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Theorem rplogsum 20676
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 20675 . 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 12840 . . . . . . 7  |-  ( ph  ->  ( phi `  N
)  e.  NN )
98adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( phi `  N )  e.  NN )
109nncnd 9762 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( phi `  N )  e.  CC )
11 fzfid 11035 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
12 inss1 3389 . . . . . . . 8  |-  ( ( 1 ... ( |_
`  x ) )  i^i  T )  C_  ( 1 ... ( |_ `  x ) )
13 ssfi 7083 . . . . . . . 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 3178 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) )  ->  p  e.  ( 1 ... ( |_ `  x ) ) )
17 elfznn 10819 . . . . . . . . 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 20356 . . . . . . . . 9  |-  ( p  e.  NN  ->  (Λ `  p )  e.  RR )
20 nndivre 9781 . . . . . . . . 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 12207 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( (Λ `  p
)  /  p )  e.  RR )
2423recnd 8861 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  T
) ( (Λ `  p
)  /  p )  e.  CC )
2510, 24mulcld 8855 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( phi `  N )  x. 
sum_ p  e.  (
( 1 ... ( |_ `  x ) )  i^i  T ) ( (Λ `  p )  /  p ) )  e.  CC )
26 relogcl 19932 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2726adantl 452 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
2827recnd 8861 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
2925, 28subcld 9157 . . 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 3389 . . . . . . . 8  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  (
1 ... ( |_ `  x ) )
31 ssfi 7083 . . . . . . . 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 3178 . . . . . . . . 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 10363 . . . . . . . . . 10  |-  ( p  e.  NN  ->  p  e.  RR+ )
3736relogcld 19974 . . . . . . . . 9  |-  ( p  e.  NN  ->  ( log `  p )  e.  RR )
3837, 36rerpdivcld 10417 . . . . . . . 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 12207 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p )  e.  RR )
4140recnd 8861 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  p
)  /  p )  e.  CC )
4210, 41mulcld 8855 . . . 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 9157 . . 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 9235 . . . . . 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 8861 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  (Λ `  p )  e.  CC )
46 0re 8838 . . . . . . . . . . . . 13  |-  0  e.  RR
47 ifcl 3601 . . . . . . . . . . . . 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 8861 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  e.  CC )
5036rpcnne0d 10399 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  (
p  e.  CC  /\  p  =/=  0 ) )
51 divsubdir 9456 . . . . . . . . . . 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 12176 . . . . . . . 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 8861 . . . . . . . . . 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 10417 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  ( if ( p  e.  Prime ,  ( log `  p
) ,  0 )  /  p )  e.  RR )
5857recnd 8861 . . . . . . . . . 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 12250 . . . . . . . 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 3390 . . . . . . . . . . . 12  |-  ( Prime  i^i  T )  C_  T
62 sslin 3395 . . . . . . . . . . . 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 3162 . . . . . . . . . . . . . . . 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 3361 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Prime  i^i  T )  =  ( T  i^i  Prime )
6766ineq2i 3367 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  =  ( ( 1 ... ( |_ `  x ) )  i^i  ( T  i^i  Prime
) )
68 inass 3379 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1 ... ( |_ `  x ) )  i^i  T )  i^i 
Prime )  =  (
( 1 ... ( |_ `  x ) )  i^i  ( T  i^i  Prime
) )
6967, 68eqtr4i 2306 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  =  ( ( ( 1 ... ( |_ `  x
) )  i^i  T
)  i^i  Prime )
7069elin2 3359 . . . . . . . . . . . . . . . . . 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 3572 . . . . . . . . . . . . . 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 5873 . . . . . . . . . . . 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 3298 . . . . . . . . . . . . . 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 9452 . . . . . . . . . . . . . 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 2315 . . . . . . . . . . 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 12198 . . . . . . . . . 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 3390 . . . . . . . . . . . . . . 15  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( Prime  i^i  T )
86 inss1 3389 . . . . . . . . . . . . . . 15  |-  ( Prime  i^i  T )  C_  Prime
8785, 86sstri 3188 . . . . . . . . . . . . . 14  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  Prime
8887, 33sseldi 3178 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  p  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  p  e.  Prime )
