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Theorem List for Metamath Proof Explorer - 36101-36200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexpgrowth 36101* Exponential growth and decay model. The derivative of a function y of variable t equals a constant k times y itself, iff y equals some constant C times the exponential of kt. This theorem and expgrowthi 36099 illustrate one of the simplest and most crucial classes of differential equations, equations that relate functions to their derivatives.

Section 6.3 of [Strang] p. 242 calls y' = ky "the most important differential equation in applied mathematics". In the field of population ecology it is known as the Malthusian growth model or exponential law, and C, k, and t correspond to initial population size, growth rate, and time respectively (https://en.wikipedia.org/wiki/Malthusian_growth_model); and in finance, the model appears in a similar role in continuous compounding with C as the initial amount of money. In exponential decay models, k is often expressed as the negative of a positive constant λ.

Here y' is given as  ( S  _D  Y
), C as  c, and ky as  ( ( S  X.  { K }
)  oF  x.  Y ).  ( S  X.  { K }
) is the constant function that maps any real or complex input to k and  oF  x. is multiplication as a function operation.

The leftward direction of the biconditional is as given in http://www.saylor.org/site/wp-content/uploads/2011/06/MA221-2.1.1.pdf pp. 1-2, which also notes the reverse direction ("While we will not prove this here, it turns out that these are the only functions that satisfy this equation."). The rightward direction is Theorem 5.1 of [LarsonHostetlerEdwards] p. 375 (which notes " C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k > 0, and exponential decay occurs when k < 0."); its proof here closely follows the proof of y' = y in https://proofwiki.org/wiki/Exponential_Growth_Equation/Special_Case.

Statements for this and expgrowthi 36099 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.)

 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  K  e.  CC )   &    |-  ( ph  ->  Y : S --> CC )   &    |-  ( ph  ->  dom  ( S  _D  Y )  =  S )   =>    |-  ( ph  ->  (
 ( S  _D  Y )  =  ( ( S  X.  { K }
 )  oF  x.  Y )  <->  E. c  e.  CC  Y  =  ( t  e.  S  |->  ( c  x.  ( exp `  ( K  x.  t ) ) ) ) ) )
 
21.27.7  The generalized binomial coefficient operation
 
Syntaxcbcc 36102 Extend class notation to include the generalized binomial coefficient operation.
 class C𝑐
 
Definitiondf-bcc 36103* Define a generalized binomial coefficient operation, which unlike df-bc 12427 allows complex numbers for the first argument. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |- C𝑐  =  ( c  e.  CC ,  k  e. 
 NN0  |->  ( ( c FallFac  k )  /  ( ! `  k ) ) )
 
Theorembccval 36104 Value of the generalized binomial coefficient,  C choose  K. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  K  e.  NN0 )   =>    |-  ( ph  ->  ( CC𝑐 K )  =  (
 ( C FallFac  K )  /  ( ! `  K ) ) )
 
Theorembcccl 36105 Closure of the generalized binomial coefficient. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  K  e.  NN0 )   =>    |-  ( ph  ->  ( CC𝑐 K )  e.  CC )
 
Theorembcc0 36106 The generalized binomial coefficient  C choose  K is zero iff  C is an integer between zero and  ( K  - 
1 ) inclusive. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  K  e.  NN0 )   =>    |-  ( ph  ->  (
 ( CC𝑐 K )  =  0  <->  C  e.  ( 0 ... ( K  -  1
 ) ) ) )
 
Theorembccp1k 36107 Generalized binomial coefficient: 
C choose  ( K  +  1 ). (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  K  e.  NN0 )   =>    |-  ( ph  ->  ( CC𝑐 ( K  +  1
 ) )  =  ( ( CC𝑐 K )  x.  (
 ( C  -  K )  /  ( K  +  1 ) ) ) )
 
Theorembccm1k 36108 Generalized binomial coefficient: 
C choose  ( K  -  1 ), when  C is not  ( K  -  1 ). (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  ( CC  \  { ( K  -  1 ) }
 ) )   &    |-  ( ph  ->  K  e.  NN )   =>    |-  ( ph  ->  ( CC𝑐 ( K  -  1
 ) )  =  ( ( CC𝑐 K )  /  (
 ( C  -  ( K  -  1 ) ) 
 /  K ) ) )
 
Theorembccn0 36109 Generalized binomial coefficient: 
C choose  0. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( CC𝑐 0 )  =  1
 )
 
