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Theorem List for Metamath Proof Explorer - 20701-20800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempntibndlem1 20701 Lemma for pntibnd 20705. (Contributed by Mario Carneiro, 10-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  L  =  ( ( 1  /  4
 )  /  ( A  +  3 ) )   =>    |-  ( ph  ->  L  e.  ( 0 (,) 1
 ) )
 
Theorempntibndlem2a 20702* Lemma for pntibndlem2 20703. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  L  =  ( ( 1  /  4
 )  /  ( A  +  3 ) )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  (
 ( R `  x )  /  x ) ) 
 <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  K  =  ( exp `  ( B  /  ( E  /  2
 ) ) )   &    |-  C  =  ( ( 2  x.  B )  +  ( log `  2 ) )   &    |-  ( ph  ->  E  e.  ( 0 (,) 1
 ) )   &    |-  ( ph  ->  Z  e.  RR+ )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ( ph  /\  u  e.  ( N [,] ( ( 1  +  ( L  x.  E ) )  x.  N ) ) ) 
 ->  ( u  e.  RR  /\  N  <_  u  /\  u  <_  ( ( 1  +  ( L  x.  E ) )  x.  N ) ) )
 
Theorempntibndlem2 20703* Lemma for pntibnd 20705. The main work, after eliminating all the quantifiers. (Contributed by Mario Carneiro, 10-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  L  =  ( ( 1  /  4
 )  /  ( A  +  3 ) )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  (
 ( R `  x )  /  x ) ) 
 <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  K  =  ( exp `  ( B  /  ( E  /  2
 ) ) )   &    |-  C  =  ( ( 2  x.  B )  +  ( log `  2 ) )   &    |-  ( ph  ->  E  e.  ( 0 (,) 1
 ) )   &    |-  ( ph  ->  Z  e.  RR+ )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  ( 1 (,)  +oo ) A. y  e.  ( x [,] ( 2  x.  x ) ) ( (ψ `  y )  -  (ψ `  x )
 )  <_  ( (
 2  x.  ( y  -  x ) )  +  ( T  x.  ( x  /  ( log `  x ) ) ) ) )   &    |-  X  =  ( ( exp `  ( T  /  ( E  / 
 4 ) ) )  +  Z )   &    |-  ( ph  ->  M  e.  (
 ( exp `  ( C  /  E ) ) [,)  +oo ) )   &    |-  ( ph  ->  Y  e.  ( X (,)  +oo ) )   &    |-  ( ph  ->  ( ( Y  <  N  /\  N  <_  ( ( M  /  2 )  x.  Y ) )  /\  ( abs `  ( ( R `  N )  /  N ) )  <_  ( E  /  2
 ) ) )   =>    |-  ( ph  ->  E. z  e.  RR+  ( ( Y  <  z  /\  ( ( 1  +  ( L  x.  E ) )  x.  z
 )  <  ( M  x.  Y ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )
 
Theorempntibndlem3 20704* Lemma for pntibnd 20705. Package up pntibndlem2 20703 in quantifiers. (Contributed by Mario Carneiro, 10-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  L  =  ( ( 1  /  4
 )  /  ( A  +  3 ) )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  (
 ( R `  x )  /  x ) ) 
 <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  K  =  ( exp `  ( B  /  ( E  /  2
 ) ) )   &    |-  C  =  ( ( 2  x.  B )  +  ( log `  2 ) )   &    |-  ( ph  ->  E  e.  ( 0 (,) 1
 ) )   &    |-  ( ph  ->  Z  e.  RR+ )   &    |-  ( ph  ->  A. m  e.  ( K [,)  +oo ) A. v  e.  ( Z (,)  +oo ) E. i  e.  NN  ( ( v  < 
 i  /\  i  <_  ( m  x.  v ) )  /\  ( abs `  ( ( R `  i )  /  i
 ) )  <_  ( E  /  2 ) ) )   =>    |-  ( ph  ->  E. x  e.  RR+  A. k  e.  (
 ( exp `  ( C  /  E ) ) [,)  +oo ) A. y  e.  ( x (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( k  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )
 
Theorempntibnd 20705* Lemma for pnt 20726. Establish smallness of  R on an interval. Lemma 10.6.2 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 10-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   =>    |- 
 E. c  e.  RR+  E. l  e.  ( 0 (,) 1 ) A. e  e.  ( 0 (,) 1 ) E. x  e.  RR+  A. k  e.  (
 ( exp `  ( c  /  e ) ) [,)  +oo ) A. y  e.  ( x (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( l  x.  e ) )  x.  z )  < 
 ( k  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( l  x.  e
 ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  e )
 
Theorempntlemd 20706 Lemma for pnt 20726. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434,  A is C^*,  B is c1,  L is λ,  D is c2, and  F is c3. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   =>    |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
 
Theorempntlemc 20707* Lemma for pnt 20726. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434,  U is α,  E is ε, and  K is K. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   =>    |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1
 )  /\  1  <  K 
 /\  ( U  -  E )  e.  RR+ )
 ) )
 
Theorempntlema 20708* Lemma for pnt 20726. Closure for the constants used in the proof. The mammoth expression  W is a number large enough to satisfy all the lower bounds needed for  Z. For comparison with Equation 10.6.27 of [Shapiro], p. 434,  Y is x2,  X is x1,  C is the big-O constant in Equation 10.6.29 of [Shapiro], p. 435, and  W is the unnamed lower bound of "for sufficiently large x" in Equation 10.6.34 of [Shapiro], p. 436. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   =>    |-  ( ph  ->  W  e.  RR+ )
 
