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Theorem rplogsum 21174
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 21173 . 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 13116 . . . . . . 7  |-  ( ph  ->  ( phi `  N
)  e.  NN )
98adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( phi `  N )  e.  NN )
109nncnd 9972 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( phi `  N )  e.  CC )
11 fzfid 11267 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
12 inss1 3521 . . . . . . . 8  |-  ( ( 1 ... ( |_
`  x ) )  i^i  T )  C_  ( 1 ... ( |_ `  x ) )
13 ssfi 7288 . . . . . . . 8  |-  ( ( ( 1 ... ( |_ `  x ) )  e.  Fin  /\  (
( 1 ... ( |_ `  x ) )  i^i  T )  C_  ( 1 ... ( |_ `  x ) ) )  ->  ( (
1 ... ( |_ `  x ) )  i^i 
T )  e.  Fin )
1411, 12, 13sylancl 644 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( (
1 ... ( |_ `  x ) )  i^i 
T )  e.  Fin )
15 simpr 448 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )
1612, 15sseldi 3306 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  p  e.  ( 1 ... ( |_ `  x ) ) )
17 elfznn 11036 . . . . . . . . 9  |-  ( p  e.  ( 1 ... ( |_ `  x
) )  ->  p  e.  NN )
1816, 17syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  p  e.  NN )
19 vmacl 20854 . . . . . . . . 9  |-  ( p  e.  NN  ->  (Λ `  p )  e.  RR )
20 nndivre 9991 . . . . . . . . 9  |-  ( ( (Λ `  p )  e.  RR  /\  p  e.  NN )  ->  (
(Λ `  p )  /  p )  e.  RR )
2119, 20mpancom 651 . . . . . . . 8  |-  ( p  e.  NN  ->  (
(Λ `  p )  /  p )  e.  RR )
2218, 21syl 16 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  (
(Λ `  p )  /  p )  e.  RR )
2314, 22fsumrecl 12483 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( (Λ `  p
)  /  p )  e.  RR )
2423recnd 9070 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( (Λ `  p
)  /  p )  e.  CC )
2510, 24mulcld 9064 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( phi `  N )  x. 
sum_ p  e.  (
( 1 ... ( |_ `  x ) )  i^i  T ) ( (Λ `  p )  /  p ) )  e.  CC )
26 relogcl 20426 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2726adantl 453 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
2827recnd 9070 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
2925, 28subcld 9367 . . 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 3521 . . . . . . . 8  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  (
1 ... ( |_ `  x ) )
31 ssfi 7288 . . . . . . . 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 644 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( (
1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  e.  Fin )
33 simpr 448 . . . . . . . . . 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 3306 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  p  e.  ( 1 ... ( |_ `  x ) ) )
3534, 17syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  p  e.  NN )
36 nnrp 10577 . . . . . . . . . 10  |-  ( p  e.  NN  ->  p  e.  RR+ )
3736relogcld 20471 . . . . . . . . 9  |-  ( p  e.  NN  ->  ( log `  p )  e.  RR )
3837, 36rerpdivcld 10631 . . . . . . . 8  |-  ( p  e.  NN  ->  (
( log `  p
)  /  p )  e.  RR )
3935, 38syl 16 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  ( ( log `  p )  /  p )  e.  RR )
4032, 39fsumrecl 12483 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p )  e.  RR )
4140recnd 9070 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p )  e.  CC )
4210, 41mulcld 9064 . . . 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 9367 . . 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 9445 . . . . . 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 9070 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  (Λ `  p )  e.  CC )
46 0re 9047 . . . . . . . . . . . . 13  |-  0  e.  RR
47 ifcl 3735 . . . . . . . . . . . . 13  |-  ( ( ( log `  p
)  e.  RR  /\  0  e.  RR )  ->  if ( p  e. 
