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Theorem vmasum 20417
Description: The sum of the von Mangoldt function over the divisors of  n. Equation 9.2.4 of [Shapiro], p. 328. (Contributed by Mario Carneiro, 15-Apr-2016.)
Assertion
Ref Expression
vmasum  |-  ( A  e.  NN  ->  sum_ n  e.  { x  e.  NN  |  x  ||  A } 
(Λ `  n )  =  ( log `  A
) )
Distinct variable group:    x, n, A

Proof of Theorem vmasum
StepHypRef Expression
1 fveq2 5458 . . 3  |-  ( n  =  ( p ^
k )  ->  (Λ `  n )  =  (Λ `  ( p ^ k
) ) )
2 fzfid 11001 . . . 4  |-  ( A  e.  NN  ->  (
1 ... A )  e. 
Fin )
3 sgmss 20306 . . . 4  |-  ( A  e.  NN  ->  { x  e.  NN  |  x  ||  A }  C_  ( 1 ... A ) )
4 ssfi 7051 . . . 4  |-  ( ( ( 1 ... A
)  e.  Fin  /\  { x  e.  NN  |  x  ||  A }  C_  ( 1 ... A
) )  ->  { x  e.  NN  |  x  ||  A }  e.  Fin )
52, 3, 4syl2anc 645 . . 3  |-  ( A  e.  NN  ->  { x  e.  NN  |  x  ||  A }  e.  Fin )
6 ssrab2 3233 . . . 4  |-  { x  e.  NN  |  x  ||  A }  C_  NN
76a1i 12 . . 3  |-  ( A  e.  NN  ->  { x  e.  NN  |  x  ||  A }  C_  NN )
8 inss1 3364 . . . 4  |-  ( ( 1 ... A )  i^i  Prime )  C_  (
1 ... A )
9 ssfi 7051 . . . 4  |-  ( ( ( 1 ... A
)  e.  Fin  /\  ( ( 1 ... A )  i^i  Prime ) 
C_  ( 1 ... A ) )  -> 
( ( 1 ... A )  i^i  Prime )  e.  Fin )
102, 8, 9sylancl 646 . . 3  |-  ( A  e.  NN  ->  (
( 1 ... A
)  i^i  Prime )  e. 
Fin )
11 pccl 12864 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
p  pCnt  A )  e.  NN0 )
1211ancoms 441 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  NN0 )
1312nn0zd 10082 . . . . . . . 8  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  ZZ )
14 fznn 10818 . . . . . . . 8  |-  ( ( p  pCnt  A )  e.  ZZ  ->  ( k  e.  ( 1 ... (
p  pCnt  A )
)  <->  ( k  e.  NN  /\  k  <_ 
( p  pCnt  A
) ) ) )
1513, 14syl 17 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( k  e.  ( 1 ... ( p 
pCnt  A ) )  <->  ( k  e.  NN  /\  k  <_ 
( p  pCnt  A
) ) ) )
1615anbi2d 687 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( ( p  e.  ( 1 ... A
)  /\  k  e.  ( 1 ... (
p  pCnt  A )
) )  <->  ( p  e.  ( 1 ... A
)  /\  ( k  e.  NN  /\  k  <_ 
( p  pCnt  A
) ) ) ) )
17 an12 775 . . . . . . 7  |-  ( ( p  e.  ( 1 ... A )  /\  ( k  e.  NN  /\  k  <_  ( p  pCnt  A ) ) )  <-> 
( k  e.  NN  /\  ( p  e.  ( 1 ... A )  /\  k  <_  (
p  pCnt  A )
) ) )
18 prmz 12723 . . . . . . . . . . . . . . 15  |-  ( p  e.  Prime  ->  p  e.  ZZ )
1918adantl 454 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  ZZ )
20 iddvdsexp 12514 . . . . . . . . . . . . . 14  |-  ( ( p  e.  ZZ  /\  k  e.  NN )  ->  p  ||  ( p ^ k ) )
2119, 20sylan 459 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  p  ||  (
p ^ k ) )
2218ad2antlr 710 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  p  e.  ZZ )
23 prmnn 12724 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  Prime  ->  p  e.  NN )
2423adantl 454 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  NN )
25 nnnn0 9939 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  k  e.  NN0 )
26 nnexpcl 11082 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  NN  /\  k  e.  NN0 )  -> 
( p ^ k
)  e.  NN )
2724, 25, 26syl2an 465 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( p ^
k )  e.  NN )
2827nnzd 10083 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( p ^
k )  e.  ZZ )
29 nnz 10012 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  A  e.  ZZ )
3029ad2antrr 709 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  A  e.  ZZ )
31 dvdstr 12524 . . . . . . . . . . . . . 14  |-  ( ( p  e.  ZZ  /\  ( p ^ k
)  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( p  ||  ( p ^ k
)  /\  ( p ^ k )  ||  A )  ->  p  ||  A ) )
3222, 28, 30, 31syl3anc 1187 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( ( p 
||  ( p ^
k )  /\  (
p ^ k ) 
||  A )  ->  p  ||  A ) )
3321, 32mpand 659 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( ( p ^ k )  ||  A  ->  p  ||  A
) )
34 simpll 733 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  A  e.  NN )
35 dvdsle 12536 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  A  e.  NN )  ->  ( p  ||  A  ->  p  <_  A )
)
3622, 34, 35syl2anc 645 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( p  ||  A  ->  p  <_  A
) )
3733, 36syld 42 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( ( p ^ k )  ||  A  ->  p  <_  A
) )
3823ad2antlr 710 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  p  e.  NN )
39 fznn 10818 . . . . . . . . . . . . 13  |-  ( A  e.  ZZ  ->  (
p  e.  ( 1 ... A )  <->  ( p  e.  NN  /\  p  <_  A ) ) )
4039baibd 880 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  p  e.  NN )  ->  ( p  e.  ( 1 ... A )  <-> 
p  <_  A )
)
4130, 38, 40syl2anc 645 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( p  e.  ( 1 ... A
)  <->  p  <_  A ) )
4237, 41sylibrd 227 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( ( p ^ k )  ||  A  ->  p  e.  ( 1 ... A ) ) )
4342pm4.71rd 619 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( ( p ^ k )  ||  A 
<->  ( p  e.  ( 1 ... A )  /\  ( p ^
k )  ||  A
) ) )
44 breq1 4000 . . . . . . . . . . 11  |-  ( x  =  ( p ^
k )  ->  (
x  ||  A  <->  ( p ^ k )  ||  A ) )
4544elrab3 2899 . . . . . . . . . 10  |-  ( ( p ^ k )  e.  NN  ->  (
( p ^ k
)  e.  { x  e.  NN  |  x  ||  A }  <->  ( p ^
k )  ||  A
) )
4627, 45syl 17 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( ( p ^ k )  e. 