Theorempntlemb 20709* Lemma for pnt 20726. Unpack all the lower bounds contained in  W, in the form they will be used. For comparison with Equation 10.6.27 of [Shapiro], p. 434,  Z is x. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   =>    |-  ( ph  ->  ( Z  e.  RR+  /\  (
 1  <  Z  /\  _e  <_  ( sqr `  Z )  /\  ( sqr `  Z )  <_  ( Z  /  Y ) )  /\  ( ( 4  /  ( L  x.  E ) )  <_  ( sqr `  Z )  /\  (
 ( ( log `  X )  /  ( log `  K ) )  +  2
 )  <_  ( (
 ( log `  Z )  /  ( log `  K ) )  /  4
 )  /\  ( ( U  x.  3 )  +  C )  <_  ( ( ( U  -  E )  x.  ( ( L  x.  ( E ^
 2 ) )  /  (; 3 2  x.  B ) ) )  x.  ( log `  Z ) ) ) ) )
 
Theorempntlemg 20710* Lemma for pnt 20726. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434,  M is j^* and  N is ĵ. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   =>    |-  ( ph  ->  ( M  e.  NN  /\  N  e.  ( ZZ>= `  M )  /\  ( ( ( log `  Z )  /  ( log `  K ) ) 
 /  4 )  <_  ( N  -  M ) ) )
 
Theorempntlemh 20711* Lemma for pnt 20726. Bounds on the subintervals in the induction. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   =>    |-  ( ( ph  /\  J  e.  ( M ... N ) )  ->  ( X  <  ( K ^ J )  /\  ( K ^ J )  <_  ( sqr `  Z )
 ) )
 
Theorempntlemn 20712* Lemma for pnt 20726. The "naive" base bound, which we will slightly improve. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   =>    |-  ( ( ph  /\  ( J  e.  NN  /\  J  <_  ( Z  /  Y ) ) )  -> 
 0  <_  ( (
 ( U  /  J )  -  ( abs `  (
 ( R `  ( Z  /  J ) ) 
 /  Z ) ) )  x.  ( log `  J ) ) )
 
Theorempntlemq 20713* Lemma for pntlemj 20715. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  O  =  ( ( ( |_ `  ( Z  /  ( K ^
 ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  ( K ^ J ) ) ) )   &    |-  ( ph  ->  V  e.  RR+ )   &    |-  ( ph  ->  ( ( ( K ^ J )  <  V  /\  ( ( 1  +  ( L  x.  E ) )  x.  V )  <  ( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( V [,] ( ( 1  +  ( L  x.  E ) )  x.  V ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  J  e.  ( M..^ N ) )   &    |-  I  =  ( ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  V ) ) )   =>    |-  ( ph  ->  I  C_  O )
 
Theorempntlemr 20714* Lemma for pntlemj 20715. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  O  =  ( ( ( |_ `  ( Z  /  ( K ^
 ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  ( K ^ J ) ) ) )   &    |-  ( ph  ->  V  e.  RR+ )   &    |-  ( ph  ->  ( ( ( K ^ J )  <  V  /\  ( ( 1  +  ( L  x.  E ) )  x.  V )  <  ( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( V [,] ( ( 1  +  ( L  x.  E ) )  x.  V ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  J  e.  ( M..^ N ) )   &    |-  I  =  ( ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  V ) ) )   =>    |-  ( ph  ->  (
 ( U  -  E )  x.  ( ( ( L  x.  E ) 
 /  8 )  x.  ( log `  Z ) ) )  <_  ( ( # `  I
 )  x.  ( ( U  -  E )  x.  ( ( log `  ( Z  /  V ) )  /  ( Z  /  V ) ) ) ) )
 
Theorempntlemj 20715* Lemma for pnt 20726. The induction step. Using pntibnd 20705, we find an interval in  K ^ J ... K ^ ( J  + 
1 ) which is sufficiently large and has a much smaller value,  R ( z )  / 
z  <_  E (instead of our original bound 
R ( z )  /  z  <_  U). (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  O  =  ( ( ( |_ `  ( Z  /  ( K ^
 ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  ( K ^ J ) ) ) )   &    |-  ( ph  ->  V  e.  RR+ )   &    |-  ( ph  ->  ( ( ( K ^ J )  <  V  /\  ( ( 1  +  ( L  x.  E ) )  x.  V )  <  ( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( V [,] ( ( 1  +  ( L  x.  E ) )  x.  V ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  J  e.  ( M..^ N ) )   &    |-  I  =  ( ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  V ) ) )   =>    |-  ( ph  ->  (
 ( U  -  E )  x.  ( ( ( L  x.  E ) 
 /  8 )  x.  ( log `  Z ) ) )  <_  sum_ n  e.  O  ( ( ( U  /  n )  -  ( abs `  ( ( R `
  ( Z  /  n ) )  /  Z ) ) )  x.  ( log `  n ) ) )
 
Theorempntlemi 20716* Lemma for pnt 20726. Eliminate some assumptions from pntlemj 20715. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  O  =  ( ( ( |_ `  ( Z  /  ( K ^
 ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  ( K ^ J ) ) ) )   =>    |-  ( ( ph  /\  J  e.  ( M..^ N ) )  ->  ( ( U  -  E )  x.  ( ( ( L  x.  E )  / 
 8 )  x.  ( log `  Z ) ) )  <_  sum_ n  e.  O  ( ( ( U  /  n )  -  ( abs `  (
 ( R `  ( Z  /  n ) ) 
 /  Z ) ) )  x.  ( log `  n ) ) )
 