Prime ,  ( log `  p ) ,  0 )  e.  RR )
4837, 46, 47sylancl 644 . . . . . . . . . . . 12  |-  ( p  e.  NN  ->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  e.  RR )
4948recnd 9070 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  e.  CC )
5036rpcnne0d 10613 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  (
p  e.  CC  /\  p  =/=  0 ) )
51 divsubdir 9666 . . . . . . . . . . 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 1184 . . . . . . . . . 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 16 . . . . . . . . 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 12452 . . . . . . . 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 9070 . . . . . . . . . 10  |-  ( p  e.  NN  ->  (
(Λ `  p )  /  p )  e.  CC )
5618, 55syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  (
(Λ `  p )  /  p )  e.  CC )
5748, 36rerpdivcld 10631 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  ( if ( p  e.  Prime ,  ( log `  p
) ,  0 )  /  p )  e.  RR )
5857recnd 9070 . . . . . . . . . 10  |-  ( p  e.  NN  ->  ( if ( p  e.  Prime ,  ( log `  p
) ,  0 )  /  p )  e.  CC )
5918, 58syl 16 . . . . . . . . 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 12526 . . . . . . . 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 3522 . . . . . . . . . . . 12  |-  ( Prime  i^i  T )  C_  T
62 sslin 3527 . . . . . . . . . . . 12  |-  ( ( Prime  i^i  T )  C_  T  ->  ( (
1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( (
1 ... ( |_ `  x ) )  i^i 
T ) )
6361, 62mp1i 12 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( (
1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( (
1 ... ( |_ `  x ) )  i^i 
T ) )
6435, 58syl 16 . . . . . . . . . . 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 3290 . . . . . . . . . . . . . . . 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 3493 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Prime  i^i  T )  =  ( T  i^i  Prime )
6766ineq2i 3499 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  =  ( ( 1 ... ( |_ `  x ) )  i^i  ( T  i^i  Prime
) )
68 inass 3511 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1 ... ( |_ `  x ) )  i^i  T )  i^i 
Prime )  =  (
( 1 ... ( |_ `  x ) )  i^i  ( T  i^i  Prime
) )
6967, 68eqtr4i 2427 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  =  ( ( ( 1 ... ( |_ `  x
) )  i^i  T
)  i^i  Prime )
7069elin2 3491 . . . . . . . . . . . . . . . . . 18  |-  ( p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  <->  ( p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
)  /\  p  e.  Prime ) )
7170simplbi2 609 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  ( ( 1 ... ( |_ `  x ) )  i^i 
T )  ->  (
p  e.  Prime  ->  p  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ) )
7271con3and 429 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  ( ( 1 ... ( |_
`  x ) )  i^i  T )  /\  -.  p  e.  (
( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) )  ->  -.  p  e.  Prime )
7365, 72sylbi 188 . . . . . . . . . . . . . . 15  |-  ( p  e.  ( ( ( 1 ... ( |_
`  x ) )  i^i  T )  \ 
( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  -.  p  e.  Prime )
7473adantl 453 . . . . . . . . . . . . . 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 3706 . . . . . . . . . . . . . 14  |-  ( -.  p  e.  Prime  ->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  =  0 )
7674, 75syl 16 . . . . . . . . . . . . 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 6055 . . . . . . . . . . . 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 3429 . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . 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 9662 . . . . . . . . . . . . . 14  |-  ( ( p  e.  CC  /\  p  =/=  0 )  -> 
( 0  /  p
)  =  0 )
8150, 80syl 16 . . . . . . . . . . . . 13  |-  ( p  e.  NN  ->  (
0  /  p )  =  0 )
8279, 81syl 16 . . . . . . . . . . . 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 2436 . . . . . . . . . . 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 12474 . . . . . . . . . 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 3522 . . . . . . . . . . . . . . 15  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( Prime  i^i  T )
86 inss1 3521 . . . . . . . . . . . . . . 15  |-  ( Prime  i^i  T )  C_  Prime
8785, 86sstri 3317 . . . . . . . . . . . . . 14  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  Prime
8887, 33sseldi 3306 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  p  e.  Prime )
89 iftrue 3705 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  if ( p  e.  Prime ,  ( log `  p ) ,  0 )  =  ( log `  p
) )
9088, 89syl 16 . . . . . . . . . . . 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 6055 . . . . . . . . . . 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 12452 . . . . . . . . . 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 2438 . . . . . . . . 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 6056 . . . . . . . 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 2440 . . . . . . 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 6056 . . . . . 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 9402 . . . . . 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 2446 . . . . 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 4255 . . . 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 9421 . . . . . . . . 9  |-  ( p  e.  NN  ->  (
(Λ `  p )  -  if ( p  e.  Prime ,  ( log `  p
) ,  0 ) )  e.  RR )
101100, 36rerpdivcld 10631 . . . . . . . 8  |-  ( p  e.  NN  ->  (
( (Λ `  p )  -  if ( p  e. 
