 
 /  ( B  +  ( ( M  +  1 )  +  1
 ) ) ) ) ) )
 
Theoremfaclim 24126* An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.)
 |-  F  =  ( n  e.  NN  |->  ( ( ( 1  +  ( 1  /  n ) ) ^ A )  /  (
 1  +  ( A 
 /  n ) ) ) )   =>    |-  ( A  e.  NN0  ->  seq  1 (  x.  ,  F )  ~~>  ( ! `  A ) )
 
18.7.6  Complex products
 
Syntaxccprod 24127 Extend class notation to include complex products.
 class k  e.  A B
 
Definitiondf-cprod 24128* Define the product of a series with an index set of integers  A. This definition takes most of the aspects of df-sum 12175 and adapts them for multiplication instead of addition. However, we insist that in the infinite case, there is a non-zero tail of the sequence. This ensures that the convergence criteria match those of infinite sums. (Contributed by Scott Fenton, 4-Dec-2017.)
 |- k  e.  A B  =  (
 iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  E. n  e.  ( ZZ>=
 `  m ) E. y ( y  =/=  0  /\  seq  n (  x.  ,  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  1 )
 ) )  ~~>  y )  /\  seq  m (  x. 
 ,  ( k  e. 
 ZZ  |->  if ( k  e.  A ,  B , 
 1 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1
 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  x.  ,  ( n  e.  NN  |->  [_ (
 f `  n )  /  k ]_ B ) ) `  m ) ) ) )
 
Theoremcprodex 24129 A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.)
 |- k  e.  A B  e.  _V
 
Theoremcprodeq1f 24130 Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)
 |-  F/_ k A   &    |-  F/_ k B   =>    |-  ( A  =  B  ->
 k  e.  A C  = k  e.  B C )
 
Theoremcprodeq1 24131* Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)
 |-  ( A  =  B  -> k  e.  A C  = k  e.  B C )
 
Theoremnfcprod1 24132* Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  F/_ k A   =>    |-  F/_ k k  e.  A B
 
Theoremnfcprod 24133* Bound-variable hypothesis builder for product: if  x is (effectively) not free in  A and  B, it is not free in ∏ k  e.  A B. (Contributed by Scott Fenton, 1-Dec-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x k  e.  A B
 
Theoremcprodeq2w 24134* Equality theorem for product, when the class expressions  B and  C are equal everywhere. Proved using only Extensionality. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  ( A. k  B  =  C  -> k  e.  A B  = k  e.  A C )
 
Theoremcprodeq2ii 24135* Equality theorem for product, with the class expressions  B and  C guarded by  _I to be always sets. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C )  -> k  e.  A B  = k  e.  A C )
 
Theoremcprodeq2 24136* Equality theorem for product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  ( A. k  e.  A  B  =  C  -> k  e.  A B  = k  e.  A C )
 
Theoremcbvcprod 24137* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  (
 j  =  k  ->  B  =  C )   &    |-  F/_ k A   &    |-  F/_ j A   &    |-  F/_ k B   &    |-  F/_ j C   =>    |- j  e.  A B  = k  e.  A C
 
Theoremcbvcprodv 24138* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  (
 j  =  k  ->  B  =  C )   =>    |- j  e.  A B  = k  e.  A C
 
Theoremcbvcprodi 24139* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  F/_ k B   &    |-  F/_ j C   &    |-  ( j  =  k  ->  B  =  C )   =>    |- j  e.  A B  = k  e.  A C
 
Theoremcprodeq1i 24140* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  A  =  B   =>    |- k  e.  A C  = k  e.  B C
 
Theoremcprodeq2i 24141* Equality inference for sum. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  (
 k  e.  A  ->  B  =  C )   =>    |- k  e.  A B  = k  e.  A C
 
Theoremcprodeq12i 24142* Equality inference for sum. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  A  =  B   &    |-  ( k  e.  A  ->  C  =  D )   =>    |- k  e.  A C  = k  e.  B D
 
Theoremcprodeq1d 24143* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  -> k  e.  A C  = k  e.  B C )
 
