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Theorem rplogsum 20678
Description: The sum of  log p  /  p over the primes  p  ==  A (mod  N) is asymptotic to  log x  /  phi ( x )  +  O ( 1 ). Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 16-Apr-2016.)
Hypotheses
Ref Expression
rpvmasum.z  |-  Z  =  (ℤ/n `  N )
rpvmasum.l  |-  L  =  ( ZRHom `  Z
)
rpvmasum.a  |-  ( ph  ->  N  e.  NN )
rpvmasum.u  |-  U  =  (Unit `  Z )
rpvmasum.b  |-  ( ph  ->  A  e.  U )
rpvmasum.t  |-  T  =  ( `' L " { A } )
Assertion
Ref Expression
rplogsum  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( ( phi `  N )  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p )  /  p
) )  -  ( log `  x ) ) )  e.  O ( 1 ) )
Distinct variable groups:    x, p, A    N, p, x    ph, p, x    T, p, x    U, p, x    Z, p, x    L, p, x

Proof of Theorem rplogsum
StepHypRef Expression
1 rpvmasum.z . . 3  |-  Z  =  (ℤ/n `  N )
2 rpvmasum.l . . 3  |-  L  =  ( ZRHom `  Z
)
3 rpvmasum.a . . 3  |-  ( ph  ->  N  e.  NN )
4 rpvmasum.u . . 3  |-  U  =  (Unit `  Z )
5 rpvmasum.b . . 3  |-  ( ph  ->  A  e.  U )
6 rpvmasum.t . . 3  |-  T  =  ( `' L " { A } )
71, 2, 3, 4, 5, 6rpvmasum 20677 . 2  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( ( phi `  N )  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i 
T ) ( (Λ `  p )  /  p
) )  -  ( log `  x ) ) )  e.  O ( 1 ) )
83phicld 12842 . . . . . . 7  |-  ( ph  ->  ( phi `  N
)  e.  NN )
98adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( phi `  N )  e.  NN )
109nncnd 9764 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( phi `  N )  e.  CC )
11 fzfid 11037 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
12 inss1 3391 . . . . . . . 8  |-  ( ( 1 ... ( |_
`  x ) )  i^i  T )  C_  ( 1 ... ( |_ `  x ) )
13 ssfi 7085 . . . . . . . 8  |-  ( ( ( 1 ... ( |_ `  x ) )  e.  Fin  /\  (
( 1 ... ( |_ `  x ) )  i^i  T )  C_  ( 1 ... ( |_ `  x ) ) )  ->  ( (
1 ... ( |_ `  x ) )  i^i 
T )  e.  Fin )
1411, 12, 13sylancl 643 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( (
1 ... ( |_ `  x ) )  i^i 
T )  e.  Fin )
15 simpr 447 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )
1612, 15sseldi 3180 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  p  e.  ( 1 ... ( |_ `  x ) ) )
17 elfznn 10821 . . . . . . . . 9  |-  ( p  e.  ( 1 ... ( |_ `  x
) )  ->  p  e.  NN )
1816, 17syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  p  e.  NN )
19 vmacl 20358 . . . . . . . . 9  |-  ( p  e.  NN  ->  (Λ `  p )  e.  RR )
20 nndivre 9783 . . . . . . . . 9  |-  ( ( (Λ `  p )  e.  RR  /\  p  e.  NN )  ->  (
(Λ `  p )  /  p )  e.  RR )
2119, 20mpancom 650 . . . . . . . 8  |-  ( p  e.  NN  ->  (
(Λ `  p )  /  p )  e.  RR )
2218, 21syl 15 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  (
(Λ `  p )  /  p )  e.  RR )
2314, 22fsumrecl 12209 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( (Λ `  p
)  /  p )  e.  RR )
2423recnd 8863 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( (Λ `  p
)  /  p )  e.  CC )
2510, 24mulcld 8857 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( phi `  N )  x. 
