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Theorem List for Intuitionistic Logic Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprodsn 11601* 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 11602* 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 11603 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 11604* Split a finite product into two parts. New proofs should use fprodsplit 11605 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 11605* 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 11606* 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 11607* 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 11608* 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 11609* 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 11610* 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 11611* 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 11612* Multiply in an additional term in a finite product. See also fprodsplitsn 11641 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 11613* 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 11614* 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 11615* 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 11616* 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 11617* 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 11618* 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 11619* 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 11620* 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 11621* Finite product closure lemma. A version of fprodcllem 11614 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 11622* Closure of a finite product of real numbers. A version of fprodrecl 11616 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 11623* Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  ( A  e.  NN0  ->  ( ! `  A )  =  prod_ k  e.  (
 1 ... A ) k )
 
Theoremfprodabs 11624* 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 11625* 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 11626* 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 11627* 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 11628* 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 11629* 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 11630* Lemma for fprod2d 11631- 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 11631* Write a double product as a product over a two-dimensional region. Compare fsum2d 11443. (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 11632* 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 11633* 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 11634* 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 11635* 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 11636* Two ways to express "the product of  A ( j ,  k ) over the triangular region  M  <_  j,  M  <_  k,  j  +  k  <_  N. Compare fisum0diag 11449. (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 11637* 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 11638* 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 11639* The quotient of two finite products. A version of fproddivap 11638 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 11640* Split a finite product into two parts. A version of fprodsplit 11605 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 11641* Separate out a term in a finite product. See also fprodunsn 11612 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 11642* 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 11643* Closure of a finite product of complex numbers. A version of fprodcl 11615 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 11644* A finite product of terms apart from zero is apart from zero. A version of fprodap0 11629 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 11645* 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 11646* 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 11647* 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 11648* 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 11649* 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 11650 Extend class notation to include the exponential function.
 class  exp
 
Syntaxceu 11651 Extend class notation to include Euler's constant  _e = 2.71828....
 class  _e
 
Syntaxcsin 11652 Extend class notation to include the sine function.
 class  sin
 
Syntaxccos 11653 Extend class notation to include the cosine function.
 class  cos
 
Syntaxctan 11654 Extend class notation to include the tangent function.
 class  tan
 
Syntaxcpi 11655 Extend class notation to include the constant pi,  pi = 3.14159....
 class  pi
 
Definitiondf-ef 11656* Define the exponential function. Its value at the complex number  A is  ( exp `  A
) and is called the "exponential of  A"; see efval 11669. (Contributed by NM, 14-Mar-2005.)
 |- 
 exp  =  ( x  e.  CC  |->  sum_ k  e.  NN0  ( ( x ^
 k )  /  ( ! `  k ) ) )
 
Definitiondf-e 11657 Define Euler's constant  _e = 2.71828.... (Contributed by NM, 14-Mar-2005.)
 |-  _e  =  ( exp `  1 )
 
Definitiondf-sin 11658 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 11659 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 11660 Define the tangent function. We define it this way for cmpt 4065, 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 11661 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 11662 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 11663 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 11664 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 11665* 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 11666* Lemma for efcl 11672. The series that defines the exponential function converges. The ratio test cvgratgt0 11541 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 11667* Lemma for efcl 11672. 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 11668* 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 11669* 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 11670 Value of Euler's constant  _e = 2.71828.... (Contributed by Steve Rodriguez, 5-Mar-2006.)
 |-  _e  =  sum_ k  e.  NN0  ( 1  /  ( ! `  k ) )
 
Theoremeff 11671 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 11672 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 11673* 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 11674* 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 11675* 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 11676 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 11677 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 11678 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 11679 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 11680 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 11681 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 11682* Lemma for efadd 11683 (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 11683 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 11684 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 11685 The exponential of a complex number is apart from zero. (Contributed by Jim Kingdon, 12-Dec-2022.)
 |-  ( A  e.  CC  ->  ( exp `  A ) #  0 )
 
Theoremefne0 11686 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 11685 (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 11687 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 11688 The exponential function maps the complex numbers to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)
 |- 
 exp : CC --> ( CC  \  { 0 } )
 
Theoremefsub 11689 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 11690 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 11691 Value of the exponential function for integers. Special case of efval 11669. 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 11692 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 11693 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 11694 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 11695* 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 11696* 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 11697* 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 11698* 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 ) ) ) )
 
Theoremefsep 11699* Separate out the next term of the power series expansion of the exponential function. The last hypothesis allows the separated terms to be rearranged as desired. (Contributed by Paul Chapman, 23-Nov-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   &    |-  N  =  ( M  +  1 )   &    |-  M  e.  NN0   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( exp `  A )  =  ( B  +  sum_ k  e.  ( ZZ>=
 `  M ) ( F `  k ) ) )   &    |-  ( ph  ->  ( B  +  ( ( A ^ M ) 
 /  ( ! `  M ) ) )  =  D )   =>    |-  ( ph  ->  ( exp `  A )  =  ( D  +  sum_ k  e.  ( ZZ>= `  N ) ( F `  k ) ) )
 
Theoremeffsumlt 11700* The partial sums of the series expansion of the exponential function at a positive real number are bounded by the value of the function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( ( A ^ n ) 
 /  ( ! `  n ) ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (  seq 0 (  +  ,  F ) `  N )  <  ( exp `  A ) )
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