Theoremcprodeq2d 24144* Equality deduction for sum. Note that unlike cprodeq2dv 24145, 
k may occur in  ph. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  ( ph  ->  A. k  e.  A  B  =  C )   =>    |-  ( ph  -> k  e.  A B  = k  e.  A C )
 
Theoremcprodeq2dv 24145* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  (
 ( ph  /\  k  e.  A )  ->  B  =  C )   =>    |-  ( ph  -> k  e.  A B  = k  e.  A C )
 
Theoremcprodeq2sdv 24146* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  -> k  e.  A B  = k  e.  A C )
 
Theorem2cprodeq2dv 24147* Equality deduction for double sum. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  (
 ( ph  /\  j  e.  A  /\  k  e.  B )  ->  C  =  D )   =>    |-  ( ph  -> j  e.  A k  e.  B C  = j  e.  A k  e.  B D )
 
Theoremcprodeq12dv 24148* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  A )  ->  C  =  D )   =>    |-  ( ph  -> k  e.  A C  = k  e.  B D )
 
Theoremcprodeq12rdv 24149* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  B )  ->  C  =  D )   =>    |-  ( ph  -> k  e.  A C  = k  e.  B D )
 
Theoremcprod2id 24150* The second class argument to a sum can be chosen so that it is always a set. (Contributed by Scott Fenton, 4-Dec-2017.)
 |- k  e.  A B  = k  e.  A (  _I  `  B )
 
Theoremprodf 24151* An infinite product of complex terms is a function from an upper set of integers to  CC. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  seq  M (  x.  ,  F ) : Z --> CC )
 
Theoremprodrblem 24152* Lemma for prodrb 24155. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  M ) )   =>    |-  ( ( ph  /\  A  C_  ( ZZ>= `  N )
 )  ->  (  seq  M (  x.  ,  F )  |`  ( ZZ>= `  N ) )  =  seq  N (  x.  ,  F ) )
 
Theoremfprodcvg 24153* The sequence of partial products of a finite product converges to the whole product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  M ) )   &    |-  ( ph  ->  A  C_  ( M ... N ) )   =>    |-  ( ph  ->  seq  M (  x.  ,  F )  ~~>  (  seq  M (  x. 
 ,  F ) `  N ) )
 
Theoremprodrblem2 24154* Lemma for prodrb 24155. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  A 
 C_  ( ZZ>= `  M ) )   &    |-  ( ph  ->  A 
 C_  ( ZZ>= `  N ) )   =>    |-  ( ( ph  /\  N  e.  ( ZZ>= `  M )
 )  ->  (  seq  M (  x.  ,  F ) 
 ~~>  C  <->  seq  N (  x. 
 ,  F )  ~~>  C )
 )
 
Theoremprodrb 24155* Rebase the starting point of a product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  A 
 C_  ( ZZ>= `  M ) )   &    |-  ( ph  ->  A 
 C_  ( ZZ>= `  N ) )   =>    |-  ( ph  ->  (  seq  M (  x.  ,  F )  ~~>  C  <->  seq  N (  x. 
 ,  F )  ~~>  C )
 )
 
Theoremprodmolem3 24156* Lemma for prodmo 24159. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  G  =  ( j  e.  NN  |->  [_ ( f `  j
 )  /  k ]_ B )   &    |-  H  =  ( j  e.  NN  |->  [_ ( K `  j ) 
 /  k ]_ B )   &    |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN ) )   &    |-  ( ph  ->  f : ( 1 ... M ) -1-1-onto-> A )   &    |-  ( ph  ->  K : ( 1 ...
 N ) -1-1-onto-> A )   =>    |-  ( ph  ->  (  seq  1 (  x.  ,  G ) `  M )  =  (  seq  1 (  x.  ,  H ) `  N ) )
 