89 iftrue 3571 . . . . . . . . . . . . 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 5873 . . . . . . . . . . 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 12176 . . . . . . . . . 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 2317 . . . . . . . . 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 5874 . . . . . . . 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 2319 . . . . . . 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 5874 . . . . . 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 9192 . . . . . 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 2325 . . . . 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 4106 . . . 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 9211 . . . . . . . . 9  |-  ( p  e.  NN  ->  (
(Λ `  p )  -  if ( p  e.  Prime ,  ( log `  p
) ,  0 ) )  e.  RR )
101100, 36rerpdivcld 10417 . . . . . . . 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 12207 . . . . . 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 8861 . . . . 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 10364 . . . . . 6  |-  RR+  C_  RR
1068nncnd 9762 . . . . . 6  |-  ( ph  ->  ( phi `  N
)  e.  CC )
107 o1const 12093 . . . . . 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 8837 . . . . . . 7  |-  1  e.  RR
111110a1i 10 . . . . . 6  |-  ( ph  ->  1  e.  RR )
112 2re 9815 . . . . . . 7  |-  2  e.  RR
113112a1i 10 . . . . . 6  |-  ( ph  ->  2  e.  RR )
114 breq1 4026 . . . . . . . . . . . . . 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 4026 . . . . . . . . . . . . . 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 20355 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  Prime  ->  (Λ `  p
)  =  ( log `  p ) )
118117adantl 452 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  NN  /\  p  e.  Prime )  -> 
(Λ `  p )  =  ( log `  p
) )
119118eqcomd 2288 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  NN  /\  p  e.  Prime )  -> 
( log `  p
)  =  (Λ `  p
) )
120 eqle 8923 . . . . . . . . . . . . . . 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 20359 . . . . . . . . . . . . . . 15  |-  ( p  e.  NN  ->  0  <_  (Λ `  p )
)
123122adantr 451 . . . . . . . . . . . . . 14  |-  ( ( p  e.  NN  /\  -.  p  e.  Prime )  ->  0  <_  (Λ `  p ) )
124114, 115, 121, 123ifbothda 3595 . . . . . . . . . . . . 13  |-  ( p  e.  NN  ->  if ( p  e.  Prime ,  ( log `  p
) ,  0 )  <_  (Λ `  p )
)
12519, 48subge0d 9362 . . . . . . . . . . . . 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 10426 . . . . . . . . . . 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 12253 . . . . . . . . 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 11905 . . . . . . . 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 12207 . . . . . . . . 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 12254 . . . . . . . . 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 3180 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR )
139138flcld 10930 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( |_ `  x )  e.  ZZ )
140 rplogsumlem2 20634 . . . . . . . . . 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 8973 . . . . . . . 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 4043 . . . . . . 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 12010 . . . . 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 12098 . . . 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 2358 . . 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 12103 . 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 1684    =/= wne 2446    \ cdif 3149    i^i cin 3151    C_ wss 3152   ifcif 3565   {csn 3640   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   ZZcz 10024   RR+crp 10354   ...cfz 10782   |_cfl 10924   abscabs 11719   O (
1 )co1 11960   sum_csu 12158   Primecprime 12758   phicphi 12832  Unitcui 15421   ZRHomczrh 16451  ℤ/nczn 16454   logclog 19912  Λcvma 20329
This theorem is referenced by:  dirith2  20677
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-rpss 6277  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-o1 11964  df-lo1 11965  df-sum 12159  df-ef 12349  df-e 12350  df-sin 12351  df-cos 12352  df-tan 12353  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-numer 12806  df-denom 12807  df-phi 12834  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-divs 13412  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-nsg 14619  df-eqg 14620  df-ghm 14681  df-gim 14723  df-ga 14744  df-cntz 14793  df-oppg 14819  df-od 14844  df-gex 14845  df-pgp 14846  df-lsm 14947  df-pj1 14948  df-cmn 15091  df-abl 15092  df-cyg 15165  df-dprd 15233  df-dpj 15234  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-rnghom 15496  df-drng 15514  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-zrh 16455  df-zn 16458  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-0p 19025  df-limc 19216  df-dv 19217  df-ply 19570  df-idp 19571  df-coe 19572  df-dgr 19573  df-quot 19671  df-log 19914  df-cxp 19915  df-em 20287  df-cht 20334  df-vma 20335  df-chp 20336  df-ppi 20337  df-mu 20338  df-dchr 20472
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