Theorempntlemf 20717* Lemma for pnt 20726. Add up the pieces in pntlemi 20716 to get an estimate slightly better than the naive lower bound  0. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   =>    |-  ( ph  ->  (
 ( U  -  E )  x.  ( ( ( L  x.  ( E ^ 2 ) ) 
 /  (; 3 2  x.  B ) )  x.  (
 ( log `  Z ) ^ 2 ) ) )  <_  sum_ n  e.  ( 1 ... ( |_ `  ( Z  /  Y ) ) ) ( ( ( U 
 /  n )  -  ( abs `  ( ( R `  ( Z  /  n ) )  /  Z ) ) )  x.  ( log `  n ) ) )
 
Theorempntlemk 20718* Lemma for pnt 20726. Evaluate the naive part of the estimate. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   =>    |-  ( ph  ->  (
 2  x.  sum_ n  e.  ( 1 ... ( |_ `  ( Z  /  Y ) ) ) ( ( U  /  n )  x.  ( log `  n ) ) )  <_  ( ( U  x.  ( ( log `  Z )  +  3 ) )  x.  ( log `  Z ) ) )
 
Theorempntlemo 20719* Lemma for pnt 20726. Combine all the estimates to establish a smaller eventual bound on  R ( Z )  /  Z. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  A. z  e.  ( 1 (,)  +oo ) ( ( ( ( abs `  ( R `  z ) )  x.  ( log `  z
 ) )  -  (
 ( 2  /  ( log `  z ) )  x.  sum_ i  e.  (
 1 ... ( |_ `  (
 z  /  Y )
 ) ) ( ( abs `  ( R `  ( z  /  i
 ) ) )  x.  ( log `  i
 ) ) ) ) 
 /  z )  <_  C )   =>    |-  ( ph  ->  ( abs `  ( ( R `
  Z )  /  Z ) )  <_  ( U  -  ( F  x.  ( U ^
 3 ) ) ) )
 
Theorempntleme 20720* Lemma for pnt 20726. Package up pntlemo 20719 in quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  (
 ( R `  z
 )  /  z )
 )  <_  U )   &    |-  ( ph  ->  A. k  e.  ( K [,)  +oo ) A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( k  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  A. z  e.  ( 1 (,)  +oo ) ( ( ( ( abs `  ( R `  z ) )  x.  ( log `  z
 ) )  -  (
 ( 2  /  ( log `  z ) )  x.  sum_ i  e.  (
 1 ... ( |_ `  (
 z  /  Y )
 ) ) ( ( abs `  ( R `  ( z  /  i
 ) ) )  x.  ( log `  i
 ) ) ) ) 
 /  z )  <_  C )   =>    |-  ( ph  ->  E. w  e.  RR+  A. v  e.  ( w [,)  +oo ) ( abs `  ( ( R `  v )  /  v
 ) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
 
Theorempntlem3 20721* Lemma for pnt 20726. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 8-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  ( ( R `  x )  /  x ) )  <_  A )   &    |-  T  =  { t  e.  ( 0 [,] A )  |  E. y  e.  RR+  A. z  e.  (
 y [,)  +oo ) ( abs `  ( ( R `  z )  /  z ) )  <_  t }   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ( ph  /\  u  e.  T ) 
 ->  ( u  -  ( C  x.  ( u ^
 3 ) ) )  e.  T )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  ~~> r  1 )
 
Theorempntlemp 20722* Lemma for pnt 20726. Wrapping up more quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  ( ( R `  x )  /  x ) )  <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  A. e  e.  ( 0 (,) 1 ) E. x  e.  RR+  A. k  e.  ( ( exp `  ( B  /  e ) ) [,)  +oo ) A. y  e.  ( x (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  e ) )  x.  z )  < 
 ( k  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  e
 ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  e ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   =>    |-  ( ph  ->  E. w  e.  RR+  A. v  e.  ( w [,)  +oo ) ( abs `  ( ( R `  v )  /  v
 ) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
 
Theorempntleml 20723* Lemma for pnt 20726. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  ( ( R `  x )  /  x ) )  <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  A. e  e.  ( 0 (,) 1 ) E. x  e.  RR+  A. k  e.  ( ( exp `  ( B  /  e ) ) [,)  +oo ) A. y  e.  ( x (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  e ) )  x.  z )  < 
 ( k  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  e
 ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  e ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  ~~> r  1 )
 
Theorempnt3 20724 The Prime Number Theorem, version 3: the second Chebyshev function tends asymptotically to  x. (Contributed by Mario Carneiro, 1-Jun-2016.)
 |-  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  ~~> r  1
 
Theorempnt2 20725 The Prime Number Theorem, version 2: the first Chebyshev function tends asymptotically to  x. (Contributed by Mario Carneiro, 1-Jun-2016.)
 |-  ( x  e.  RR+  |->  ( ( theta `  x )  /  x ) )  ~~> r  1
 
Theorempnt 20726 The Prime Number Theorem: the number of prime numbers less than  x tends asymptotically to  x  /  log (
x ) as  x goes to infinity. (Contributed by Mario Carneiro, 1-Jun-2016.)
 |-  ( x  e.  (
 1 (,)  +oo )  |->  ( (π `  x )  /  ( x  /  ( log `  x ) ) ) )  ~~> r  1
 
13.4.13  Ostrowski's theorem
 
Theoremabvcxp 20727* Raising an absolute value to a power less than one yields another absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  G  =  ( x  e.  B  |->  ( ( F `  x )  ^ c  S ) )   =>    |-  ( ( F  e.  A  /\  S  e.  (
 0 (,] 1 ) ) 
 ->  G  e.  A )
 
Theorempadicfval 20728* Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   =>    |-  ( P  e.  Prime  ->  ( J `  P )  =  ( x  e. 
 QQ  |->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  x ) ) ) ) )
 
Theorempadicval 20729* Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   =>    |-  ( ( P  e.  Prime  /\  X  e.  QQ )  ->  ( ( J `
  P ) `  X )  =  if ( X  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  X ) ) ) )
 
