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Theorem List for Intuitionistic Logic Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfprodmul 11601* The product of two finite products. (Contributed by Scott Fenton, 14-Dec-2017.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  k  e.  A )  ->  C  e.  CC )   =>    |-  ( ph  ->  prod_ k  e.  A  ( B  x.  C )  =  ( prod_ k  e.  A  B  x.  prod_ k  e.  A  C ) )
 
Theoremprodsnf 11602* A product of a singleton is the term. A version of prodsn 11603 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  F/_ k B   &    |-  ( k  =  M  ->  A  =  B )   =>    |-  ( ( M  e.  V  /\  B  e.  CC )  ->  prod_ k  e.  { M } A  =  B )
 
Theoremprodsn 11603* A product of a singleton is the term. (Contributed by Scott Fenton, 14-Dec-2017.)
 |-  ( k  =  M  ->  A  =  B )   =>    |-  ( ( M  e.  V  /\  B  e.  CC )  ->  prod_ k  e.  { M } A  =  B )
 
Theoremfprod1 11604* A finite product of only one term is the term itself. (Contributed by Scott Fenton, 14-Dec-2017.)
 |-  ( k  =  M  ->  A  =  B )   =>    |-  ( ( M  e.  ZZ  /\  B  e.  CC )  ->  prod_ k  e.  ( M ... M ) A  =  B )
 
Theoremclimprod1 11605 The limit of a product over one. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   =>    |-  ( ph  ->  seq
 M (  x.  ,  ( Z  X.  { 1 } ) )  ~~>  1 )
 
Theoremfprodsplitdc 11606* Split a finite product into two parts. New proofs should use fprodsplit 11607 which is the same but with one fewer hypothesis. (Contributed by Scott Fenton, 16-Dec-2017.) (New usage is discouraged.)
 |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  U  =  ( A  u.  B ) )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ph  ->  A. j  e.  U DECID  j  e.  A )   &    |-  ( ( ph  /\  k  e.  U )  ->  C  e.  CC )   =>    |-  ( ph  ->  prod_ k  e.  U  C  =  (
 prod_ k  e.  A  C  x.  prod_ k  e.  B  C ) )
 
Theoremfprodsplit 11607* Split a finite product into two parts. (Contributed by Scott Fenton, 16-Dec-2017.)
 |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  U  =  ( A  u.  B ) )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ( ph  /\  k  e.  U )  ->  C  e.  CC )   =>    |-  ( ph  ->  prod_ k  e.  U  C  =  (
 prod_ k  e.  A  C  x.  prod_ k  e.  B  C ) )
 
Theoremfprodm1 11608* Separate out the last term in a finite product. (Contributed by Scott Fenton, 16-Dec-2017.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( k  =  N  ->  A  =  B )   =>    |-  ( ph  ->  prod_ k  e.  ( M ... N ) A  =  ( prod_ k  e.  ( M
 ... ( N  -  1 ) ) A  x.  B ) )
 
Theoremfprod1p 11609* Separate out the first term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( k  =  M  ->  A  =  B )   =>    |-  ( ph  ->  prod_ k  e.  ( M ... N ) A  =  ( B  x.  prod_ k  e.  (
 ( M  +  1 ) ... N ) A ) )
 
Theoremfprodp1 11610* Multiply in the last term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... ( N  +  1 )
 ) )  ->  A  e.  CC )   &    |-  ( k  =  ( N  +  1 )  ->  A  =  B )   =>    |-  ( ph  ->  prod_ k  e.  ( M ... ( N  +  1 )
 ) A  =  (
 prod_ k  e.  ( M ... N ) A  x.  B ) )
 
Theoremfprodm1s 11611* Separate out the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )   =>    |-  ( ph  ->  prod_ k  e.  ( M ... N ) A  =  ( prod_ k  e.  ( M
 ... ( N  -  1 ) ) A  x.  [_ N  /  k ]_ A ) )
 
Theoremfprodp1s 11612* Multiply in the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... ( N  +  1 )
 ) )  ->  A  e.  CC )   =>    |-  ( ph  ->  prod_ k  e.  ( M ... ( N  +  1 )
 ) A  =  (
 prod_ k  e.  ( M ... N ) A  x.  [_ ( N  +  1 )  /  k ]_ A ) )
 
