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Theorem rplogsum 21221
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 21220 . 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 13161 . . . . . . 7  |-  ( ph  ->  ( phi `  N
)  e.  NN )
98adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( phi `  N )  e.  NN )
109nncnd 10016 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( phi `  N )  e.  CC )
11 fzfid 11312 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
12 inss1 3561 . . . . . . . 8  |-  ( ( 1 ... ( |_
`  x ) )  i^i  T )  C_  ( 1 ... ( |_ `  x ) )
13 ssfi 7329 . . . . . . . 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 3346 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  p  e.  ( 1 ... ( |_ `  x ) ) )
17 elfznn 11080 . . . . . . . . 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 20901 . . . . . . . . 9  |-  ( p  e.  NN  ->  (Λ `  p )  e.  RR )
20 nndivre 10035 . . . . . . . . 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 12528 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( (Λ `  p
)  /  p )  e.  RR )
2423recnd 9114 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( (Λ `  p
)  /  p )  e.  CC )
2510, 24mulcld 9108 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( phi `  N )  x. 
sum_ p  e.  (
( 1 ... ( |_ `  x ) )  i^i  T ) ( (Λ `  p )  /  p ) )  e.  CC )
26 relogcl 20473 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2726adantl 453 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
2827recnd 9114 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
2925, 28subcld 9411 . . 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 3561 . . . . . . . 8  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  (
1 ... ( |_ `  x ) )
31 ssfi 7329 . . . . . . . 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 3346 . . . . . . . . 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 10621 . . . . . . . . . 10  |-  ( p  e.  NN  ->  p  e.  RR+ )
3736relogcld 20518 . . . . . . . . 9  |-  ( p  e.  NN  ->  ( log `  p )  e.  RR )
3837, 36rerpdivcld 10675 . . . . . . . 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 12528 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p )  e.  RR )
4140recnd 9114 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p )  e.  CC )
4210, 41mulcld 9108 . . . 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 9411 . . 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 9489 . . . . . 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 9114 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  (Λ `  p )  e.  CC )
46 0re 9091 . . . . . . . . . . . . 13  |-  0  e.  RR
47 ifcl 3775 . . . . . . . . . . . . 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 9114 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  e.  CC )
5036rpcnne0d 10657 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  (
p  e.  CC  /\  p  =/=  0 ) )
51 divsubdir 9710 . . . . . . . . . . 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 12497 . . . . . . . 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 9114 . . . . . . . . . 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 10675 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  ( if ( p  e.  Prime ,  ( log `  p
) ,  0 )  /  p )  e.  RR )
5857recnd 9114 . . . . . . . . . 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 12571 . . . . . . . 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 3562 . . . . . . . . . . . 12  |-  ( Prime  i^i  T )  C_  T
62 sslin 3567 . . . . . . . . . . . 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 3330 . . . . . . . . . . . . . . . 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 3533 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Prime  i^i  T )  =  ( T  i^i  Prime )
6766ineq2i 3539 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  =  ( ( 1 ... ( |_ `  x ) )  i^i  ( T  i^i  Prime
) )
68 inass 3551 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1 ... ( |_ `  x ) )  i^i  T )  i^i 
Prime )  =  (
( 1 ... ( |_ `  x ) )  i^i  ( T  i^i  Prime
) )
6967, 68eqtr4i 2459 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  =  ( ( ( 1 ... ( |_ `  x
) )  i^i  T
)  i^i  Prime )
7069elin2 3531 . . . . . . . . . . . . . . . . . 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 3746 . . . . . . . . . . . . . 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 6096 . . . . . . . . . . . 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 3469 . . . . . . . . . . . . . 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 9706 . . . . . . . . . . . . . 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 2468 . . . . . . . . . . 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 12519 . . . . . . . . . 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 3562 . . . . . . . . . . . . . . 15  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( Prime  i^i  T )
86 inss1 3561 . . . . . . . . . . . . . . 15  |-  ( Prime  i^i  T )  C_  Prime
8785, 86sstri 3357 . . . . . . . . . . . . . 14  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  Prime
8887, 33sseldi 3346 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  p  e.  Prime )
89 iftrue 3745 . . . . . . . . . . . . 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 6096 . . . . . . . . . . 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 12497 . . . . . . . . . 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 2470 . . . . . . . . 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 6097 . . . . . . . 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 2472 . . . . . . 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 6097 . . . . . 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 9446 . . . . . 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 2478 . . . . 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 4295 . . . 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 9465 . . . . . . . . 9  |-  ( p  e.  NN  ->  (