Theoremostth2lem1 20730* Lemma for ostth2 20749, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 20749. If a power is upper bounded by a linear term, the exponent must be less than one. Or in big-O notation, 
n  e.  o ( A ^ n ) for any 
1  <  A. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  (
 ( ph  /\  n  e. 
 NN )  ->  ( A ^ n )  <_  ( n  x.  B ) )   =>    |-  ( ph  ->  A  <_  1 )
 
Theoremqrngbas 20731 The base set of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |- 
 QQ  =  ( Base `  Q )
 
Theoremqdrng 20732 The rationals form a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  Q  e.  DivRing
 
Theoremqrng0 20733 The zero element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  0  =  ( 0g
 `  Q )
 
Theoremqrng1 20734 The unit element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  1  =  ( 1r
 `  Q )
 
Theoremqrngneg 20735 The additive inverse in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  ( X  e.  QQ  ->  ( ( inv g `  Q ) `  X )  =  -u X )
 
Theoremqrngdiv 20736 The division operation in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  ( ( X  e.  QQ  /\  Y  e.  QQ  /\  Y  =/=  0 ) 
 ->  ( X (/r `  Q ) Y )  =  ( X  /  Y ) )
 
Theoremqabvle 20737 By using induction on  N, we show a long-range inequality coming from the triangle inequality. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   =>    |-  ( ( F  e.  A  /\  N  e.  NN0 )  ->  ( F `  N )  <_  N )
 
Theoremqabvexp 20738 Induct the product rule abvmul 15557 to find the absolute value of a power. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   =>    |-  ( ( F  e.  A  /\  M  e.  QQ  /\  N  e.  NN0 )  ->  ( F `  ( M ^ N ) )  =  ( ( F `
  M ) ^ N ) )
 
Theoremostthlem1 20739* Lemma for ostth 20751. If two absolute values agree on the positive integers greater than one, then they agree for all rational numbers and thus are equal as functions. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  G  e.  A )   &    |-  ( ( ph  /\  n  e.  ( ZZ>= `  2 )
 )  ->  ( F `  n )  =  ( G `  n ) )   =>    |-  ( ph  ->  F  =  G )
 
Theoremostthlem2 20740* Lemma for ostth 20751. Refine ostthlem1 20739 so that it is sufficient to only show equality on the primes. (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  G  e.  A )   &    |-  ( ( ph  /\  p  e.  Prime )  ->  ( F `  p )  =  ( G `  p ) )   =>    |-  ( ph  ->  F  =  G )
 
Theoremqabsabv 20741 The regular absolute value function on the rationals is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   =>    |-  ( abs  |`  QQ )  e.  A
 
Theorempadicabv 20742* The p-adic absolute value (with arbitrary base) is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  F  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( N ^ ( P  pCnt  x ) ) ) )   =>    |-  ( ( P  e.  Prime  /\  N  e.  (
 0 (,) 1 ) ) 
 ->  F  e.  A )
 
Theorempadicabvf 20743* The p-adic absolute value is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   =>    |-  J : Prime --> A
 
Theorempadicabvcxp 20744* All positive powers of the p-adic absolute value are absolute values. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   =>    |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( y  e. 
 QQ  |->  ( ( ( J `  P ) `
  y )  ^ c  R ) )  e.  A )
 
Theoremostth1 20745* - Lemma for ostth 20751: trivial case. (Not that the proof is trivial, but that we are proving that the function is trivial.) If  F is equal to  1 on the primes, then by complete induction and the multiplicative property abvmul 15557 of the absolute value,  F is equal to  1 on all the integers, and ostthlem1 20739 extends this to the other rational numbers. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  A. n  e.  NN  -.  1  <  ( F `
  n ) )   &    |-  ( ph  ->  A. n  e. 
 Prime  -.  ( F `  n )  <  1 )   =>    |-  ( ph  ->  F  =  K )
 
Theoremostth2lem2 20746* Lemma for ostth2 20749. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  2
 ) )   &    |-  S  =  ( ( log `  ( F `  M ) ) 
 /  ( log `  M ) )   &    |-  T  =  if ( ( F `  M )  <_  1 ,  1 ,  ( F `
  M ) )   =>    |-  ( ( ph  /\  X  e.  NN0  /\  Y  e.  ( 0 ... (
 ( M ^ X )  -  1 ) ) )  ->  ( F `  Y )  <_  (
 ( M  x.  X )  x.  ( T ^ X ) ) )
 
Theoremostth2lem3 20747* Lemma for ostth2 20749. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  2
 ) )   &    |-  S  =  ( ( log `  ( F `  M ) ) 
 /  ( log `  M ) )   &    |-  T  =  if ( ( F `  M )  <_  1 ,  1 ,  ( F `
  M ) )   &    |-  U  =  ( ( log `  N )  /  ( log `  M )
 )   =>    |-  ( ( ph  /\  X  e.  NN )  ->  (
 ( ( F `  N )  /  ( T  ^ c  U ) ) ^ X ) 
 <_  ( X  x.  (
 ( M  x.  T )  x.  ( U  +  1 ) ) ) )
 
Theoremostth2lem4 20748* Lemma for ostth2 20749. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  2
 ) )   &    |-  S  =  ( ( log `  ( F `  M ) ) 
 /  ( log `  M ) )   &    |-  T  =  if ( ( F `  M )  <_  1 ,  1 ,  ( F `
  M ) )   &    |-  U  =  ( ( log `  N )  /  ( log `  M )
 )   =>    |-  ( ph  ->  (
 1  <  ( F `  M )  /\  R  <_  S ) )
 