Theoremprodsns 11613* A product of the singleton is the term. (Contributed by Scott Fenton, 25-Dec-2017.)
 |-  ( ( M  e.  V  /\  [_ M  /  k ]_ A  e.  CC )  ->  prod_ k  e.  { M } A  =  [_ M  /  k ]_ A )
 
Theoremfprodunsn 11614* Multiply in an additional term in a finite product. See also fprodsplitsn 11643 which is the same but with a  F/ k
ph hypothesis in place of the distinct variable condition between  ph and  k. (Contributed by Jim Kingdon, 16-Aug-2024.)
 |-  F/_ k D   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  -.  B  e.  A )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( k  =  B  ->  C  =  D )   =>    |-  ( ph  ->  prod_ k  e.  ( A  u.  { B } ) C  =  ( prod_ k  e.  A  C  x.  D ) )
 
Theoremfprodcl2lem 11615* Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.) (Revised by Jim Kingdon, 17-Aug-2024.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  x.  y )  e.  S )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  S )   &    |-  ( ph  ->  A  =/=  (/) )   =>    |-  ( ph  ->  prod_ k  e.  A  B  e.  S )
 
Theoremfprodcllem 11616* Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  x.  y )  e.  S )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  S )   &    |-  ( ph  ->  1  e.  S )   =>    |-  ( ph  ->  prod_
 k  e.  A  B  e.  S )
 
Theoremfprodcl 11617* Closure of a finite product of complex numbers. (Contributed by Scott Fenton, 14-Dec-2017.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  prod_ k  e.  A  B  e.  CC )
 
Theoremfprodrecl 11618* Closure of a finite product of real numbers. (Contributed by Scott Fenton, 14-Dec-2017.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   =>    |-  ( ph  ->  prod_ k  e.  A  B  e.  RR )
 
Theoremfprodzcl 11619* Closure of a finite product of integers. (Contributed by Scott Fenton, 14-Dec-2017.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  ZZ )   =>    |-  ( ph  ->  prod_ k  e.  A  B  e.  ZZ )
 
Theoremfprodnncl 11620* Closure of a finite product of positive integers. (Contributed by Scott Fenton, 14-Dec-2017.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  NN )   =>    |-  ( ph  ->  prod_ k  e.  A  B  e.  NN )
 
Theoremfprodrpcl 11621* Closure of a finite product of positive reals. (Contributed by Scott Fenton, 14-Dec-2017.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR+ )   =>    |-  ( ph  ->  prod_ k  e.  A  B  e.  RR+ )
 
Theoremfprodnn0cl 11622* Closure of a finite product of nonnegative integers. (Contributed by Scott Fenton, 14-Dec-2017.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  NN0 )   =>    |-  ( ph  ->  prod_ k  e.  A  B  e.  NN0 )
 
Theoremfprodcllemf 11623* Finite product closure lemma. A version of fprodcllem 11616 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |- 
 F/ k ph   &    |-  ( ph  ->  S 
 C_  CC )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  x.  y )  e.  S )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  S )   &    |-  ( ph  ->  1  e.  S )   =>    |-  ( ph  ->  prod_
 k  e.  A  B  e.  S )
 
Theoremfprodreclf 11624* Closure of a finite product of real numbers. A version of fprodrecl 11618 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |- 
 F/ k ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  RR )   =>    |-  ( ph  ->  prod_ k  e.  A  B  e.  RR )
 
Theoremfprodfac 11625* Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  ( A  e.  NN0  ->  ( ! `  A )  =  prod_ k  e.  (
 1 ... A ) k )
 
Theoremfprodabs 11626* The absolute value of a finite product. (Contributed by Scott Fenton, 25-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  prod_ k  e.  ( M ... N ) A )  =  prod_ k  e.  ( M ... N ) ( abs `  A ) )
 
Theoremfprodeq0 11627* Any finite product containing a zero term is itself zero. (Contributed by Scott Fenton, 27-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  A  e.  CC )   &    |-  ( ( ph  /\  k  =  N ) 
 ->  A  =  0 )   =>    |-  ( ( ph  /\  K  e.  ( ZZ>= `  N )
 )  ->  prod_ k  e.  ( M ... K ) A  =  0
 )
 