Theorembccn1 36110 Generalized binomial coefficient: 
C choose  1. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( CC𝑐 1 )  =  C )
 
Theorembccbc 36111 The binomial coefficient and generalized binomial coefficient are equal when their arguments are nonnegative integers. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  K  e.  NN0 )   =>    |-  ( ph  ->  ( NC𝑐 K )  =  ( N  _C  K ) )
 
21.27.8  Binomial series
 
Theoremuzmptshftfval 36112* When  F is a maps-to function on some set of upper integers  Z that returns a set  B,  ( F  shift  N ) is another maps-to function on the shifted set of upper integers  W. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  F  =  ( x  e.  Z  |->  B )   &    |-  B  e.  _V   &    |-  ( x  =  ( y  -  N )  ->  B  =  C )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  W  =  ( ZZ>= `  ( M  +  N ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( F  shift  N )  =  ( y  e.  W  |->  C ) )
 
Theoremdvradcnv2 36113* The radius of convergence of the (formal) derivative  H of the power series  G is (at least) as large as the radius of convergence of  G. This version of dvradcnv 23110 uses a shifted version of  H to match the sum form of  ( CC  _D  F
) in pserdv2 23119 (and shows how to use uzmptshftfval 36112 to shift a maps-to function on a set of upper integers). (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  H  =  ( n  e.  NN  |->  ( ( n  x.  ( A `  n ) )  x.  ( X ^ ( n  -  1 ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  ( abs `  X )  <  R )   =>    |-  ( ph  ->  seq 1
 (  +  ,  H )  e.  dom  ~~>  )
 
Theorembinomcxplemwb 36114 Lemma for binomcxp 36123. The lemma in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  K  e.  NN )   =>    |-  ( ph  ->  (
 ( ( C  -  K )  x.  ( CC𝑐 K ) )  +  ( ( C  -  ( K  -  1
 ) )  x.  ( CC𝑐 ( K  -  1
 ) ) ) )  =  ( C  x.  ( CC𝑐 K ) ) )
 
Theorembinomcxplemnn0 36115* Lemma for binomcxp 36123. When  C is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom 13795 equals this generalized infinite sum: the generalized binomial coefficient and exponentiation operators give exactly the same values in the standard index set  ( 0 ... C
), and when the index set is widened beyond  C the additional values are just zeroes. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ( ph  /\  C  e.  NN0 )  ->  ( ( A  +  B )  ^c  C )  =  sum_ k  e. 
 NN0  ( ( CC𝑐 k )  x.  ( ( A  ^c  ( C  -  k ) )  x.  ( B ^ k ) ) ) )
 
Theorembinomcxplemrat 36116* Lemma for binomcxp 36123. As  k increases, this ratio's absolute value converges to one. Part of equation "Since continuity of the absolute value..." in the Wikibooks proof (proven for the inverse ratio, which we later show is no problem). (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( k  e.  NN0  |->  ( abs `  ( ( C  -  k )  /  (
 k  +  1 ) ) ) )  ~~>  1 )
 
Theorembinomcxplemfrat 36117* Lemma for binomcxp 36123. binomcxplemrat 36116 implies that when  C is not a nonnegative integer, the absolute value of the ratio  ( ( F `
 ( k  +  1 ) )  / 
( F `  k
) ) converges to one. The rest of equation "Since continuity of the absolute value..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   &    |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )   =>    |-  ( ( ph  /\ 
 -.  C  e.  NN0 )  ->  ( k  e. 
 NN0  |->  ( abs `  (
 ( F `  (
 k  +  1 ) )  /  ( F `
  k ) ) ) )  ~~>  1 )
 
Theorembinomcxplemradcnv 36118* Lemma for binomcxp 36123. By binomcxplemfrat 36117 and radcnvrat 36056 the radius of convergence of power series  sum_ k  e.  NN0 ( ( F `  k )  x.  (
b ^ k ) ) is one. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   &    |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )   &    |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
 k ) ) ) )   &    |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( S `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   =>    |-  (
 ( ph  /\  -.  C  e.  NN0 )  ->  R  =  1 )
 