Theoremostth2 20749* - Lemma for ostth 20751: regular case. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   =>    |-  ( ph  ->  E. a  e.  ( 0 (,] 1
 ) F  =  ( y  e.  QQ  |->  ( ( abs `  y
 )  ^ c  a ) ) )
 
Theoremostth3 20750* - Lemma for ostth 20751: p-adic case. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  A. n  e.  NN  -.  1  <  ( F `
  n ) )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  ( F `  P )  <  1 )   &    |-  R  =  -u ( ( log `  ( F `  P ) )  /  ( log `  P ) )   &    |-  S  =  if (
 ( F `  P )  <_  ( F `  p ) ,  ( F `  p ) ,  ( F `  P ) )   =>    |-  ( ph  ->  E. a  e.  RR+  F  =  ( y  e.  QQ  |->  ( ( ( J `  P ) `  y
 )  ^ c  a ) ) )
 
Theoremostth 20751* Ostrowski's theorem, which classifies all absolute values on  QQ. Any such absolute value must either be the trivial absolute value  K, a constant exponent  0  <  a  <_  1 times the regular absolute value, or a positive exponent times the p-adic absolute value. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   =>    |-  ( F  e.  A  <->  ( F  =  K  \/  E. a  e.  ( 0 (,] 1 ) F  =  ( y  e. 
 QQ  |->  ( ( abs `  y )  ^ c  a ) )  \/ 
 E. a  e.  RR+  E. g  e.  ran  J  F  =  ( y  e.  QQ  |->  ( ( g `
  y )  ^ c  a ) ) ) )
 
PART 14  MISCELLANEA
 
14.1  Definitional Examples
 
Theoremex-or 20752 Example for df-or 361. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
 |-  ( 2  =  3  \/  4  =  4 )
 
Theoremex-an 20753 Example for df-an 362. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
 |-  ( 2  =  2 
 /\  3  =  3 )
 
Theoremex-dif 20754 Example for df-dif 3130. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( { 1 ,  3 }  \  {
 1 ,  8 } )  =  { 3 }
 
Theoremex-un 20755 Example for df-un 3132. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( { 1 ,  3 }  u.  {
 1 ,  8 } )  =  { 1 ,  3 ,  8 }
 
Theoremex-in 20756 Example for df-in 3134. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( { 1 ,  3 }  i^i  {
 1 ,  8 } )  =  { 1 }
 
Theoremex-uni 20757 Example for df-uni 3802. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
 |- 
 U. { { 1 ,  3 } ,  { 1 ,  8 } }  =  {
 1 ,  3 ,  8 }
 
Theoremex-ss 20758 Example for df-ss 3141. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |- 
 { 1 ,  2 }  C_  { 1 ,  2 ,  3 }
 
Theoremex-pss 20759 Example for df-pss 3143. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |- 
 { 1 ,  2 }  C.  { 1 ,  2 ,  3 }
 
Theoremex-pw 20760 Example for df-pw 3601. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
 |-  ( A  =  {
 3 ,  5 ,  7 }  ->  ~P A  =  ( ( { (/) }  u.  { { 3 } ,  { 5 } ,  { 7 } }
 )  u.  ( { { 3 ,  5 } ,  { 3 ,  7 } ,  { 5 ,  7 } }  u.  { { 3 ,  5 ,  7 } }
 ) ) )
 
Theoremex-pr 20761 Example for df-pr 3621. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( A  e.  {
 1 ,  -u 1 }  ->  ( A ^
 2 )  =  1 )
 
Theoremex-br 20762 Example for df-br 3998. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( R  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  3 R
 9 )
 
Theoremex-opab 20763* Example for df-opab 4052. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( R  =  { <. x ,  y >.  |  ( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1
 )  =  y ) }  ->  3 R
 4 )
 
Theoremex-eprel 20764 Example for df-eprel 4277. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  5  _E  { 1 ,  5 }
 
Theoremex-id 20765 Example for df-id 4281. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( 5  _I  5  /\  -.  4  _I  5
 )
 
Theoremex-po 20766 Example for df-po 4286. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  (  <  Po  RR  /\ 
 -.  <_  Po  RR )
 
Theoremex-xp 20767 Example for df-xp 4675. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( { 1 ,  5 }  X.  {
 2 ,  7 } )  =  ( { <. 1 ,  2 >. ,  <. 1 ,  7
 >. }  u.  { <. 5 ,  2 >. ,  <. 5 ,  7 >. } )
 
Theoremex-cnv 20768 Example for df-cnv 4677. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  `' { <. 2 ,  6
 >. ,  <. 3 ,  9
 >. }  =  { <. 6 ,  2 >. ,  <. 9 ,  3 >. }
 
Theoremex-co 20769 Example for df-co 4678. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ( exp  o.  cos ) `  0 )  =  _e
 
Theoremex-dm 20770 Example for df-dm 4679. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  dom  F  =  { 2 ,  3 } )
 
Theoremex-rn 20771 Example for df-rn 4680. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  ran  F  =  { 6 ,  9 } )
 
Theoremex-res 20772 Example for df-res 4681. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ( F  =  { <. 2 ,  6
 >. ,  <. 3 ,  9
 >. }  /\  B  =  { 1 ,  2 } )  ->  ( F  |`  B )  =  { <. 2 ,  6
 >. } )
 
Theoremex-ima 20773 Example for df-ima 4682. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ( F  =  { <. 2 ,  6
 >. ,  <. 3 ,  9
 >. }  /\  B  =  { 1 ,  2 } )  ->  ( F " B )  =  { 6 } )
 
Theoremex-fv 20774 Example for df-fv 4689. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  ( F `  3 )  =  9 )
 
Theoremex-1st 20775 Example for df-1st 6056. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( 1st `  <. 3 ,  4 >. )  =  3
 
Theoremex-2nd 20776 Example for df-2nd 6057. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( 2nd `  <. 3 ,  4 >. )  =  4
 
Theorem1kp2ke3k 20777 Example for df-dec 10093, 1000 + 2000 = 3000.