(Λ `  p )  -  if ( p  e.  Prime ,  ( log `  p
) ,  0 ) )  e.  RR )
101100, 36rerpdivcld 10675 . . . . . . . 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 12528 . . . . . 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 9114 . . . . 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 10622 . . . . . 6  |-  RR+  C_  RR
1068nncnd 10016 . . . . . 6  |-  ( ph  ->  ( phi `  N
)  e.  CC )
107 o1const 12413 . . . . . 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 9090 . . . . . . 7  |-  1  e.  RR
111110a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  RR )
112 2re 10069 . . . . . . 7  |-  2  e.  RR
113112a1i 11 . . . . . 6  |-  ( ph  ->  2  e.  RR )
114 breq1 4215 . . . . . . . . . . . . . 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 4215 . . . . . . . . . . . . . 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 20900 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  Prime  ->  (Λ `  p
)  =  ( log `  p ) )
118117adantl 453 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  NN  /\  p  e.  Prime )  -> 
(Λ `  p )  =  ( log `  p
) )
119118eqcomd 2441 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  NN  /\  p  e.  Prime )  -> 
( log `  p
)  =  (Λ `  p
) )
120 eqle 9176 . . . . . . . . . . . . . . 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 20904 . . . . . . . . . . . . . . 15  |-  ( p  e.  NN  ->  0  <_  (Λ `  p )
)
123122adantr 452 . . . . . . . . . . . . . 14  |-  ( ( p  e.  NN  /\  -.  p  e.  Prime )  ->  0  <_  (Λ `  p ) )
124114, 115, 121, 123ifbothda 3769 . . . . . . . . . . . . 13  |-  ( p  e.  NN  ->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  <_  (Λ `  p )
)
12519, 48subge0d 9616 . . . . . . . . . . . . 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 10684 . . . . . . . . . . 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 12574 . . . . . . . . 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 12225 . . . . . . . 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 12528 . . . . . . . . 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 12575 . . . . . . . . 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 3348 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR )
139138flcld 11207 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( |_ `  x )  e.  ZZ )
140 rplogsumlem2 21179 . . . . . . . . . 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 9227 . . . . . . . 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 4232 . . . . . . 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 12330 . . . . 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 12418 . . . 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 2511 . . 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 12423 . 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 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317    i^i cin 3319    C_ wss 3320   ifcif 3739   {csn 3814   class class class wbr 4212    e. cmpt 4266   `'ccnv 4877   "cima 4881   ` cfv 5454  (class class class)co 6081   Fincfn 7109   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995    <_ cle 9121    - cmin 9291    / cdiv 9677   NNcn 10000   2c2 10049   ZZcz 10282   RR+crp 10612   ...cfz 11043   |_cfl 11201   abscabs 12039   O (
1 )co1 12280   sum_csu 12479   Primecprime 13079   phicphi 13153  Unitcui 15744   ZRHomczrh 16778  ℤ/nczn 16781   logclog 20452  Λcvma 20874
This theorem is referenced by:  dirith2  21222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-disj 4183  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-tpos 6479  df-rpss 6522  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-omul 6729  df-er 6905  df-ec 6907  df-qs 6911  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-acn 7829  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ioc 10921  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-word 11723  df-concat 11724  df-s1 11725  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-o1 12284  df-lo1 12285  df-sum 12480  df-ef 12670  df-e 12671  df-sin 12672  df-cos 12673  df-tan 12674  df-pi 12675  df-dvds 12853  df-gcd 13007  df-prm 13080  df-numer 13127  df-denom 13128  df-phi 13155  df-pc 13211  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-divs 13735  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-nsg 14942  df-eqg 14943  df-ghm 15004  df-gim 15046  df-ga 15067  df-cntz 15116  df-oppg 15142  df-od 15167  df-gex 15168  df-pgp 15169  df-lsm 15270  df-pj1 15271  df-cmn 15414  df-abl 15415  df-cyg 15488  df-dprd 15556  df-dpj 15557  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-dvr 15788  df-rnghom 15819  df-drng 15837  df-subrg 15866  df-lmod 15952  df-lss 16009  df-lsp 16048  df-sra 16244  df-rgmod 16245  df-lidl 16246  df-rsp 16247  df-2idl 16303  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-zrh 16782  df-zn 16785  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-cmp 17450  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cncf 18908  df-0p 19562  df-limc 19753  df-dv 19754  df-ply 20107  df-idp 20108  df-coe 20109  df-dgr 20110  df-quot 20208  df-log 20454  df-cxp 20455  df-em 20831  df-cht 20879  df-vma 20880  df-chp 20881  df-ppi 20882  df-mu 20883  df-dchr 21017
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