Theorembinomcxplemdvbinom 36119* Lemma for binomcxp 36123. By the power and chain rules, calculate the derivative of  ( ( 1  +  b )  ^c  -u C ), with respect to  b in the disk of convergence 
D. We later multiply the derivative in the later binomcxplemdvsum 36121 by this derivative to show that  ( ( 1  +  b )  ^c  C ) (with a non-negated  C) and the later sum, since both at  b  =  0 equal one, are the same. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   &    |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )   &    |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
 k ) ) ) )   &    |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( S `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  E  =  ( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
 ) )  x.  (
 b ^ ( k  -  1 ) ) ) ) )   &    |-  D  =  ( `' abs " (
 0 [,) R ) )   =>    |-  ( ( ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  (
 b  e.  D  |->  ( ( 1  +  b
 )  ^c  -u C ) ) )  =  ( b  e.  D  |->  ( -u C  x.  ( ( 1  +  b )  ^c 
 ( -u C  -  1
 ) ) ) ) )
 
Theorembinomcxplemcvg 36120* Lemma for binomcxp 36123. The sum in binomcxplemnn0 36115 and its derivative (see the next theorem, binomcxplemdvsum 36121) converge, as long as their base  J is within the disk of convergence. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   &    |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )   &    |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
 k ) ) ) )   &    |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( S `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  E  =  ( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
 ) )  x.  (
 b ^ ( k  -  1 ) ) ) ) )   &    |-  D  =  ( `' abs " (
 0 [,) R ) )   =>    |-  ( ( ph  /\  J  e.  D )  ->  (  seq 0 (  +  ,  ( S `  J ) )  e.  dom  ~~>  /\  seq 1
 (  +  ,  ( E `  J ) )  e.  dom  ~~>  ) )
 
Theorembinomcxplemdvsum 36121* Lemma for binomcxp 36123. The derivative of the generalized sum in binomcxplemnn0 36115. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   &    |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )   &    |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
 k ) ) ) )   &    |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( S `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  E  =  ( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
 ) )  x.  (
 b ^ ( k  -  1 ) ) ) ) )   &    |-  D  =  ( `' abs " (
 0 [,) R ) )   &    |-  P  =  ( b  e.  D  |->  sum_ k  e.  NN0  ( ( S `  b ) `  k
 ) )   =>    |-  ( ph  ->  ( CC  _D  P )  =  ( b  e.  D  |->  sum_
 k  e.  NN  (
 ( E `  b
 ) `  k )
 ) )
 
Theorembinomcxplemnotnn0 36122* Lemma for binomcxp 36123. When  C is not a nonnegative integer, the generalized sum in binomcxplemnn0 36115 —which we will call  P —is a convergent power series: its base  b is always of smaller absolute value than the radius of convergence.

pserdv2 23119 gives the derivative of  P, which by dvradcnv 23110 also converges in that radius. When  A is fixed at one,  ( A  +  b ) times that derivative equals  ( C  x.  P
) and fraction  ( P  / 
( ( A  +  b )  ^c  C ) ) is always defined with derivative zero, so the fraction is a constant—specifically one, because  ( ( 1  +  0 )  ^c  C )  =  1. Thus  ( ( 1  +  b )  ^c  C )  =  ( P `  b ).

Finally, let  b be  ( B  /  A ), and multiply both the binomial  ( ( 1  +  ( B  /  A ) )  ^c  C ) and the sum  ( P `  ( B  /  A
) ) by  ( A  ^c  C ) to get the result. (Contributed by Steve Rodriguez, 22-Apr-2020.)

 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   &    |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )   &    |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
 k ) ) ) )   &    |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( S `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  E  =  ( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
 ) )  x.  (
 b ^ ( k  -  1 ) ) ) ) )   &    |-  D  =  ( `' abs " (
 0 [,) R ) )   &    |-  P  =  ( b  e.  D  |->  sum_ k  e.  NN0  ( ( S `  b ) `  k
 ) )   =>    |-  ( ( ph  /\  -.  C  e.  NN0 )  ->  ( ( A  +  B )  ^c  C )  =  sum_ k  e. 
 NN0  ( ( CC𝑐 k )  x.  ( ( A  ^c  ( C  -  k ) )  x.  ( B ^ k ) ) ) )
 