This proof disproves (by counter-example) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with  ( 2  +  1 )  =  3, commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 10093 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

 |-  (;;; 1 0 0 0  + ;;; 2 0 0 0 )  = ;;; 3 0 0 0
 
Theoremex-fl 20778 Example for df-fl 10892. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( ( |_ `  (
 3  /  2 )
 )  =  1  /\  ( |_ `  -u (
 3  /  2 )
 )  =  -u 2
 )
 
Theoremex-dvds 20779 3 divides into 6. A demonstration of df-divides 12495. (Contributed by David A. Wheeler, 19-May-2015.)
 |-  3  ||  6
 
14.2  Natural deduction examples

These are examples of how natural deduction rules can be applied in metamath (both as line-for-line translations of ND rules, and as a way to apply deduction forms without being limited to applying ND rules). For more information, see natded 4 and http://us.metamath.org/mpegif/mmnatded.html.

 
Theoremex-natded5.2 20780 Theorem 5.2 of [Clemente] p. 15, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows:
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
15  ( ( ps  /\  ch )  ->  th )  ( ph  ->  ( ( ps  /\  ch )  ->  th ) ) Given $e.
22  ( ch  ->  ps )  ( ph  ->  ( ch  ->  ps ) ) Given $e.
31  ch  ( ph  ->  ch ) Given $e.
43  ps  ( ph  ->  ps )  ->E 2,3 mpd 16, the MPE equivalent of  ->E, 1,2
54  ( ps  /\  ch )  ( ph  ->  ( ps  /\  ch ) )  /\I 4,3 jca 520, the MPE equivalent of  /\I, 3,1
66  th  ( ph  ->  th )  ->E 1,5 mpd 16, the MPE equivalent of  ->E, 4,5

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. Below is the final metamath proof (which reorders some steps). A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.2-2 20781. A proof without context is shown in ex-natded5.2i 20782. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   &    |-  ( ph  ->  ( ch  ->  ps ) )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  th )
 
Theoremex-natded5.2-2 20781 A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 20780 and ex-natded5.2i 20782. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   &    |-  ( ph  ->  ( ch  ->  ps ) )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  th )
 
Theoremex-natded5.2i 20782 The same as ex-natded5.2 20780 and ex-natded5.2-2 20781 but with no context. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ps  /\  ch )  ->  th )   &    |-  ( ch  ->  ps )   &    |-  ch   =>    |- 
 th
 
Theoremex-natded5.3 20783 Theorem 5.3 of [Clemente] p. 16, translated line by line using an interpretation of natural deduction in Metamath. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.3-2 20784. A proof without context is shown in ex-natded5.3i 20785. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer . The original proof, which uses Fitch style, was written as follows:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3  ( ps  ->  ch )  ( ph  ->  ( ps  ->  ch ) ) Given $e; adantr 453 to move it into the ND hypothesis
25;6  ( ch  ->  th )  ( ph  ->  ( ch  ->  th ) ) Given $e; adantr 453 to move it into the ND hypothesis
31 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption simpr 449, to access the new assumption
44 ...  ch  ( ( ph  /\  ps )  ->  ch )  ->E 1,3 mpd 16, the MPE equivalent of  ->E, 1.3. adantr 453 was used to transform its dependency (we could also use imp 420 to get this directly from 1)
57 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 16, the MPE equivalent of  ->E, 4,6. adantr 453 was used to transform its dependency
68 ...  ( ch  /\  th )  ( ( ph  /\  ps )  ->  ( ch  /\  th ) )  /\I 4,5 jca 520, the MPE equivalent of  /\I, 4,7
79  ( ps  ->  ( ch  /\  th ) )  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )  ->I 3,6 ex 425, the MPE equivalent of  ->I, 8

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremex-natded5.3-2 20784 A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 20783 and ex-natded5.3i 20785. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremex-natded5.3i 20785 The same as ex-natded5.3 20783 and ex-natded5.3-2 20784 but with no context. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ps  ->  ch )   &    |-  ( ch  ->  th )   =>    |-  ( ps  ->  ( ch  /\  th ) )
 
Theoremex-natded5.5 20786 Theorem 5.5 of [Clemente] p. 18, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3  ( ps  ->  ch )  ( ph  ->  ( ps  ->  ch ) ) Given $e; adantr 453 to move it into the ND hypothesis
25  -.  ch  ( ph  ->  -.  ch ) Given $e; we'll use adantr 453 to move it into the ND hypothesis
31 ...|  ps  ( ph  ->  ps ) ND hypothesis assumption simpr 449
44 ...  ch  ( ( ph  /\  ps )  ->  ch )  ->E 1,3 mpd 16 1,3
56 ...  -.  ch  ( ( ph  /\  ps )  ->  -.  ch ) IT 2 adantr 453 5
67  -.  ps  ( ph  ->  -.  ps )  /\I 3,4,5 pm2.65da 562 4,6

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 453; simpr 449 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof is mtod 170; a proof without context is shown in mto 169.

(Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  -.  ps )
 
Theoremex-natded5.7 20787 Theorem 5.7 of [Clemente] p. 19, translated line by line using the interpretation of natural deduction in Metamath. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.7-2 20788. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer . The original proof, which uses Fitch style, was written as follows:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
16  ( ps  \/  ( ch  /\  th ) )  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) ) Given $e. No need for adantr 453 because we do not move this into an ND hypothesis
21 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption (new scope) simpr 449
32 ...  ( ps  \/  ch )  ( ( ph  /\  ps )  ->  ( ps  \/  ch ) )  \/IL 2 orcd 383, the MPE equivalent of  \/IL, 1
43 ...|  ( ch  /\  th )  ( ( ph  /\  ( ch  /\  th ) )  ->  ( ch  /\  th ) ) ND hypothesis assumption (new scope) simpr 449
54 ...  ch  ( ( ph  /\  ( ch  /\  th ) )  ->  ch )  /\EL 4 simpld 447, the MPE equivalent of  /\EL, 3
66 ...  ( ps  \/  ch )  ( ( ph  /\  ( ch  /\  th ) )  ->  ( ps  \/  ch ) )  \/IR 5 olcd 384, the MPE equivalent of  \/IR, 4
77  ( ps  \/  ch )  ( ph  ->  ( ps  \/  ch ) )  \/E 1,3,6 mpjaodan 764, the MPE equivalent of  \/E, 2,5,6

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) )   =>    |-  ( ph  ->  ( ps  \/  ch )
 )
 
Theoremex-natded5.7-2 20788 A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with ex-natded5.7 20787. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) )   =>    |-  ( ph  ->  ( ps  \/  ch )
 )
 
Theoremex-natded5.8 20789 Theorem 5.8 of [Clemente] p. 20, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
110;11  ( ( ps  /\  ch )  ->  -.  th )  ( ph  ->  ( ( ps  /\  ch )  ->  -.  th ) ) Given $e; adantr 453 to move it into the ND hypothesis
23;4  ( ta  ->  th )  ( ph  ->  ( ta  ->  th ) ) Given $e; adantr 453 to move it into the ND hypothesis
37;8  ch  ( ph  ->  ch ) Given $e; adantr 453 to move it into the ND hypothesis
41;2  ta  ( ph  ->  ta ) Given $e. adantr 453 to move it into the ND hypothesis
56 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND Hypothesis/Assumption simpr 449. New ND hypothesis scope, each reference outside the scope must change antedent  ph to  ( ph  /\  ps ).
69 ...  ( ps  /\  ch )  ( ( ph  /\  ps )  ->  ( ps  /\  ch ) )  /\I 5,3 jca 520 ( /\I), 6,8 (adantr 453 to bring in scope)
75 ...  -.  th  ( ( ph  /\  ps )  ->  -.  th )  ->E 1,6 mpd 16 ( ->E), 2,4
812 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 16 ( ->E), 9,11; note the contradiction with ND line 7 (MPE line 5)
913  -.  ps  ( ph  ->  -.  ps )  -.I 5,7,8 pm2.65da 562 ( -.I), 5,12; proof by contradiction. MPE step 6 (ND#5) does not need a reference here, because the assumption is embedded in the antecedents

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 453; simpr 449 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.8-2 20790.

(Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  -.  th )
 )   &    |-  ( ph  ->  ( ta  ->  th ) )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ta )   =>    |-  ( ph  ->  -.  ps )
 
Theoremex-natded5.8-2 20790 A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 20789. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  -.  th )
 )   &    |-  ( ph  ->  ( ta  ->  th ) )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ta )   =>    |-  ( ph  ->  -.  ps )
 
Theoremex-natded5.13 20791 Theorem 5.13 of [Clemente] p. 20, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.13-2 20792. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
115  ( ps  \/  ch )  ( ph  ->  ( ps  \/  ch ) ) Given $e.
2;32  ( ps  ->  th )  ( ph  ->  ( ps  ->  th ) ) Given $e. adantr 453 to move it into the ND hypothesis
39  ( -.  ta  ->  -.  ch )  ( ph  ->  ( -.  ta  ->  -.  ch ) ) Given $e. ad2antrr 709 to move it into the ND sub-hypothesis
41 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption simpr 449
54 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 16 1,3
65 ...  ( th  \/  ta )  ( ( ph  /\  ps )  ->  ( th  \/  ta ) )  \/I 5 orcd 383 4
76 ...|  ch  ( ( ph  /\  ch )  ->  ch ) ND hypothesis assumption simpr 449
88 ... ...|  -.  ta  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ta ) (sub) ND hypothesis assumption simpr 449
911 ... ...  -.  ch  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ch )  ->E 3,8 mpd 16 8,10
107 ... ...  ch  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  ch ) IT 7 adantr 453 6
1112 ...  -.  -.  ta  ( ( ph  /\  ch )  ->  -.  -.  ta )  -.I 8,9,10 pm2.65da 562 7,11
1213 ...  ta  ( ( ph  /\  ch )  ->  ta )  -.E 11 notnotrd 107 12
1314 ...  ( th  \/  ta )  ( ( ph  /\  ch )  ->  ( th  \/  ta ) )  \/I 12 olcd 384 13
1416  ( th  \/  ta )  ( ph  ->  ( th  \/  ta ) )  \/E 1,6,13 mpjaodan 764 5,14,15

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 453; simpr 449 is useful when you want to depend directly on the new assumption). (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  ( ps  \/  ch ) )   &    |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ph  ->  ( -.  ta  ->  -.  ch ) )   =>    |-  ( ph  ->  ( th  \/  ta ) )
 
Theoremex-natded5.13-2 20792 A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 20791. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ( ps  \/  ch ) )   &    |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ph  ->  ( -.  ta  ->  -.  ch ) )   =>    |-  ( ph  ->  ( th  \/  ta ) )
 