Theoremfprodshft 11628* Shift the index of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ( ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( j  =  ( k  -  K )  ->  A  =  B )   =>    |-  ( ph  ->  prod_ j  e.  ( M ... N ) A  =  prod_ k  e.  ( ( M  +  K ) ... ( N  +  K ) ) B )
 
Theoremfprodrev 11629* Reversal of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ( ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )   &    |-  ( j  =  ( K  -  k
 )  ->  A  =  B )   =>    |-  ( ph  ->  prod_ j  e.  ( M ... N ) A  =  prod_ k  e.  ( ( K  -  N ) ... ( K  -  M ) ) B )
 
Theoremfprodconst 11630* The product of constant terms ( k is not free in  B). (Contributed by Scott Fenton, 12-Jan-2018.)
 |-  ( ( A  e.  Fin  /\  B  e.  CC )  -> 
 prod_ k  e.  A  B  =  ( B ^ ( `  A )
 ) )
 
Theoremfprodap0 11631* A finite product of nonzero terms is nonzero. (Contributed by Scott Fenton, 15-Jan-2018.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B #  0 )   =>    |-  ( ph  ->  prod_ k  e.  A  B #  0 )
 
Theoremfprod2dlemstep 11632* Lemma for fprod2d 11633- induction step. (Contributed by Scott Fenton, 30-Jan-2018.)
 |-  ( z  =  <. j ,  k >.  ->  D  =  C )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  j  e.  A )  ->  B  e.  Fin )   &    |-  ( ( ph  /\  ( j  e.  A  /\  k  e.  B ) )  ->  C  e.  CC )   &    |-  ( ph  ->  -.  y  e.  x )   &    |-  ( ph  ->  ( x  u.  { y } )  C_  A )   &    |-  ( ph  ->  x  e.  Fin )   &    |-  ( ps 
 <-> 
 prod_ j  e.  x  prod_ k  e.  B  C  =  prod_ z  e.  U_  j  e.  x  ( { j }  X.  B ) D )   =>    |-  ( ( ph  /\  ps )  ->  prod_ j  e.  ( x  u.  { y }
 ) prod_ k  e.  B  C  =  prod_ z  e.  U_  j  e.  ( x  u.  { y }
 ) ( { j }  X.  B ) D )
 
Theoremfprod2d 11633* Write a double product as a product over a two-dimensional region. Compare fsum2d 11445. (Contributed by Scott Fenton, 30-Jan-2018.)
 |-  ( z  =  <. j ,  k >.  ->  D  =  C )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  j  e.  A )  ->  B  e.  Fin )   &    |-  ( ( ph  /\  ( j  e.  A  /\  k  e.  B ) )  ->  C  e.  CC )   =>    |-  ( ph  ->  prod_ j  e.  A  prod_ k  e.  B  C  =  prod_ z  e.  U_  j  e.  A  ( { j }  X.  B ) D )
 
Theoremfprodxp 11634* Combine two products into a single product over the cartesian product. (Contributed by Scott Fenton, 1-Feb-2018.)
 |-  ( z  =  <. j ,  k >.  ->  D  =  C )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ( ph  /\  (
 j  e.  A  /\  k  e.  B )
 )  ->  C  e.  CC )   =>    |-  ( ph  ->  prod_ j  e.  A  prod_ k  e.  B  C  =  prod_ z  e.  ( A  X.  B ) D )
 
Theoremfprodcnv 11635* Transform a product region using the converse operation. (Contributed by Scott Fenton, 1-Feb-2018.)
 |-  ( x  =  <. j ,  k >.  ->  B  =  D )   &    |-  ( y  = 
 <. k ,  j >.  ->  C  =  D )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  Rel  A )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  prod_ x  e.  A  B  =  prod_ y  e.  `'  A C )
 
Theoremfprodcom2fi 11636* Interchange order of multiplication. Note that  B ( j ) and  D ( k ) are not necessarily constant expressions. (Contributed by Scott Fenton, 1-Feb-2018.) (Proof shortened by JJ, 2-Aug-2021.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  C  e.  Fin )   &    |-  (
 ( ph  /\  j  e.  A )  ->  B  e.  Fin )   &    |-  ( ( ph  /\  k  e.  C ) 
 ->  D  e.  Fin )   &    |-  ( ph  ->  ( ( j  e.  A  /\  k  e.  B )  <->  ( k  e.  C  /\  j  e.  D ) ) )   &    |-  ( ( ph  /\  (
 j  e.  A  /\  k  e.  B )
 )  ->  E  e.  CC )   =>    |-  ( ph  ->  prod_ j  e.  A  prod_ k  e.  B  E  =  prod_ k  e.  C  prod_ j  e.  D  E )
 