Prime ,  ( log `  p ) ,  0 ) )  /  p
)  e.  RR )
10218, 101syl 16 . . . . . . 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 12483 . . . . . 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 9070 . . . . 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 10578 . . . . . 6  |-  RR+  C_  RR
1068nncnd 9972 . . . . . 6  |-  ( ph  ->  ( phi `  N
)  e.  CC )
107 o1const 12368 . . . . . 6  |-  ( (
RR+  C_  RR  /\  ( phi `  N )  e.  CC )  ->  (
x  e.  RR+  |->  ( phi `  N ) )  e.  O ( 1 ) )
108105, 106, 107sylancr 645 . . . . 5  |-  ( ph  ->  ( x  e.  RR+  |->  ( phi `  N ) )  e.  O ( 1 ) )
109105a1i 11 . . . . . 6  |-  ( ph  -> 
RR+  C_  RR )
110 1re 9046 . . . . . . 7  |-  1  e.  RR
111110a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  RR )
112 2re 10025 . . . . . . 7  |-  2  e.  RR
113112a1i 11 . . . . . 6  |-  ( ph  ->  2  e.  RR )
114 breq1 4175 . . . . . . . . . . . . . 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 4175 . . . . . . . . . . . . . 14  |-  ( 0  =  if ( p  e.  Prime ,  ( log `  p ) ,  0 )  ->  ( 0  <_  (Λ `  p )  <->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  <_  (Λ `  p )
) )
11637adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  NN  /\  p  e.  Prime )  -> 
( log `  p
)  e.  RR )
117 vmaprm 20853 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  Prime  ->  (Λ `  p
)  =  ( log `  p ) )
118117adantl 453 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  NN  /\  p  e.  Prime )  -> 
(Λ `  p )  =  ( log `  p
) )
119118eqcomd 2409 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  NN  /\  p  e.  Prime )  -> 
( log `  p
)  =  (Λ `  p
) )
120 eqle 9132 . . . . . . . . . . . . . . 15  |-  ( ( ( log `  p
)  e.  RR  /\  ( log `  p )  =  (Λ `  p
) )  ->  ( log `  p )  <_ 
(Λ `  p ) )
121116, 119, 120syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( p  e.  NN  /\  p  e.  Prime )  -> 
( log `  p
)  <_  (Λ `  p
) )
122 vmage0 20857 . . . . . . . . . . . . . . 15  |-  ( p  e.  NN  ->  0  <_  (Λ `  p )
)
123122adantr 452 . . . . . . . . . . . . . 14  |-  ( ( p  e.  NN  /\  -.  p  e.  Prime )  ->  0  <_  (Λ `  p ) )
124114, 115, 121, 123ifbothda 3729 . . . . . . . . . . . . 13  |-  ( p  e.  NN  ->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  <_  (Λ `  p )
)
12519, 48subge0d 9572 . . . . . . . . . . . . 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 224 . . . . . . . . . . . 12  |-  ( p  e.  NN  ->  0  <_  ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) ) )
127100, 36, 126divge0d 10640 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  0  <_  ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p ) )
12818, 127syl 16 . . . . . . . . . 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 12529 . . . . . . . . 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 12180 . . . . . . . 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 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( 1 ... ( |_ `  x ) ) )  ->  p  e.  NN )
132131, 101syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
(Λ `  p )  -  if ( p  e.  Prime ,  ( log `  p
) ,  0 ) )  /  p )  e.  RR )
13311, 132fsumrecl 12483 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( 1 ... ( |_ `  x ) ) ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p )  e.  RR )
134112a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  2  e.  RR )
135131, 127syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p ) )
13612a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( (
1 ... ( |_ `  x ) )  i^i 
T )  C_  (
1 ... ( |_ `  x ) ) )
13711, 132, 135, 136fsumless 12530 . . . . . . . . 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 3308 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR )
139138flcld 11162 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( |_ `  x )  e.  ZZ )
140 rplogsumlem2 21132 . . . . . . . . . 10  |-  ( ( |_ `  x )  e.  ZZ  ->  sum_ p  e.  ( 1 ... ( |_ `  x ) ) ( ( (Λ `  p
)  -  if ( p  e.  Prime ,  ( log `  p ) ,  0 ) )  /  p )  <_ 
2 )
141139, 140syl 16 . . . . . . . . 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 9183 . . . . . . . 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 4192 . . . . . . 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 698 . . . . . 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 12285 . . . . 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 12373 . . . 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 2479 . . 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 12378 . 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 202 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 359    = wceq 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277    i^i cin 3279    C_ wss 3280   ifcif 3699   {csn 3774   class class class wbr 4172    e. cmpt 4226   `'ccnv 4836   "cima 4840   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    <_ cle 9077    - cmin 9247    / cdiv 9633   NNcn 9956   2c2 10005   ZZcz 10238   RR+crp 10568   ...cfz 10999   |_cfl 11156   abscabs 11994   O (
1 )co1 12235   sum_csu 12434   Primecprime 13034   phicphi 13108  Unitcui 15699   ZRHomczrh 16733  ℤ/nczn 16736   logclog 20405  Λcvma 20827
This theorem is referenced by:  dirith2  21175
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-rpss 6481  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-o1 12239  df-lo1 12240  df-sum 12435  df-ef 12625  df-e 12626  df-sin 12627  df-cos 12628  df-tan 12629  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-numer 13082  df-denom 13083  df-phi 13110  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-divs 13690  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-nsg 14897  df-eqg 14898  df-ghm 14959  df-gim 15001  df-ga 15022  df-cntz 15071  df-oppg 15097  df-od 15122  df-gex 15123  df-pgp 15124  df-lsm 15225  df-pj1 15226  df-cmn 15369  df-abl 15370  df-cyg 15443  df-dprd 15511  df-dpj 15512  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-rnghom 15774  df-drng 15792  df-subrg 15821  df-lmod 15907  df-lss 15964  df-lsp 16003  df-sra 16199  df-rgmod 16200  df-lidl 16201  df-rsp 16202  df-2idl 16258  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-zrh 16737  df-zn 16740  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-0p 19515  df-limc 19706  df-dv 19707  df-ply 20060  df-idp 20061  df-coe 20062  df-dgr 20063  df-quot 20161  df-log 20407  df-cxp 20408  df-em 20784  df-cht 20832  df-vma 20833  df-chp 20834  df-ppi 20835  df-mu 20836  df-dchr 20970
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