Theorembinomcxp 36123* Generalize the binomial theorem binom 13795 to positive real summand  A, real summand  B, and complex exponent  C. Proof in https://en.wikibooks.org/wiki/Advanced_Calculus; see also https://en.wikipedia.org/wiki/Binomial_series, https://en.wikipedia.org/wiki/Binomial_theorem (sections "Newton's generalized binomial theorem" and "Future generalizations"), and proof "General Binomial Theorem" in https://proofwiki.org/wiki/Binomial_Theorem. (Contributed by Steve Rodriguez, 22-Apr-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( abs `  B )  <  ( abs `  A ) )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( ( A  +  B )  ^c  C )  =  sum_ k  e.  NN0  ( ( CC𝑐 k )  x.  ( ( A 
 ^c  ( C  -  k ) )  x.  ( B ^
 k ) ) ) )
 
21.28  Mathbox for Andrew Salmon
 
21.28.1  Principia Mathematica * 10
 
Theorempm10.12 36124* Theorem *10.12 in [WhiteheadRussell] p. 146. In *10, this is treated as an axiom, and the proofs in *10 are based on this theorem. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x ( ph  \/  ps )  ->  ( ph  \/  A. x ps )
 )
 
Theorempm10.14 36125 Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  (
 ( A. x ph  /\  A. x ps )  ->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps )
 )
 
Theorempm10.251 36126 Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x  -.  ph  ->  -. 
 A. x ph )
 
Theorempm10.252 36127 Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.)
 |-  ( -.  E. x ph  <->  A. x  -.  ph )
 
Theorempm10.253 36128 Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( -.  A. x ph  <->  E. x  -.  ph )
 
Theoremalbitr 36129 Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ( ph  <->  ps )  /\  A. x ( ps  <->  ch ) )  ->  A. x ( ph  <->  ch ) )
 
Theorempm10.42 36130 Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  (
 ( E. x ph  \/  E. x ps )  <->  E. x ( ph  \/  ps ) )
 
Theorempm10.52 36131* Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x ph  ->  ( A. x ( ph  ->  ps )  <->  ps ) )
 
Theorempm10.53 36132 Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x ph  ->  A. x ( ph  ->  ps ) )
 
Theorempm10.541 36133* Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ch  \/  ps )
 ) 
 <->  ( ch  \/  A. x ( ph  ->  ps ) ) )
 
Theorempm10.542 36134* Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ch  ->  ps )
 ) 
 <->  ( ch  ->  A. x ( ph  ->  ps )
 ) )
 
Theorempm10.55 36135 Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( E. x (
 ph  /\  ps )  /\  A. x ( ph  ->  ps ) )  <->  ( E. x ph 
 /\  A. x ( ph  ->  ps ) ) )
 
Theorempm10.56 36136 Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ( ph  ->  ps )  /\  E. x ( ph  /\  ch ) )  ->  E. x ( ps  /\  ch )
 )
 
Theorempm10.57 36137 Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ps  \/  ch )
 )  ->  ( A. x ( ph  ->  ps )  \/  E. x ( ph  /\  ch )
 ) )
 
21.28.2  Principia Mathematica * 11
 
Theorem2alanimi 36138 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( A. x A. y ph  /\  A. x A. y ps )  ->  A. x A. y ch )
 
Theorem2al2imi 36139 Removes two universal qunatifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x A. y ph  ->  ( A. x A. y ps  ->  A. x A. y ch ) )
 
Theorempm11.11 36140 Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ph   =>    |- 
 A. z A. w [ z  /  x ] [ w  /  y ] ph
 
Theorempm11.12 36141* Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x A. y (
 ph  \/  ps )  ->  ( ph  \/  A. x A. y ps )
 )
 
Theorem19.21vv 36142* Compare theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1754. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ps  ->  ph )  <->  ( ps  ->  A. x A. y ph ) )
 
Theorem2alim 36143 Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  ->  ps )  ->  ( A. x A. y ph  ->  A. x A. y ps ) )
 
Theorem2albi 36144 Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph 
 <->  ps )  ->  ( A. x A. y ph  <->  A. x A. y ps )
 )
 
Theorem2exim 36145 Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  ->  ps )  ->  ( E. x E. y ph  ->  E. x E. y ps ) )
 
Theorem2exbi 36146 Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph 
 <->  ps )  ->  ( E. x E. y ph  <->  E. x E. y ps )
 )
 
Theoremspsbce-2 36147 Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  ->  E. x E. y ph )
 
Theorem19.33-2 36148 Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x A. y ph  \/  A. x A. y ps )  ->  A. x A. y ( ph  \/  ps ) )
 
Theorem19.36vv 36149* Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)
 |-  ( E. x E. y (
 ph  ->  ps )  <->  ( A. x A. y ph  ->  ps )
 )
 