Theoremprodmolem2a 24157* Lemma for prodmo 24159. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  G  =  ( j  e.  NN  |->  [_ ( f `  j
 )  /  k ]_ B )   &    |-  H  =  ( j  e.  NN  |->  [_ ( K `  j ) 
 /  k ]_ B )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  C_  ( ZZ>=
 `  M ) )   &    |-  ( ph  ->  f :
 ( 1 ... N )
 -1-1-onto-> A )   &    |-  ( ph  ->  K 
 Isom  <  ,  <  (
 ( 1 ... ( # `
  A ) ) ,  A ) )   =>    |-  ( ph  ->  seq  M (  x.  ,  F )  ~~>  (  seq  1 (  x. 
 ,  G ) `  N ) )
 
Theoremprodmolem2 24158* Lemma for prodmo 24159. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  G  =  ( j  e.  NN  |->  [_ ( f `  j
 )  /  k ]_ B )   =>    |-  ( ( ph  /\  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  E. n  e.  ( ZZ>= `  m ) E. y ( y  =/=  0  /\  seq  n (  x.  ,  F )  ~~>  y )  /\  seq  m (  x.  ,  F )  ~~>  x ) )  ->  ( E. m  e.  NN  E. f ( f : ( 1 ... m )
 -1-1-onto-> A  /\  z  =  ( 
 seq  1 (  x. 
 ,  G ) `  m ) )  ->  x  =  z )
 )
 
Theoremprodmo 24159* A product has at most one limit. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  G  =  ( j  e.  NN  |->  [_ ( f `  j
 )  /  k ]_ B )   =>    |-  ( ph  ->  E* x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  E. n  e.  ( ZZ>= `  m ) E. y ( y  =/=  0  /\  seq  n (  x.  ,  F )  ~~>  y )  /\  seq  m (  x.  ,  F )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1
 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  x.  ,  G ) `  m ) ) ) )
 
Theoremzprod 24160* Series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 5-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq 
 n (  x.  ,  F )  ~~>  y ) )   &    |-  ( ph  ->  A  C_  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
 1 ) )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  -> k  e.  A B  =  (  ~~>  ` 
 seq  M (  x.  ,  F ) ) )
 
Theoremiprod 24161* Series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 5-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq 
 n (  x.  ,  F )  ~~>  y ) )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  B  e.  CC )   =>    |-  ( ph  -> k  e.  Z B  =  (  ~~>  `  seq  M (  x.  ,  F ) ) )
 
Theoremzprodn0 24162* Non-zero series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 6-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  X  =/=  0 )   &    |-  ( ph  ->  seq 
 M (  x.  ,  F )  ~~>  X )   &    |-  ( ph  ->  A  C_  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
 1 ) )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  -> k  e.  A B  =  X )
 
Theoremiprodn0 24163* Non-zero series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 6-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  X  =/=  0 )   &    |-  ( ph  ->  seq 
 M (  x.  ,  F )  ~~>  X )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  B  e.  CC )   =>    |-  ( ph  -> k  e.  Z B  =  X )
 
Theoremfprod 24164* The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.)
 |-  (
 k  =  ( F `
  n )  ->  B  =  C )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 M ) -1-1-onto-> A )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  n  e.  ( 1 ... M ) )  ->  ( G `
  n )  =  C )   =>    |-  ( ph  -> k  e.  A B  =  ( 
 seq  1 (  x. 
 ,  G ) `  M ) )
 
Theoremprodf1 24165 The value of the partial products in a one-valued infinite product. (Contributed by Scott Fenton, 5-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( N  e.  Z  ->  ( 
 seq  M (  x.  ,  ( Z  X.  { 1 } ) ) `  N )  =  1
 )
 
Theoremprodf1f 24166 A one-valued infinite product is equal to the constant one function. (Contributed by Scott Fenton, 5-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  seq 
 M (  x.  ,  ( Z  X.  { 1 } ) )  =  ( Z  X.  {
 1 } ) )
 
Theoremprodfclim1 24167 The constant one product converges to one. (Contributed by Scott Fenton, 5-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  seq 
 M (  x.  ,  ( Z  X.  { 1 } ) )  ~~>  1 )
 