sum_ p  e.  (
( 1 ... ( |_ `  x ) )  i^i  T ) ( (Λ `  p )  /  p ) )  e.  CC )
26 relogcl 19934 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2726adantl 452 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
2827recnd 8863 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
2925, 28subcld 9159 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( (
( phi `  N
)  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( (Λ `  p
)  /  p ) )  -  ( log `  x ) )  e.  CC )
30 inss1 3391 . . . . . . . 8  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  (
1 ... ( |_ `  x ) )
31 ssfi 7085 . . . . . . . 8  |-  ( ( ( 1 ... ( |_ `  x ) )  e.  Fin  /\  (
( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  C_  (
1 ... ( |_ `  x ) ) )  ->  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  e.  Fin )
3211, 30, 31sylancl 643 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( (
1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  e.  Fin )
33 simpr 447 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )
3430, 33sseldi 3180 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  p  e.  ( 1 ... ( |_ `  x ) ) )
3534, 17syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  p  e.  NN )
36 nnrp 10365 . . . . . . . . . 10  |-  ( p  e.  NN  ->  p  e.  RR+ )
3736relogcld 19976 . . . . . . . . 9  |-  ( p  e.  NN  ->  ( log `  p )  e.  RR )
3837, 36rerpdivcld 10419 . . . . . . . 8  |-  ( p  e.  NN  ->  (
( log `  p
)  /  p )  e.  RR )
3935, 38syl 15 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  ( ( log `  p )  /  p )  e.  RR )
4032, 39fsumrecl 12209 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p )  e.  RR )
4140recnd 8863 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p )  e.  CC )
4210, 41mulcld 8857 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( phi `  N )  x. 
sum_ p  e.  (
( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p )  /  p ) )  e.  CC )
4342, 28subcld 9159 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( (
( phi `  N
)  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p ) )  -  ( log `  x ) )  e.  CC )
4410, 24, 41subdid 9237 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( phi `  N )  x.  ( sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  T ) ( (Λ `  p )  /  p )  -  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p )  /  p
) ) )  =  ( ( ( phi `  N )  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i 
T ) ( (Λ `  p )  /  p
) )  -  (
( phi `  N
)  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p ) ) ) )
4519recnd 8863 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  (Λ `  p )  e.  CC )
46 0re 8840 . . . . . . . . . . . . 13  |-  0  e.  RR
47 ifcl 3603 . . . . . . . . . . . . 13  |-  ( ( ( log `  p
)  e.  RR  /\  0  e.  RR )  ->  if ( p  e. 
Prime ,  ( log `  p ) ,  0 )  e.  RR )
4837, 46, 47sylancl 643 . . . . . . . . . . . 12  |-  ( p  e.  NN  ->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  e.  RR )
4948recnd 8863 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  e.  CC )
5036rpcnne0d 10401 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  (
p  e.  CC  /\  p  =/=  0 ) )
51 divsubdir 9458 . . . . . . . . . . 11  |-  ( ( (Λ `  p )  e.  CC  /\  if ( p  e.  Prime ,  ( log `  p ) ,  0 )  e.  CC  /\  ( p  e.  CC  /\  p  =/=  0 ) )  -> 
( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p )  =  ( ( (Λ `  p
)  /  p )  -  ( if ( p  e.  Prime ,  ( log `  p ) ,  0 )  /  p ) ) )
5245, 49, 50, 51syl3anc 1182 . . . . . . . . . 10  |-  ( p  e.  NN  ->  (
( (Λ `  p )  -  if ( p  e. 
Prime ,  ( log `  p ) ,  0 ) )  /  p
)  =  ( ( (Λ `  p )  /  p )  -  ( if ( p  e.  Prime ,  ( log `  p
) ,  0 )  /  p ) ) )
5318, 52syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  (
( (Λ `  p )  -  if ( p  e. 