Theoremfprodcom 11637* Interchange product order. (Contributed by Scott Fenton, 2-Feb-2018.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  (
 ( ph  /\  ( j  e.  A  /\  k  e.  B ) )  ->  C  e.  CC )   =>    |-  ( ph  ->  prod_ j  e.  A  prod_ k  e.  B  C  =  prod_ k  e.  B  prod_ j  e.  A  C )
 
Theoremfprod0diagfz 11638* Two ways to express "the product of  A ( j ,  k ) over the triangular region  M  <_  j,  M  <_  k,  j  +  k  <_  N. Compare fisum0diag 11451. (Contributed by Scott Fenton, 2-Feb-2018.)
 |-  ( ( ph  /\  (
 j  e.  ( 0
 ... N )  /\  k  e.  ( 0 ... ( N  -  j
 ) ) ) ) 
 ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  prod_ j  e.  ( 0 ... N ) prod_ k  e.  (
 0 ... ( N  -  j ) ) A  =  prod_ k  e.  (
 0 ... N ) prod_
 j  e.  ( 0
 ... ( N  -  k ) ) A )
 
Theoremfprodrec 11639* The finite product of reciprocals is the reciprocal of the product. (Contributed by Jim Kingdon, 28-Aug-2024.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B #  0 )   =>    |-  ( ph  ->  prod_ k  e.  A  ( 1  /  B )  =  (
 1  /  prod_ k  e.  A  B ) )
 
Theoremfproddivap 11640* The quotient of two finite products. (Contributed by Scott Fenton, 15-Jan-2018.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  k  e.  A )  ->  C  e.  CC )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C #  0 )   =>    |-  ( ph  ->  prod_
 k  e.  A  ( B  /  C )  =  ( prod_ k  e.  A  B  /  prod_ k  e.  A  C ) )
 
Theoremfproddivapf 11641* The quotient of two finite products. A version of fproddivap 11640 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |- 
 F/ k ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  A )  ->  C #  0 )   =>    |-  ( ph  ->  prod_ k  e.  A  ( B  /  C )  =  ( prod_ k  e.  A  B  / 
 prod_ k  e.  A  C ) )
 
Theoremfprodsplitf 11642* Split a finite product into two parts. A version of fprodsplit 11607 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |- 
 F/ k ph   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  U  =  ( A  u.  B ) )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ( ph  /\  k  e.  U )  ->  C  e.  CC )   =>    |-  ( ph  ->  prod_ k  e.  U  C  =  (
 prod_ k  e.  A  C  x.  prod_ k  e.  B  C ) )
 
Theoremfprodsplitsn 11643* Separate out a term in a finite product. See also fprodunsn 11614 which is the same but with a distinct variable condition in place of  F/ k ph. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |- 
 F/ k ph   &    |-  F/_ k D   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  -.  B  e.  A )   &    |-  ( ( ph  /\  k  e.  A )  ->  C  e.  CC )   &    |-  ( k  =  B  ->  C  =  D )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  prod_
 k  e.  ( A  u.  { B }
 ) C  =  (
 prod_ k  e.  A  C  x.  D ) )
 
Theoremfprodsplit1f 11644* Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |- 
 F/ k ph   &    |-  ( ph  ->  F/_ k D )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  A )   &    |-  (
 ( ph  /\  k  =  C )  ->  B  =  D )   =>    |-  ( ph  ->  prod_ k  e.  A  B  =  ( D  x.  prod_ k  e.  ( A  \  { C } ) B ) )
 
Theoremfprodclf 11645* Closure of a finite product of complex numbers. A version of fprodcl 11617 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |- 
 F/ k ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  prod_ k  e.  A  B  e.  CC )
 
Theoremfprodap0f 11646* A finite product of terms apart from zero is apart from zero. A version of fprodap0 11631 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by Jim Kingdon, 30-Aug-2024.)
 |- 
 F/ k ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B #  0 )   =>    |-  ( ph  ->  prod_
 k  e.  A  B #  0 )
 