Theoremcprod0 24168 A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)
 |- k  e.  (/) A  =  1
 
Theoremprod1 24169* Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.)
 |-  (
 ( A  C_  ( ZZ>=
 `  M )  \/  A  e.  Fin )  ->
 k  e.  A 1  =  1 )
 
Theoremprod2id 24170* The second class argument to a product can be chosen so that it is always a set. (Contributed by Scott Fenton, 7-Dec-2017.)
 |- k  e.  A B  = k  e.  A (  _I  `  B )
 
Theoremprodfc 24171* A lemma to facilitate conversions from the function form to the class-variable form of a product. (Contributed by Scott Fenton, 7-Dec-2017.)
 |- j  e.  A ( ( k  e.  A  |->  B ) `
  j )  =
 k  e.  A B
 
Theoremfprodf1o 24172* Re-index a finite product using a bijection. (Contributed by Scott Fenton, 7-Dec-2017.)
 |-  (
 k  =  G  ->  B  =  D )   &    |-  ( ph  ->  C  e.  Fin )   &    |-  ( ph  ->  F : C -1-1-onto-> A )   &    |-  ( ( ph  /\  n  e.  C ) 
 ->  ( F `  n )  =  G )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  -> k  e.  A B  = n  e.  C D )
 
18.7.7  Greatest common divisor and divisibility
 
Theorempdivsq 24173 Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( P  e.  Prime  /\  M  e.  ZZ )  ->  ( P  ||  M  <->  P 
 ||  ( M ^
 2 ) ) )
 
Theoremdvdspw 24174 Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  ->  ( K  ||  M  ->  K  ||  ( M ^ N ) ) )
 
Theoremgcd32 24175 Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( A  gcd  B )  gcd  C )  =  ( ( A 
 gcd  C )  gcd  B ) )
 
Theoremgcdabsorb 24176 Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  gcd  B )  =  ( A  gcd  B ) )
 
18.7.8  Properties of relationships
 
Theorembrtp 24177 A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  X  e.  _V   &    |-  Y  e.  _V   =>    |-  ( X { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } Y  <->  ( ( X  =  A  /\  Y  =  B )  \/  ( X  =  C  /\  Y  =  D )  \/  ( X  =  E  /\  Y  =  F ) ) )
 
Theoremdftr6 24178 A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)
 |-  A  e.  _V   =>    |-  ( Tr  A  <->  A  e.  ( _V  \  ran  ( (  _E  o.  _E  )  \  _E  ) ) )
 
Theoremcoep 24179* Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A (  _E  o.  R ) B  <->  E. x  e.  B  A R x )
 
Theoremcoepr 24180* Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A ( R  o.  `'  _E  ) B  <->  E. x  e.  A  x R B )
 
Theoremdffr5 24181 A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)
 |-  ( R  Fr  A  <->  ( ~P A  \  { (/) } )  C_  ran  (  _E  \  (  _E  o.  `' R ) ) )
 
Theoremdfso2 24182 Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)
 |-  ( R  Or  A  <->  ( R  Po  A  /\  ( A  X.  A )  C_  ( R  u.  (  _I  u.  `' R ) ) ) )
 
Theoremdfpo2 24183 Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.)
 |-  ( R  Po  A  <->  ( ( R  i^i  (  _I  |`  A ) )  =  (/)  /\  (
 ( R  i^i  ( A  X.  A ) )  o.  ( R  i^i  ( A  X.  A ) ) )  C_  R ) )
 
Theorembr8 24184* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  (
 e  =  E  ->  ( ta  <->  et ) )   &    |-  (
 f  =  F  ->  ( et  <->  ze ) )   &    |-  (
 g  =  G  ->  ( ze  <->  si ) )   &    |-  ( h  =  H  ->  (
 si 
 <->  rh ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  E. e  e.  P  E. f  e.  P  E. g  e.  P  E. h  e.  P  ( p  = 
 <. <. a ,  b >. ,  <. c ,  d >.
 >.  /\  q  =  <. <.
 e ,  f >. , 
 <. g ,  h >. >.  /\  ph ) }   =>    |-  ( ( ( X  e.  S  /\  A  e.  Q  /\  B  e.  Q )  /\  ( C  e.  Q  /\  D  e.  Q  /\  E  e.  Q )  /\  ( F  e.  Q  /\  G  e.  Q  /\  H  e.  Q )
 )  ->  ( <. <. A ,  B >. , 
 <. C ,  D >. >. R <. <. E ,  F >. ,  <. G ,  H >.
 >. 
 <->  rh ) )
 