Prime ,  ( log `  p ) ,  0 ) )  /  p
)  =  ( ( (Λ `  p )  /  p )  -  ( if ( p  e.  Prime ,  ( log `  p
) ,  0 )  /  p ) ) )
5453sumeq2dv 12178 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( ( (Λ `  p )  -  if ( p  e.  Prime ,  ( log `  p
) ,  0 ) )  /  p )  =  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  T ) ( ( (Λ `  p
)  /  p )  -  ( if ( p  e.  Prime ,  ( log `  p ) ,  0 )  /  p ) ) )
5521recnd 8863 . . . . . . . . . 10  |-  ( p  e.  NN  ->  (
(Λ `  p )  /  p )  e.  CC )
5618, 55syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  (
(Λ `  p )  /  p )  e.  CC )
5748, 36rerpdivcld 10419 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  ( if ( p  e.  Prime ,  ( log `  p
) ,  0 )  /  p )  e.  RR )
5857recnd 8863 . . . . . . . . . 10  |-  ( p  e.  NN  ->  ( if ( p  e.  Prime ,  ( log `  p
) ,  0 )  /  p )  e.  CC )
5918, 58syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  ( if ( p  e.  Prime ,  ( log `  p
) ,  0 )  /  p )  e.  CC )
6014, 56, 59fsumsub 12252 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( ( (Λ `  p )  /  p
)  -  ( if ( p  e.  Prime ,  ( log `  p
) ,  0 )  /  p ) )  =  ( sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( (Λ `  p
)  /  p )  -  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  T ) ( if ( p  e. 
Prime ,  ( log `  p ) ,  0 )  /  p ) ) )
61 inss2 3392 . . . . . . . . . . . 12  |-  ( Prime  i^i  T )  C_  T
62 sslin 3397 . . . . . . . . . . . 12  |-  ( ( Prime  i^i  T )  C_  T  ->  ( (
1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( (
1 ... ( |_ `  x ) )  i^i 
T ) )
6361, 62mp1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( (
1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( (
1 ... ( |_ `  x ) )  i^i 
T ) )
6435, 58syl 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  ( if ( p  e.  Prime ,  ( log `  p
) ,  0 )  /  p )  e.  CC )
65 eldif 3164 . . . . . . . . . . . . . . . 16  |-  ( p  e.  ( ( ( 1 ... ( |_
`  x ) )  i^i  T )  \ 
( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  <->  ( p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
)  /\  -.  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ) )
66 incom 3363 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Prime  i^i  T )  =  ( T  i^i  Prime )
6766ineq2i 3369 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  =  ( ( 1 ... ( |_ `  x ) )  i^i  ( T  i^i  Prime
) )
68 inass 3381 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1 ... ( |_ `  x ) )  i^i  T )  i^i 
Prime )  =  (
( 1 ... ( |_ `  x ) )  i^i  ( T  i^i  Prime
) )
6967, 68eqtr4i 2308 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  =  ( ( ( 1 ... ( |_ `  x
) )  i^i  T
)  i^i  Prime )
7069elin2 3361 . . . . . . . . . . . . . . . . . 18  |-  ( p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  <->  ( p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
)  /\  p  e.  Prime ) )
7170simplbi2 608 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  ( ( 1 ... ( |_ `  x ) )  i^i 
T )  ->  (
p  e.  Prime  ->  p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ) )
7271con3and 428 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  ( ( 1 ... ( |_
`  x ) )  i^i  T )  /\  -.  p  e.  (
( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) )  ->  -.  p  e.  Prime )
7365, 72sylbi 187 . . . . . . . . . . . . . . 15  |-  ( p  e.  ( ( ( 1 ... ( |_
`  x ) )  i^i  T )  \ 
( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  -.  p  e.  Prime )
7473adantl 452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( ( 1 ... ( |_ `  x ) )  i^i 
T )  \  (
( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ) )  ->  -.  p  e.  Prime )
75 iffalse 3574 . . . . . . . . . . . . . 14  |-  ( -.  p  e.  Prime  ->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  =  0 )
7674, 75syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( ( 1 ... ( |_ `  x ) )  i^i 
T )  \  (
( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ) )  ->  if ( p  e.  Prime ,  ( log `  p ) ,  0 )  =  0 )
7776oveq1d 5875 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( ( 1 ... ( |_ `  x ) )  i^i 
T )  \  (
( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ) )  ->  ( if ( p  e.  Prime ,  ( log `  p ) ,  0 )  /  p )  =  ( 0  /  p ) )
78 eldifi 3300 . . . . . . . . . . . . . 14  |-  ( p  e.  ( ( ( 1 ... ( |_
`  x ) )  i^i  T )  \ 
( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )
7978, 18sylan2 460 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( ( 1 ... ( |_ `  x ) )  i^i 
T )  \  (
( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ) )  ->  p  e.  NN )
80 div0 9454 . . . . . . . . . . . . . 14  |-  ( ( p  e.  CC  /\  p  =/=  0 )  -> 
( 0  /  p
)  =  0 )
8150, 80syl 15 . . . . . . . . . . . . 13  |-  ( p  e.  NN  ->  (
0  /  p )  =  0 )
8279, 81syl 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( ( 1 ... ( |_ `  x ) )  i^i 
T )  \  (
( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ) )  ->  ( 0  /  p )  =  0 )
8377, 82eqtrd 2317 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( ( 1 ... ( |_ `  x ) )  i^i 
T )  \  (
( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ) )  ->  ( if ( p  e.  Prime ,  ( log `  p ) ,  0 )  /  p )  =  0 )
8463, 64, 83, 14fsumss 12200 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( if ( p  e.  Prime ,  ( log `  p ) ,  0 )  /  p )  =  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  T ) ( if ( p  e. 
Prime ,  ( log `  p ) ,  0 )  /  p ) )
85 inss2 3392 . . . . . . . . . . . . . . 15  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( Prime  i^i  T )
86 inss1 3391 . . . . . . . . . . . . . . 15  |-  ( Prime  i^i  T )  C_  Prime
8785, 86sstri 3190 . . . . . . . . . . . . . 14  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  Prime
8887, 33sseldi 3180 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  p  e.  Prime )
89 iftrue 3573 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  if ( p  e.  Prime ,  ( log `  p ) ,  0 )  =  ( log `  p
) )
9088, 89syl 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  if (
p  e.  Prime ,  ( log `  p ) ,  0 )  =  ( log `  p
) )
9190oveq1d 5875 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  ( if ( p  e.  Prime ,  ( log `  p
) ,  0 )  /  p )  =  ( ( log `  p
)  /  p ) )
9291sumeq2dv 12178 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( if ( p  e.  Prime ,  ( log `  p ) ,  0 )  /  p )  =  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p )  /  p ) )
9384, 92eqtr3d 2319 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( if ( p  e.  Prime ,  ( log `  p ) ,  0 )  /  p )  =  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p )  /  p
) )
9493oveq2d 5876 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i 
T ) ( (Λ `  p )  /  p
)  -  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( if ( p  e.  Prime ,  ( log `  p ) ,  0 )  /  p ) )  =  ( sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  T ) ( (Λ `  p )  /  p )  -  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p )  /  p
) ) )
9554, 60, 943eqtrd 2321 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( ( (Λ `  p )  -  if ( p  e.  Prime ,  ( log `  p
) ,  0 ) )  /  p )  =  ( sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( (Λ `  p
)  /  p )  -  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p )  /  p ) ) )
9695oveq2d 5876 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( phi `  N )  x. 
sum_ p  e.  (
( 1 ... ( |_ `  x ) )  i^i  T ) ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p ) )  =  ( ( phi `  N )  x.  ( sum_ p  e.  ( ( 1 ... ( |_
`  x ) )  i^i  T ) ( (Λ `  p )  /  p )  -  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p )  /  p
) ) ) )
9725, 42, 28nnncan2d 9194 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( (
( ( phi `  N )  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i 
T ) ( (Λ `  p )  /  p
) )  -  ( log `  x ) )  -  ( ( ( phi `  N )  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p )  /  p ) )  -  ( log `  x
) ) )  =  ( ( ( phi `  N )  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i 
T ) ( (Λ `  p )  /  p
) )  -  (
( phi `  N
)  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p ) ) ) )
9844, 96, 973eqtr4d 2327 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( phi `  N )  x. 
sum_ p  e.  (
( 1 ... ( |_ `  x ) )  i^i  T ) ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p ) )  =  ( ( ( ( phi `  N
)  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( (Λ `  p
)  /  p ) )  -  ( log `  x ) )  -  ( ( ( phi `  N )  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p )  /  p
) )  -  ( log `  x ) ) ) )
9998mpteq2dva 4108 . . . 4  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( phi `  N )  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i 
T ) ( ( (Λ `  p )  -  if ( p  e. 