Theoremfprodge0 11647* If all the terms of a finite product are nonnegative, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |- 
 F/ k ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  0  <_  B )   =>    |-  ( ph  ->  0  <_  prod_
 k  e.  A  B )
 
Theoremfprodeq0g 11648* Any finite product containing a zero term is itself zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |- 
 F/ k ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  A )   &    |-  (
 ( ph  /\  k  =  C )  ->  B  =  0 )   =>    |-  ( ph  ->  prod_
 k  e.  A  B  =  0 )
 
Theoremfprodge1 11649* If all of the terms of a finite product are greater than or equal to  1, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |- 
 F/ k ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  1  <_  B )   =>    |-  ( ph  ->  1  <_  prod_
 k  e.  A  B )
 
Theoremfprodle 11650* If all the terms of two finite products are nonnegative and compare, so do the two products. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |- 
 F/ k ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  0  <_  B )   &    |-  ( ( ph  /\  k  e.  A )  ->  C  e.  RR )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  <_  C )   =>    |-  ( ph  ->  prod_ k  e.  A  B  <_  prod_ k  e.  A  C )
 
Theoremfprodmodd 11651* If all factors of two finite products are equal modulo  M, the products are equal modulo  M. (Contributed by AV, 7-Jul-2021.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  A )  ->  C  e.  ZZ )   &    |-  ( ph  ->  M  e.  NN )   &    |-  (
 ( ph  /\  k  e.  A )  ->  ( B  mod  M )  =  ( C  mod  M ) )   =>    |-  ( ph  ->  ( prod_ k  e.  A  B  mod  M )  =  (
 prod_ k  e.  A  C  mod  M ) )
 
4.9  Elementary trigonometry
 
4.9.1  The exponential, sine, and cosine functions
 
Syntaxce 11652 Extend class notation to include the exponential function.
 class  exp
 
Syntaxceu 11653 Extend class notation to include Euler's constant  _e = 2.71828....
 class  _e
 
Syntaxcsin 11654 Extend class notation to include the sine function.
 class  sin
 
Syntaxccos 11655 Extend class notation to include the cosine function.
 class  cos
 
Syntaxctan 11656 Extend class notation to include the tangent function.
 class  tan
 
Syntaxcpi 11657 Extend class notation to include the constant pi,  pi = 3.14159....
 class  pi
 
Definitiondf-ef 11658* Define the exponential function. Its value at the complex number  A is  ( exp `  A
) and is called the "exponential of  A"; see efval 11671. (Contributed by NM, 14-Mar-2005.)
 |- 
 exp  =  ( x  e.  CC  |->  sum_ k  e.  NN0  ( ( x ^
 k )  /  ( ! `  k ) ) )
 
Definitiondf-e 11659 Define Euler's constant  _e = 2.71828.... (Contributed by NM, 14-Mar-2005.)
 |-  _e  =  ( exp `  1 )
 
Definitiondf-sin 11660 Define the sine function. (Contributed by NM, 14-Mar-2005.)
 |- 
 sin  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) ) 
 /  ( 2  x.  _i ) ) )
 
Definitiondf-cos 11661 Define the cosine function. (Contributed by NM, 14-Mar-2005.)
 |- 
 cos  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x ) )  +  ( exp `  ( -u _i  x.  x ) ) ) 
 /  2 ) )
 
Definitiondf-tan 11662 Define the tangent function. We define it this way for cmpt 4066, which requires the form  ( x  e.  A  |->  B ). (Contributed by Mario Carneiro, 14-Mar-2014.)
 |- 
 tan  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  ( ( sin `  x )  /  ( cos `  x ) ) )
 
Definitiondf-pi 11663 Define the constant pi,  pi = 3.14159..., which is the smallest positive number whose sine is zero. Definition of  pi in [Gleason] p. 311. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by AV, 14-Sep-2020.)
 |-  pi  = inf ( (
 RR+  i^i  ( `' sin " { 0 } )
 ) ,  RR ,  <  )
 
Theoremeftcl 11664 Closure of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 11-Sep-2007.)
 |-  ( ( A  e.  CC  /\  K  e.  NN0 )  ->  ( ( A ^ K )  /  ( ! `  K ) )  e.  CC )
 
Theoremreeftcl 11665 The terms of the series expansion of the exponential function at a real number are real. (Contributed by Paul Chapman, 15-Jan-2008.)
 |-  ( ( A  e.  RR  /\  K  e.  NN0 )  ->  ( ( A ^ K )  /  ( ! `  K ) )  e.  RR )
 