Theorembr6 24185* Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  (
 e  =  E  ->  ( ta  <->  et ) )   &    |-  (
 f  =  F  ->  ( et  <->  ze ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  E. e  e.  P  E. f  e.  P  ( p  = 
 <. a ,  <. b ,  c >. >.  /\  q  =  <. d ,  <. e ,  f >. >.  /\  ph ) }   =>    |-  (
 ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q  /\  C  e.  Q )  /\  ( D  e.  Q  /\  E  e.  Q  /\  F  e.  Q )
 )  ->  ( <. A ,  <. B ,  C >.
 >. R <. D ,  <. E ,  F >. >.  <->  ze ) )
 
Theorembr4 24186* Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( p  = 
 <. a ,  b >.  /\  q  =  <. c ,  d >.  /\  ph ) }   =>    |-  ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q )  /\  ( C  e.  Q  /\  D  e.  Q ) )  ->  ( <. A ,  B >. R <. C ,  D >.  <->  ta ) )
 
Theoremdfres3 24187 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  ran  A )
 )
 
Theoremcnvco1 24188 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
 |-  `' ( `' A  o.  B )  =  ( `' B  o.  A )
 
Theoremcnvco2 24189 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
 |-  `' ( A  o.  `' B )  =  ( B  o.  `' A )
 
Theoremeldm3 24190 Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)
 |-  ( A  e.  dom  B  <->  ( B  |`  { A } )  =/=  (/) )
 
Theoremelrn3 24191 Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)
 |-  ( A  e.  ran  B  <->  ( B  i^i  ( _V  X.  { A } ) )  =/=  (/) )
 
18.7.9  Properties of functions and mappings
 
Theoremfunpsstri 24192 A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)
 |-  (
 ( Fun  H  /\  ( F  C_  H  /\  G  C_  H )  /\  ( dom  F  C_  dom  G  \/  dom  G  C_  dom  F ) )  ->  ( F 
 C.  G  \/  F  =  G  \/  G  C.  F ) )
 
Theoremfundmpss 24193 If a class  F is a proper subset of a function  G, then  dom  F  C.  dom  G. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  ( Fun  G  ->  ( F  C.  G  ->  dom  F  C.  dom 
 G ) )
 
Theoremfvresval 24194 The value of a function at a restriction is either null or the same as the function itself. (Contributed by Scott Fenton, 4-Sep-2011.)
 |-  (
 ( ( F  |`  B ) `
  A )  =  ( F `  A )  \/  ( ( F  |`  B ) `  A )  =  (/) )
 
Theoremmptrel 24195 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Rel  ( x  e.  A  |->  B )
 
Theoremfunsseq 24196 Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  (
 ( Fun  F  /\  Fun 
 G  /\  dom  F  =  dom  G )  ->  ( F  =  G  <->  F  C_  G ) )
 
Theoremfununiq 24197 The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( Fun  F  ->  ( ( A F B  /\  A F C ) 
 ->  B  =  C ) )
 
Theoremfunbreq 24198 An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( ( Fun  F  /\  A F B ) 
 ->  ( A F C  <->  B  =  C ) )
 
Theoremmpteq12d 24199 An equality inference for the maps to notation. Compare mpteq12dv 4114. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/ x ph   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
 
Theoremfprb 24200* A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  =/=  B  ->  ( F : { A ,  B } --> R  <->  E. x  e.  R  E. y  e.  R  F  =  { <. A ,  x >. ,  <. B ,  y >. } ) )
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