Theoremex-natded9.20 20793 Theorem 9.20 of [Clemente] p. 43, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
11  ( ps  /\  ( ch  \/  th ) )  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) ) Given $e
22  ps  ( ph  ->  ps )  /\EL 1 simpld 447 1
311  ( ch  \/  th )  ( ph  ->  ( ch  \/  th ) )  /\ER 1 simprd 451 1
44 ...|  ch  ( ( ph  /\  ch )  ->  ch ) ND hypothesis assumption simpr 449
55 ...  ( ps  /\  ch )  ( ( ph  /\  ch )  ->  ( ps  /\  ch ) )  /\I 2,4 jca 520 3,4
66 ...  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ( ph  /\  ch )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/IR 5 orcd 383 5
78 ...|  th  ( ( ph  /\  th )  ->  th ) ND hypothesis assumption simpr 449
89 ...  ( ps  /\  th )  ( ( ph  /\  th )  ->  ( ps  /\  th ) )  /\I 2,7 jca 520 7,8
910 ...  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ( ph  /\  th )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/IL 8 olcd 384 9
1012  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/E 3,6,9 mpjaodan 764 6,10,11

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 453; simpr 449 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof is ex-natded9.20-2 20794. (Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

 |-  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
 
Theoremex-natded9.20-2 20794 A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 20793. (Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)
 |-  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
 
Theoremex-natded9.26 20795* Theorem 9.26 of [Clemente] p. 45, translated line by line using an interpretation of natural deduction in Metamath. This proof has some additional complications due to the fact that Metamath's existential elimination rule does not change bound variables, so we need to verify that  x is bound in the conclusion. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
13  E. x A. y ps ( x ,  y )  ( ph  ->  E. x A. y ps ) Given $e.
26 ...|  A. y ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  A. y ps ) ND hypothesis assumption simpr 449. Later statements will have this scope.
37;5,4 ...  ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  ps )  A.E 2,y a4sbcd 2979 ( A.E), 5,6. To use it we need a1i 12 and vex 2766. This could be immediately done with 19.21bi 1774, but we want to show the general approach for substitution.
412;8,9,10,11 ...  E. x ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  E. x ps )  E.I 3,a a4esbcd 3048 ( E.I), 11. To use it we need sylibr 205, which in turn requires sylib 190 and two uses of sbcid 2982. This could be more immediately done using 19.8a 1758, but we want to show the general approach for substitution.
513;1,2  E. x ps ( x ,  y )  ( ph  ->  E. x ps )  E.E 1,2,4,a exlimdd 1934 ( E.E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1629 and nfe1 1566 (MPE# 1,2)
614  A. y E. x ps ( x ,  y )  ( ph  ->  A. y E. x ps )  A.I 5 alrimiv 2013 ( A.I), 13

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. Below is the final metamath proof (which reorders some steps).

Note that in the original proof,  ps ( x ,  y ) has explicit parameters. In Metamath, these parameters are always implicit, and the parameters upon which a wff variable can depend are recorded in the "allowed substitution hints" below.

A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded9.26-2 20796.

(Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by David A. Wheeler, 18-Feb-2017.)

 |-  ( ph  ->  E. x A. y ps )   =>    |-  ( ph  ->  A. y E. x ps )
 
Theoremex-natded9.26-2 20796* A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 20795. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  E. x A. y ps )   =>    |-  ( ph  ->  A. y E. x ps )
 
14.3  Humor
 
14.3.1  April Fool's theorem
 
Theoremavril1 20797 Poisson d'Avril's Theorem. This theorem is noted for its Selbstdokumentieren property, which means, literally, "self-documenting" and recalls the principle of quidquid germanus dictum sit, altum viditur, often used in set theory. Starting with the seemingly simple yet profound fact that any object  x equals itself (proved by Tarski in 1965; see Lemma 6 of [Tarski] p. 68), we demonstrate that the power set of the real numbers, as a relation on the value of the imaginary unit, does not conjoin with an empty relation on the product of the additive and multiplicative identity elements, leading to this startling conclusion that has left even seasoned professional mathematicians scratching their heads. (Contributed by Prof. Loof Lirpa, 1-Apr-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

A reply to skeptics can be found at http://us.metamath.org/mpegif/mmnotes.txt, under the 1-Apr-2006 entry.

 |- 
 -.  ( A ~P RR ( _i `  1
 )  /\  F (/) ( 0  x.  1 ) )
 
Theorem2bornot2b 20798 The law of excluded middle. Act III, Theorem 1 of Shakespeare, Hamlet, Prince of Denmark (1602). Its author leaves its proof as an exercise for the reader - "To be, or not to be: that is the question" - starting a trend that has become standard in modern-day textbooks, serving to make the frustrated reader feel inferior, or in some cases to mask the fact that the author does not know its solution. (Contributed by Prof. Loof Lirpa, 1-Apr-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 2  x.  B  \/  -.  2  x.  B )
 
Theoremhelloworld 20799 The classic "Hello world" benchmark has been translated into 314 computer programming languages - see http://www.roesler-ac.de/wolfram/hello.htm. However, for many years it eluded a proof that it is more than just a conjecture, even though a wily mathematician once claimed, "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." Using an IBM 709 mainframe, a team of mathematicians led by Prof. Loof Lirpa, at the New College of Tahiti, were finally able put it rest with a remarkably short proof only 4 lines long. (Contributed by Prof. Loof Lirpa, 1-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 -.  ( h  e.  ( L L 0 )  /\  W (/) ( R. 1 d ) )
 
Theorem1p1e2apr1 20800 One plus one equals two. Using proof-shortening techniques pioneered by Mr. Mel O'Cat, along with the latest supercomputer technology, Prof. Loof Lirpa and colleagues were able to shorten Whitehead and Russell's 360-page proof that 1+1=2 in Principia Mathematica to this remarkable proof only two steps long, thus establishing a new world's record for this famous theorem. (Contributed by Prof. Loof Lirpa, 1-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 1  +  1 )  =  2
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