Theoremeftabs 11666 The absolute value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 23-Nov-2007.)
 |-  ( ( A  e.  CC  /\  K  e.  NN0 )  ->  ( abs `  (
 ( A ^ K )  /  ( ! `  K ) ) )  =  ( ( ( abs `  A ) ^ K )  /  ( ! `  K ) ) )
 
Theoremeftvalcn 11667* The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 8-Dec-2022.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( F `  N )  =  (
 ( A ^ N )  /  ( ! `  N ) ) )
 
Theoremefcllemp 11668* Lemma for efcl 11674. The series that defines the exponential function converges. The ratio test cvgratgt0 11543 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  ( 2  x.  ( abs `  A ) )  <  K )   =>    |-  ( ph  ->  seq 0
 (  +  ,  F )  e.  dom  ~~>  )
 
Theoremefcllem 11669* Lemma for efcl 11674. The series that defines the exponential function converges. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( A  e.  CC  ->  seq 0 (  +  ,  F )  e.  dom  ~~>  )
 
Theoremef0lem 11670* The series defining the exponential function converges in the (trivial) case of a zero argument. (Contributed by Steve Rodriguez, 7-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( A  =  0  ->  seq 0 (  +  ,  F )  ~~>  1 )
 
Theoremefval 11671* Value of the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
 |-  ( A  e.  CC  ->  ( exp `  A )  =  sum_ k  e. 
 NN0  ( ( A ^ k )  /  ( ! `  k ) ) )
 
Theoremesum 11672 Value of Euler's constant  _e = 2.71828.... (Contributed by Steve Rodriguez, 5-Mar-2006.)
 |-  _e  =  sum_ k  e.  NN0  ( 1  /  ( ! `  k ) )
 
Theoremeff 11673 Domain and codomain of the exponential function. (Contributed by Paul Chapman, 22-Oct-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
 |- 
 exp : CC --> CC
 
Theoremefcl 11674 Closure law for the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
 |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
 
Theoremefval2 11675* Value of the exponential function. (Contributed by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( A  e.  CC  ->  ( exp `  A )  =  sum_ k  e. 
 NN0  ( F `  k ) )
 
Theoremefcvg 11676* The series that defines the exponential function converges to it. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( A  e.  CC  ->  seq 0 (  +  ,  F )  ~~>  ( exp `  A ) )
 
Theoremefcvgfsum 11677* Exponential function convergence in terms of a sequence of partial finite sums. (Contributed by NM, 10-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  sum_ k  e.  ( 0 ... n ) ( ( A ^ k )  /  ( ! `  k ) ) )   =>    |-  ( A  e.  CC  ->  F  ~~>  ( exp `  A ) )
 
Theoremreefcl 11678 The exponential function is real if its argument is real. (Contributed by NM, 27-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR )
 
Theoremreefcld 11679 The exponential function is real if its argument is real. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( exp `  A )  e. 
 RR )
 
Theoremere 11680 Euler's constant  _e = 2.71828... is a real number. (Contributed by NM, 19-Mar-2005.) (Revised by Steve Rodriguez, 8-Mar-2006.)
 |-  _e  e.  RR
 
Theoremege2le3 11681 Euler's constant  _e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 20-Mar-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
 |-  F  =  ( n  e.  NN  |->  ( 2  x.  ( ( 1 
 /  2 ) ^ n ) ) )   &    |-  G  =  ( n  e.  NN0  |->  ( 1  /  ( ! `  n ) ) )   =>    |-  ( 2  <_  _e  /\  _e  <_  3 )
 
Theoremef0 11682 Value of the exponential function at 0. Equation 2 of [Gleason] p. 308. (Contributed by Steve Rodriguez, 27-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  ( exp `  0
 )  =  1
 
Theoremefcj 11683 The exponential of a complex conjugate. Equation 3 of [Gleason] p. 308. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  ( A  e.  CC  ->  ( exp `  ( * `  A ) )  =  ( * `  ( exp `  A )
 ) )
 