Theorem19.31vv 36150* Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  \/  ps )  <->  (
 A. x A. y ph  \/  ps ) )
 
Theorem19.37vv 36151* Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y ( ps  ->  ph )  <->  ( ps  ->  E. x E. y ph ) )
 
Theorem19.28vv 36152* Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ps  /\  ph )  <->  ( ps  /\  A. x A. y ph ) )
 
Theorempm11.52 36153 Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y (
 ph  /\  ps )  <->  -. 
 A. x A. y
 ( ph  ->  -.  ps ) )
 
Theorem2exanali 36154 Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x E. y
 ( ph  /\  -.  ps ) 
 <-> 
 A. x A. y
 ( ph  ->  ps )
 )
 
Theoremaaanv 36155* Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2005. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ph  /\  A. y ps )  <->  A. x A. y
 ( ph  /\  ps )
 )
 
Theorempm11.57 36156* Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ph  <->  A. x A. y
 ( ph  /\  [ y  /  x ] ph )
 )
 
Theorempm11.58 36157* Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x ph  <->  E. x E. y
 ( ph  /\  [ y  /  x ] ph )
 )
 
Theorempm11.59 36158* Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
 |-  ( A. x ( ph  ->  ps )  ->  A. y A. x ( ( ph  /\ 
 [ y  /  x ] ph )  ->  ( ps  /\  [ y  /  x ] ps ) ) )
 
Theorempm11.6 36159* Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
 |-  ( E. x ( E. y
 ( ph  /\  ps )  /\  ch )  <->  E. y ( E. x ( ph  /\  ch )  /\  ps ) )
 
Theorempm11.61 36160* Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. y A. x (
 ph  ->  ps )  ->  A. x ( ph  ->  E. y ps ) )
 
Theorempm11.62 36161* Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ( ph  /\  ps )  ->  ch )  <->  A. x ( ph  ->  A. y ( ps 
 ->  ch ) ) )
 
Theorempm11.63 36162 Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x E. y ph  ->  A. x A. y
 ( ph  ->  ps )
 )
 
Theorempm11.7 36163 Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y (
 ph  \/  ph )  <->  E. x E. y ph )
 
Theorempm11.71 36164* Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( E. x ph  /\ 
 E. y ch )  ->  ( ( A. x ( ph  ->  ps )  /\  A. y ( ch 
 ->  th ) )  <->  A. x A. y
 ( ( ph  /\  ch )  ->  ( ps  /\  th ) ) ) )
 
21.28.3  Predicate Calculus
 
Theoremsbeqal1 36165* If  x  =  y always implies 
x  =  z, then  y  =  z is true. (Contributed by Andrew Salmon, 2-Jun-2011.)
 |-  ( A. x ( x  =  y  ->  x  =  z )  ->  y  =  z )
 
Theoremsbeqal1i 36166* Suppose you know  x  =  y implies  x  =  z, assuming  x and  z are distinct. Then,  y  =  z. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( x  =  y  ->  x  =  z )   =>    |-  y  =  z
 
Theoremsbeqal2i 36167* If  x  =  y implies  x  =  z, then we can infer  z  =  y. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( x  =  y  ->  x  =  z )   =>    |-  z  =  y
 
Theoremsbeqalbi 36168* When both  x and  z and  y and  z are both distinct, then the converse of sbeqal1 holds as well. (Contributed by Andrew Salmon, 2-Jun-2011.)
 |-  ( x  =  y  <->  A. z ( z  =  x  ->  z  =  y ) )
 
Theoremaxc5c4c711 36169 Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1641 as the inference rule. This proof extends the idea of axc5c711 31954 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.)
 |-  (
 ( A. x A. y  -.  A. x A. y
 ( A. y ph  ->  ps )  ->  ( ph  ->  A. y ( A. y ph  ->  ps )
 ) )  ->  ( A. y ph  ->  A. y ps ) )
 
Theoremaxc5c4c711toc5 36170 Re-derivation of sp 1885 from axc5c4c711 36169. Note that ax6 2032 is used for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 1774 instead of ax6 2032, so that this re-derivation requires only ax6v 1774 and propositional calculus. (Revised by BJ, 14-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x ph  ->  ph )
 