Prime ,  ( log `  p ) ,  0 ) )  /  p
) ) )  =  ( x  e.  RR+  |->  ( ( ( ( phi `  N )  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  T ) ( (Λ `  p )  /  p ) )  -  ( log `  x ) )  -  ( ( ( phi `  N
)  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p ) )  -  ( log `  x ) ) ) ) )
10019, 48resubcld 9213 . . . . . . . . 9  |-  ( p  e.  NN  ->  (
(Λ `  p )  -  if ( p  e.  Prime ,  ( log `  p
) ,  0 ) )  e.  RR )
101100, 36rerpdivcld 10419 . . . . . . . 8  |-  ( p  e.  NN  ->  (
( (Λ `  p )  -  if ( p  e. 
Prime ,  ( log `  p ) ,  0 ) )  /  p
)  e.  RR )
10218, 101syl 15 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  (
( (Λ `  p )  -  if ( p  e. 
Prime ,  ( log `  p ) ,  0 ) )  /  p
)  e.  RR )
10314, 102fsumrecl 12209 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( ( (Λ `  p )  -  if ( p  e.  Prime ,  ( log `  p
) ,  0 ) )  /  p )  e.  RR )
104103recnd 8863 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( ( (Λ `  p )  -  if ( p  e.  Prime ,  ( log `  p
) ,  0 ) )  /  p )  e.  CC )
105 rpssre 10366 . . . . . 6  |-  RR+  C_  RR
1068nncnd 9764 . . . . . 6  |-  ( ph  ->  ( phi `  N
)  e.  CC )
107 o1const 12095 . . . . . 6  |-  ( (
RR+  C_  RR  /\  ( phi `  N )  e.  CC )  ->  (
x  e.  RR+  |->  ( phi `  N ) )  e.  O ( 1 ) )
108105, 106, 107sylancr 644 . . . . 5  |-  ( ph  ->  ( x  e.  RR+  |->  ( phi `  N ) )  e.  O ( 1 ) )
109105a1i 10 . . . . . 6  |-  ( ph  -> 
RR+  C_  RR )
110 1re 8839 . . . . . . 7  |-  1  e.  RR
111110a1i 10 . . . . . 6  |-  ( ph  ->  1  e.  RR )
112 2re 9817 . . . . . . 7  |-  2  e.  RR
113112a1i 10 . . . . . 6  |-  ( ph  ->  2  e.  RR )
114 breq1 4028 . . . . . . . . . . . . . 14  |-  ( ( log `  p )  =  if ( p  e.  Prime ,  ( log `  p ) ,  0 )  ->  ( ( log `  p )  <_ 
(Λ `  p )  <->  if (
p  e.  Prime ,  ( log `  p ) ,  0 )  <_ 
(Λ `  p ) ) )
115 breq1 4028 . . . . . . . . . . . . . 14  |-  ( 0  =  if ( p  e.  Prime ,  ( log `  p ) ,  0 )  ->  ( 0  <_  (Λ `  p )  <->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  <_  (Λ `  p )
) )
11637adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  NN  /\  p  e.  Prime )  -> 
( log `  p
)  e.  RR )
117 vmaprm 20357 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  Prime  ->  (Λ `  p
)  =  ( log `  p ) )
118117adantl 452 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  NN  /\  p  e.  Prime )  -> 
(Λ `  p )  =  ( log `  p
) )
119118eqcomd 2290 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  NN  /\  p  e.  Prime )  -> 
( log `  p
)  =  (Λ `  p
) )
120 eqle 8925 . . . . . . . . . . . . . . 15  |-  ( ( ( log `  p
)  e.  RR  /\  ( log `  p )  =  (Λ `  p
) )  ->  ( log `  p )  <_ 
(Λ `  p ) )
121116, 119, 120syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( p  e.  NN  /\  p  e.  Prime )  -> 
( log `  p
)  <_  (Λ `  p
) )
122 vmage0 20361 . . . . . . . . . . . . . . 15  |-  ( p  e.  NN  ->  0  <_  (Λ `  p )
)
123122adantr 451 . . . . . . . . . . . . . 14  |-  ( ( p  e.  NN  /\  -.  p  e.  Prime )  ->  0  <_  (Λ `  p ) )
124114, 115, 121, 123ifbothda 3597 . . . . . . . . . . . . 13  |-  ( p  e.  NN  ->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  <_  (Λ `  p )
)
12519, 48subge0d 9364 . . . . . . . . . . . . 13  |-  ( p  e.  NN  ->  (
0  <_  ( (Λ `  p )  -  if ( p  e.  Prime ,  ( log `  p
) ,  0 ) )  <->  if ( p  e. 