Theoremefaddlem 11684* Lemma for efadd 11685 (exponential function addition law). (Contributed by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   &    |-  G  =  ( n  e.  NN0  |->  ( ( B ^ n )  /  ( ! `  n ) ) )   &    |-  H  =  ( n  e.  NN0  |->  ( ( ( A  +  B ) ^ n )  /  ( ! `  n ) ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( exp `  ( A  +  B ) )  =  ( ( exp `  A )  x.  ( exp `  B ) ) )
 
Theoremefadd 11685 Sum of exponents law for exponential function. (Contributed by NM, 10-Jan-2006.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( exp `  ( A  +  B )
 )  =  ( ( exp `  A )  x.  ( exp `  B ) ) )
 
Theoremefcan 11686 Cancellation law for exponential function. Equation 27 of [Rudin] p. 164. (Contributed by NM, 13-Jan-2006.)
 |-  ( A  e.  CC  ->  ( ( exp `  A )  x.  ( exp `  -u A ) )  =  1
 )
 
Theoremefap0 11687 The exponential of a complex number is apart from zero. (Contributed by Jim Kingdon, 12-Dec-2022.)
 |-  ( A  e.  CC  ->  ( exp `  A ) #  0 )
 
Theoremefne0 11688 The exponential of a complex number is nonzero. Corollary 15-4.3 of [Gleason] p. 309. The same result also holds with not equal replaced by apart, as seen at efap0 11687 (which will be more useful in most contexts). (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  ( A  e.  CC  ->  ( exp `  A )  =/=  0 )
 
Theoremefneg 11689 The exponential of the opposite is the inverse of the exponential. (Contributed by Mario Carneiro, 10-May-2014.)
 |-  ( A  e.  CC  ->  ( exp `  -u A )  =  ( 1  /  ( exp `  A ) ) )
 
Theoremeff2 11690 The exponential function maps the complex numbers to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)
 |- 
 exp : CC --> ( CC  \  { 0 } )
 
Theoremefsub 11691 Difference of exponents law for exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( exp `  ( A  -  B ) )  =  ( ( exp `  A )  /  ( exp `  B ) ) )
 
Theoremefexp 11692 The exponential of an integer power. Corollary 15-4.4 of [Gleason] p. 309, restricted to integers. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( N  x.  A ) )  =  ( ( exp `  A ) ^ N ) )
 
Theoremefzval 11693 Value of the exponential function for integers. Special case of efval 11671. Equation 30 of [Rudin] p. 164. (Contributed by Steve Rodriguez, 15-Sep-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  ( N  e.  ZZ  ->  ( exp `  N )  =  ( _e ^ N ) )
 
Theoremefgt0 11694 The exponential of a real number is greater than 0. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( A  e.  RR  ->  0  <  ( exp `  A ) )
 
Theoremrpefcl 11695 The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.)
 |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR+ )
 
Theoremrpefcld 11696 The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( exp `  A )  e.  RR+ )
 
Theoremeftlcvg 11697* The tail series of the exponential function are convergent. (Contributed by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  seq M (  +  ,  F )  e.  dom  ~~>  )
 
Theoremeftlcl 11698* Closure of the sum of an infinite tail of the series defining the exponential function. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  sum_ k  e.  ( ZZ>=
 `  M ) ( F `  k )  e.  CC )
 
Theoremreeftlcl 11699* Closure of the sum of an infinite tail of the series defining the exponential function. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   =>    |-  ( ( A  e.  RR  /\  M  e.  NN0 )  ->  sum_ k  e.  ( ZZ>=
 `  M ) ( F `  k )  e.  RR )
 
Theoremeftlub 11700* An upper bound on the absolute value of the infinite tail of the series expansion of the exponential function on the closed unit disk. (Contributed by Paul Chapman, 19-Jan-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   &    |-  G  =  ( n  e.  NN0  |->  ( ( ( abs `  A ) ^ n )  /  ( ! `  n ) ) )   &    |-  H  =  ( n  e.  NN0  |->  ( ( ( ( abs `  A ) ^ M )  /  ( ! `  M ) )  x.  ( ( 1  /  ( M  +  1 ) ) ^ n ) ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  <_ 
 1 )   =>    |-  ( ph  ->  ( abs `  sum_ k  e.  ( ZZ>=
 `  M ) ( F `  k ) )  <_  ( (
 ( abs `  A ) ^ M )  x.  (
 ( M  +  1 )  /  ( ( ! `  M )  x.  M ) ) ) )
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