Theoremaxc5c4c711toc4 36171 Re-derivation of axc4 1886 from axc5c4c711 36169. Note that only propositional calculus is required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x ( A. x ph 
 ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremaxc5c4c711toc7 36172 Re-derivation of axc7 1887 from axc5c4c711 36169. Note that neither axc7 1887 nor ax-11 1868 are required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremaxc5c4c711to11 36173 Re-derivation of ax-11 1868 from axc5c4c711 36169. Note that ax-11 1868 is not required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremaxc11next 36174* This theorem shows that, given axext4 2386, we can derive a version of axc11n 2077. However, it is weaker than axc11n 2077 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  z  ->  A. z  z  =  x )
 
21.28.4  Principia Mathematica * 13 and * 14
 
Theorempm13.13a 36175 One result of theorem *13.13 in [WhiteheadRussell] p. 178. A note on the section - to make the theorems more usable, and because inequality is notation for set theory (it is not defined in the predicate calculus section), this section will use classes instead of sets. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  A )  ->  [. A  /  x ]. ph )
 
Theorempm13.13b 36176 Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( [. A  /  x ].
 ph  /\  x  =  A )  ->  ph )
 
Theorempm13.14 36177 Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( [. A  /  x ].
 ph  /\  -.  ph )  ->  x  =/=  A )
 
Theorempm13.192 36178* Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
 |-  ( E. y ( A. x ( x  =  A  <->  x  =  y )  /\  ph )  <->  [. A  /  y ]. ph )
 
Theorempm13.193 36179 Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  y )  <->  ( [ y  /  x ] ph  /\  x  =  y ) )
 
Theorempm13.194 36180 Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  y )  <->  ( [ y  /  x ] ph  /\  ph  /\  x  =  y ) )
 
Theorempm13.195 36181* Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3304. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
 |-  ( E. y ( y  =  A  /\  ph )  <->  [. A  /  y ]. ph )
 
Theorempm13.196a 36182* Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( -.  ph  <->  A. y ( [
 y  /  x ] ph  ->  y  =/=  x ) )
 
Theorem2sbc6g 36183* Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A. z A. w ( ( z  =  A  /\  w  =  B )  ->  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
 
Theorem2sbc5g 36184* Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
 
Theoremiotain 36185 Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( E! x ph  ->  |^| { x  |  ph }  =  (
 iota x ph ) )
 
Theoremiotaexeu 36186 The iota class exists. This theorem does not require ax-nul 4527 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  e. 
 _V )
 
Theoremiotasbc 36187* Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define  iota in terms of a function of  ( iota x ph ). Their definition differs in that a function of  ( iota x ph ) evaluates to "false" when there isn't a single  x that satisfies  ph. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( A. x ( ph  <->  x  =  y
 )  /\  ps )
 ) )
 
Theoremiotasbc2 36188* Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  (
 ( E! x ph  /\ 
 E! x ps )  ->  ( [. ( iota
 x ph )  /  y ]. [. ( iota x ps )  /  z ]. ch  <->  E. y E. z
 ( A. x ( ph  <->  x  =  y )  /\  A. x ( ps  <->  x  =  z
 )  /\  ch )
 ) )
 
Theorempm14.12 36189* Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  A. x A. y ( ( ph  /\  [. y  /  x ].
 ph )  ->  x  =  y ) )
 
Theorempm14.122a 36190* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  (
 A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  [. A  /  x ]. ph )
 ) )
 
Theorempm14.122b 36191* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  ( ( A. x (
 ph  ->  x  =  A )  /\  [. A  /  x ].
 ph )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
 
Theorempm14.122c 36192* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  (
 A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
 
Theorempm14.123a 36193* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  [. A  /  z ]. [. B  /  w ]. ph ) ) )
 
Theorempm14.123b 36194* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  [. A  /  z ]. [. B  /  w ]. ph )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  E. z E. w ph ) ) )
 
Theorempm14.123c 36195* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  E. z E. w ph ) ) )
 
Theorempm14.18 36196 Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ps  ->  [. ( iota x ph )  /  x ]. ps ) )
 
Theoremiotaequ 36197* Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( iota x x  =  y )  =  y
 
Theoremiotavalb 36198* Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5546. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y
 ) 
 <->  ( iota x ph )  =  y )
 )
 
Theoremiotasbc5 36199* Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( y  =  ( iota x ph )  /\  ps ) ) )
 
Theorempm14.24 36200* Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  A. y
 ( [. y  /  x ].
 ph 
 <->  y  =  ( iota
 x ph ) ) )
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38873
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