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Theorem rplogsum 20672
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 20671 . 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 12836 . . . . . . 7  |-  ( ph  ->  ( phi `  N
)  e.  NN )
98adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( phi `  N )  e.  NN )
109nncnd 9758 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( phi `  N )  e.  CC )
11 fzfid 11031 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
12 inss1 3390 . . . . . . . 8  |-  ( ( 1 ... ( |_
`  x ) )  i^i  T )  C_  ( 1 ... ( |_ `  x ) )
13 ssfi 7079 . . . . . . . 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 3179 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  p  e.  ( 1 ... ( |_ `  x ) ) )
17 elfznn 10815 . . . . . . . . 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 20352 . . . . . . . . 9  |-  ( p  e.  NN  ->  (Λ `  p )  e.  RR )
20 nndivre 9777 . . . . . . . . 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 12203 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( (Λ `  p
)  /  p )  e.  RR )
2423recnd 8857 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( (Λ `  p
)  /  p )  e.  CC )
2510, 24mulcld 8851 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( phi `  N )  x. 
sum_ p  e.  (
( 1 ... ( |_ `  x ) )  i^i  T ) ( (Λ `  p )  /  p ) )  e.  CC )
26 relogcl 19928 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2726adantl 452 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
2827recnd 8857 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
2925, 28subcld 9153 . . 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 3390 . . . . . . . 8  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  (
1 ... ( |_ `  x ) )
31 ssfi 7079 . . . . . . . 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 3179 . . . . . . . . 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 10359 . . . . . . . . . 10  |-  ( p  e.  NN  ->  p  e.  RR+ )
3736relogcld 19970 . . . . . . . . 9  |-  ( p  e.  NN  ->  ( log `  p )  e.  RR )
3837, 36rerpdivcld 10413 . . . . . . . 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 12203 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p )  e.  RR )
4140recnd 8857 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p )  e.  CC )
4210, 41mulcld 8851 . . . 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 9153 . . 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 9231 . . . . . 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 8857 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  (Λ `  p )  e.  CC )
46 0re 8834 . . . . . . . . . . . . 13  |-  0  e.  RR
47 ifcl 3602 . . . . . . . . . . . . 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 8857 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  e.  CC )
5036rpcnne0d 10395 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  (
p  e.  CC  /\  p  =/=  0 ) )
51 divsubdir 9452 . . . . . . . . . . 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 12172 . . . . . . . 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 8857 . . . . . . . . . 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 10413 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  ( if ( p  e.  Prime ,  ( log `  p
) ,  0 )  /  p )  e.  RR )
5857recnd 8857 . . . . . . . . . 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 12246 . . . . . . . 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 3391 . . . . . . . . . . . 12  |-  ( Prime  i^i  T )  C_  T
62 sslin 3396 . . . . . . . . . . . 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 3163 . . . . . . . . . . . . . . . 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 3362 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Prime  i^i  T )  =  ( T  i^i  Prime )
6766ineq2i 3368 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  =  ( ( 1 ... ( |_ `  x ) )  i^i  ( T  i^i  Prime
) )
68 inass 3380 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1 ... ( |_ `  x ) )  i^i  T )  i^i 
Prime )  =  (
( 1 ... ( |_ `  x ) )  i^i  ( T  i^i  Prime
) )
6967, 68eqtr4i 2307 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  =  ( ( ( 1 ... ( |_ `  x
) )  i^i  T
)  i^i  Prime )
7069elin2 3360 . . . . . . . . . . . . . . . . . 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 3573 . . . . . . . . . . . . . 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 5835 . . . . . . . . . . . 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 3299 . . . . . . . . . . . . . 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 9448 . . . . . . . . . . . . . 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 2316 . . . . . . . . . . 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 12194 . . . . . . . . . 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 3391 . . . . . . . . . . . . . . 15  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( Prime  i^i  T )
86 inss1 3390 . . . . . . . . . . . . . . 15  |-  ( Prime  i^i  T )  C_  Prime
8785, 86sstri 3189 . . . . . . . . . . . . . 14  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  Prime
8887, 33sseldi 3179 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  p  e.  Prime )
89 iftrue 3572 . . . . . . . . . . . . 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 5835 . . . . . . . . . . 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 12172 . . . . . . . . . 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 2318 . . . . . . . . 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 5836 . . . . . . . 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 2320 . . . . . . 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 5836 . . . . . 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 9188 . . . . . 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 2326 . . . . 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 4107 . . . 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 9207 . . . . . . . . 9  |-  ( p  e.  NN  ->  (
(Λ `  p )  -  if ( p  e.  Prime ,  ( log `  p
) ,  0 ) )  e.  RR )