Prime ,  ( log `  p ) ,  0 )  <_  (Λ `  p
) ) )
126124, 125mpbird 223 . . . . . . . . . . . 12  |-  ( p  e.  NN  ->  0  <_  ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) ) )
127100, 36, 126divge0d 10428 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  0  <_  ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p ) )
12818, 127syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  0  <_  ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p ) )
12914, 102, 128fsumge0 12255 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  0  <_  sum_
p  e.  ( ( 1 ... ( |_
`  x ) )  i^i  T ) ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p ) )
130103, 129absidd 11907 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ p  e.  (
( 1 ... ( |_ `  x ) )  i^i  T ) ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p ) )  =  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  T ) ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p ) )
13117adantl 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( 1 ... ( |_ `  x ) ) )  ->  p  e.  NN )
132131, 101syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
(Λ `  p )  -  if ( p  e.  Prime ,  ( log `  p
) ,  0 ) )  /  p )  e.  RR )
13311, 132fsumrecl 12209 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( 1 ... ( |_ `  x ) ) ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p )  e.  RR )
134112a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  2  e.  RR )
135131, 127syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p ) )
13612a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( (
1 ... ( |_ `  x ) )  i^i 
T )  C_  (
1 ... ( |_ `  x ) ) )
13711, 132, 135, 136fsumless 12256 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( ( (Λ `  p )  -  if ( p  e.  Prime ,  ( log `  p
) ,  0 ) )  /  p )  <_  sum_ p  e.  ( 1 ... ( |_
`  x ) ) ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p ) )
138109sselda 3182 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR )
139138flcld 10932 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( |_ `  x )  e.  ZZ )
140 rplogsumlem2 20636 . . . . . . . . . 10  |-  ( ( |_ `  x )  e.  ZZ  ->  sum_ p  e.  ( 1 ... ( |_ `  x ) ) ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p )  <_ 
2 )
141139, 140syl 15 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( 1 ... ( |_ `  x ) ) ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p )  <_ 
2 )
142103, 133, 134, 137, 141letrd 8975 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( ( (Λ `  p )  -  if ( p  e.  Prime ,  ( log `  p
) ,  0 ) )  /  p )  <_  2 )
143130, 142eqbrtrd 4045 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ p  e.  (
( 1 ... ( |_ `  x ) )  i^i  T ) ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p ) )  <_  2 )
144143adantrr 697 . . . . . 6  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i 
T ) ( ( (Λ `  p )  -  if ( p  e. 
Prime ,  ( log `  p ) ,  0 ) )  /  p
) )  <_  2
)
145109, 104, 111, 113, 144elo1d 12012 . . . . 5  |-  ( ph  ->  ( x  e.  RR+  |->  sum_
p  e.  ( ( 1 ... ( |_
`  x ) )  i^i  T ) ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p ) )  e.  O ( 1 ) )
14610, 104, 108, 145o1mul2 12100 . . . 4  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( phi `  N )  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i 
T ) ( ( (Λ `  p )  -  if ( p  e. 