101100, 36rerpdivcld 10413 . . . . . . . 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 12203 . . . . . 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 8857 . . . . 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 10360 . . . . . 6  |-  RR+  C_  RR
1068nncnd 9758 . . . . . 6  |-  ( ph  ->  ( phi `  N
)  e.  CC )
107 o1const 12089 . . . . . 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 8833 . . . . . . 7  |-  1  e.  RR
111110a1i 10 . . . . . 6  |-  ( ph  ->  1  e.  RR )
112 2re 9811 . . . . . . 7  |-  2  e.  RR
113112a1i 10 . . . . . 6  |-  ( ph  ->  2  e.  RR )
114 breq1 4027 . . . . . . . . . . . . . 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 4027 . . . . . . . . . . . . . 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 20351 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  Prime  ->  (Λ `  p
)  =  ( log `  p ) )
118117adantl 452 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  NN  /\  p  e.  Prime )  -> 
(Λ `  p )  =  ( log `  p
) )
119118eqcomd 2289 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  NN  /\  p  e.  Prime )  -> 
( log `  p
)  =  (Λ `  p
) )
120 eqle 8919 . . . . . . . . . . . . . . 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 20355 . . . . . . . . . . . . . . 15  |-  ( p  e.  NN  ->  0  <_  (Λ `  p )
)
123122adantr 451 . . . . . . . . . . . . . 14  |-  ( ( p  e.  NN  /\  -.  p  e.  Prime )  ->  0  <_  (Λ `  p ) )
124114, 115, 121, 123ifbothda 3596 . . . . . . . . . . . . 13  |-  ( p  e.  NN  ->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  <_  (Λ `  p )
)
12519, 48subge0d 9358 . . . . . . . . . . . . 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 10422 . . . . . . . . . . 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 12249 . . . . . . . . 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 11901 . . . . . . . 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 12203 . . . . . . . . 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 12250 . . . . . . . . 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 3181 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR )
139138flcld 10926 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( |_ `  x )  e.  ZZ )
140 rplogsumlem2 20630 . . . . . . . . . 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 8969 . . . . . . . 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 4044 . . . . . . 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 12006 . . . . 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 12094 . . . 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 2359 . . 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 12099 . 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 1623    e. wcel 1685    =/= wne 2447    \ cdif 3150    i^i cin 3152    C_ wss 3153   ifcif 3566   {csn 3641   class class class wbr 4024    e. cmpt 4078   `'ccnv 4687   "cima 4691   ` cfv 5221  (class class class)co 5820   Fincfn 6859   CCcc 8731   RRcr 8732   0cc0 8733   1c1 8734    x. cmul 8738    <_ cle 8864    - cmin 9033    / cdiv 9419   NNcn 9742   2c2 9791   ZZcz 10020   RR+crp 10350   ...cfz 10778   |_cfl 10920   abscabs 11715   O (
1 )co1 11956   sum_csu 12154   Primecprime 12754   phicphi 12828  Unitcui 15417   ZRHomczrh 16447  ℤ/nczn 16450   logclog 19908  Λcvma 20325
This theorem is referenced by:  dirith2  20673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-disj 3995  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-of 6040  df-1st 6084  df-2nd 6085  df-tpos 6196  df-rpss 6239  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-omul 6480  df-er 6656  df-ec 6658  df-qs 6662  df-map 6770  df-pm 6771  df-ixp 6814  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-fi 7161  df-sup 7190  df-oi 7221  df-card 7568  df-acn 7571  df-cda 7790  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-q 10313  df-rp 10351  df-xneg 10448  df-xadd 10449  df-xmul 10450  df-ioo 10656  df-ioc 10657  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-fl 10921  df-mod 10970  df-seq 11043  df-exp 11101  df-fac 11285  df-bc 11312  df-hash 11334  df-word 11405  df-concat 11406  df-s1 11407  df-shft 11558  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-limsup 11941  df-clim 11958  df-rlim 11959  df-o1 11960  df-lo1 11961  df-sum 12155  df-ef 12345  df-e 12346  df-sin 12347  df-cos 12348  df-tan 12349  df-pi 12350  df-dvds 12528  df-gcd 12682  df-prm 12755  df-numer 12802  df-denom 12803  df-phi 12830  df-pc 12886  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-starv 13219  df-sca 13220  df-vsca 13221  df-tset 13223  df-ple 13224  df-ds 13226  df-hom 13228  df-cco 13229  df-rest 13323  df-topn 13324  df-topgen 13340  df-pt 13341  df-prds 13344  df-xrs 13399  df-0g 13400  df-gsum 13401  df-qtop 13406  df-imas 13407  df-divs 13408  df-xps 13409  df-mre 13484  df-mrc 13485  df-acs 13487  df-mnd 14363  df-mhm 14411  df-submnd 14412  df-grp 14485  df-minusg 14486  df-sbg 14487  df-mulg 14488  df-subg 14614  df-nsg 14615  df-eqg 14616  df-ghm 14677  df-gim 14719  df-ga 14740  df-cntz 14789  df-oppg 14815  df-od 14840  df-gex 14841  df-pgp 14842  df-lsm 14943  df-pj1 14944  df-cmn 15087  df-abl 15088  df-cyg 15161  df-dprd 15229  df-dpj 15230  df-mgp 15322  df-rng 15336  df-cring 15337  df-ur 15338  df-oppr 15401  df-dvdsr 15419  df-unit 15420  df-invr 15450  df-dvr 15461  df-rnghom 15492  df-drng 15510  df-subrg 15539  df-lmod 15625  df-lss 15686  df-lsp 15725  df-sra 15921  df-rgmod 15922  df-lidl 15923  df-rsp 15924  df-2idl 15980  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  df-zrh 16451  df-zn 16454  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-cld 16752  df-ntr 16753  df-cls 16754  df-nei 16831  df-lp 16864  df-perf 16865  df-cn 16953  df-cnp 16954  df-haus 17039  df-cmp 17110  df-tx 17253  df-hmeo 17442  df-fbas 17516  df-fg 17517  df-fil 17537  df-fm 17629  df-flim 17630  df-flf 17631  df-xms 17881  df-ms 17882  df-tms 17883  df-cncf 18378  df-0p 19021  df-limc 19212  df-dv 19213  df-ply 19566  df-idp 19567  df-coe 19568  df-dgr 19569  df-quot 19667  df-log 19910  df-cxp 19911  df-em 20283  df-cht 20330  df-vma 20331  df-chp 20332  df-ppi 20333  df-mu 20334  df-dchr 20468
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