Prime ,  ( log `  p ) ,  0 ) )  /  p
) ) )  e.  O ( 1 ) )
14799, 146eqeltrrd 2360 . . 3  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( ( ( phi `  N )  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  T ) ( (Λ `  p )  /  p ) )  -  ( log `  x ) )  -  ( ( ( phi `  N
)  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p ) )  -  ( log `  x ) ) ) )  e.  O ( 1 ) )
14829, 43, 147o1dif 12105 . 2  |-  ( ph  ->  ( ( x  e.  RR+  |->  ( ( ( phi `  N )  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  T ) ( (Λ `  p )  /  p ) )  -  ( log `  x ) ) )  e.  O
( 1 )  <->  ( x  e.  RR+  |->  ( ( ( phi `  N )  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p )  /  p ) )  -  ( log `  x
) ) )  e.  O ( 1 ) ) )
1497, 148mpbid 201 1  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( ( phi `  N )  x.  sum_ p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p )  /  p
) )  -  ( log `  x ) ) )  e.  O ( 1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686    =/= wne 2448    \ cdif 3151    i^i cin 3153    C_ wss 3154   ifcif 3567   {csn 3642   class class class wbr 4025    e. cmpt 4079   `'ccnv 4690   "cima 4694   ` cfv 5257  (class class class)co 5860   Fincfn 6865   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    x. cmul 8744    <_ cle 8870    - cmin 9039    / cdiv 9425   NNcn 9748   2c2 9797   ZZcz 10026   RR+crp 10356   ...cfz 10784   |_cfl 10926   abscabs 11721   O (
1 )co1 11962   sum_csu 12160   Primecprime 12760   phicphi 12834  Unitcui 15423   ZRHomczrh 16453  ℤ/nczn 16456   logclog 19914  Λcvma 20331
This theorem is referenced by:  dirith2  20679
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-disj 3996  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-tpos 6236  df-rpss 6279  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-omul 6486  df-er 6662  df-ec 6664  df-qs 6668  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-acn 7577  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ioc 10663  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-word 11411  df-concat 11412  df-s1 11413  df-shft 11564  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-limsup 11947  df-clim 11964  df-rlim 11965  df-o1 11966  df-lo1 11967  df-sum 12161  df-ef 12351  df-e 12352  df-sin 12353  df-cos 12354  df-tan 12355  df-pi 12356  df-dvds 12534  df-gcd 12688  df-prm 12761  df-numer 12808  df-denom 12809  df-phi 12836  df-pc 12892  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-divs 13414  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-mhm 14417  df-submnd 14418  df-grp 14491  df-minusg 14492  df-sbg 14493  df-mulg 14494  df-subg 14620  df-nsg 14621  df-eqg 14622  df-ghm 14683  df-gim 14725  df-ga 14746  df-cntz 14795  df-oppg 14821  df-od 14846  df-gex 14847  df-pgp 14848  df-lsm 14949  df-pj1 14950  df-cmn 15093  df-abl 15094  df-cyg 15167  df-dprd 15235  df-dpj 15236  df-mgp 15328  df-rng 15342  df-cring 15343  df-ur 15344  df-oppr 15407  df-dvdsr 15425  df-unit 15426  df-invr 15456  df-dvr 15467  df-rnghom 15498  df-drng 15516  df-subrg 15545  df-lmod 15631  df-lss 15692  df-lsp 15731  df-sra 15927  df-rgmod 15928  df-lidl 15929  df-rsp 15930  df-2idl 15986  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-zrh 16457  df-zn 16460  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-cmp 17116  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-0p 19027  df-limc 19218  df-dv 19219  df-ply 19572  df-idp 19573  df-coe 19574  df-dgr 19575  df-quot 19673  df-log 19916  df-cxp 19917  df-em 20289  df-cht 20336  df-vma 20337  df-chp 20338  df-ppi 20339  df-mu 20340  df-dchr 20474
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