{ x  e.  NN  |  x  ||  A }  <->  ( p ^ k ) 
||  A ) )
47 simplr 734 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  p  e.  Prime )
4825adantl 454 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  k  e.  NN0 )
49 pcdvdsb 12883 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  A  e.  ZZ  /\  k  e. 
NN0 )  ->  (
k  <_  ( p  pCnt  A )  <->  ( p ^ k )  ||  A ) )
5047, 30, 48, 49syl3anc 1187 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( k  <_ 
( p  pCnt  A
)  <->  ( p ^
k )  ||  A
) )
5150anbi2d 687 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( ( p  e.  ( 1 ... A )  /\  k  <_  ( p  pCnt  A
) )  <->  ( p  e.  ( 1 ... A
)  /\  ( p ^ k )  ||  A ) ) )
5243, 46, 513bitr4rd 279 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( ( p  e.  ( 1 ... A )  /\  k  <_  ( p  pCnt  A
) )  <->  ( p ^ k )  e. 
{ x  e.  NN  |  x  ||  A }
) )
5352pm5.32da 625 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( ( k  e.  NN  /\  ( p  e.  ( 1 ... A )  /\  k  <_  ( p  pCnt  A
) ) )  <->  ( k  e.  NN  /\  ( p ^ k )  e. 
{ x  e.  NN  |  x  ||  A }
) ) )
5417, 53syl5bb 250 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( ( p  e.  ( 1 ... A
)  /\  ( k  e.  NN  /\  k  <_ 
( p  pCnt  A
) ) )  <->  ( k  e.  NN  /\  ( p ^ k )  e. 
{ x  e.  NN  |  x  ||  A }
) ) )
5516, 54bitrd 246 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( ( p  e.  ( 1 ... A
)  /\  k  e.  ( 1 ... (
p  pCnt  A )
) )  <->  ( k  e.  NN  /\  ( p ^ k )  e. 
{ x  e.  NN  |  x  ||  A }
) ) )
5655pm5.32da 625 . . . 4  |-  ( A  e.  NN  ->  (
( p  e.  Prime  /\  ( p  e.  ( 1 ... A )  /\  k  e.  ( 1 ... ( p 
pCnt  A ) ) ) )  <->  ( p  e. 
Prime  /\  ( k  e.  NN  /\  ( p ^ k )  e. 
{ x  e.  NN  |  x  ||  A }
) ) ) )
57 elin 3333 . . . . . 6  |-  ( p  e.  ( ( 1 ... A )  i^i 
Prime )  <->  ( p  e.  ( 1 ... A
)  /\  p  e.  Prime ) )
5857anbi1i 679 . . . . 5  |-  ( ( p  e.  ( ( 1 ... A )  i^i  Prime )  /\  k  e.  ( 1 ... (
p  pCnt  A )
) )  <->  ( (
p  e.  ( 1 ... A )  /\  p  e.  Prime )  /\  k  e.  ( 1 ... ( p  pCnt  A ) ) ) )
59 anass 633 . . . . 5  |-  ( ( ( p  e.  ( 1 ... A )  /\  p  e.  Prime )  /\  k  e.  ( 1 ... ( p 
pCnt  A ) ) )  <-> 
( p  e.  ( 1 ... A )  /\  ( p  e. 
Prime  /\  k  e.  ( 1 ... ( p 
pCnt  A ) ) ) ) )
60 an12 775 . . . . 5  |-  ( ( p  e.  ( 1 ... A )  /\  ( p  e.  Prime  /\  k  e.  ( 1 ... ( p  pCnt  A ) ) ) )  <-> 
( p  e.  Prime  /\  ( p  e.  ( 1 ... A )  /\  k  e.  ( 1 ... ( p 
pCnt  A ) ) ) ) )
6158, 59, 603bitri 264 . . . 4  |-  ( ( p  e.  ( ( 1 ... A )  i^i  Prime )  /\  k  e.  ( 1 ... (
p  pCnt  A )
) )  <->  ( p  e.  Prime  /\  ( p  e.  ( 1 ... A
)  /\  k  e.  ( 1 ... (
p  pCnt  A )
) ) ) )
62 anass 633 . . . 4  |-  ( ( ( p  e.  Prime  /\  k  e.  NN )  /\  ( p ^
k )  e.  {
x  e.  NN  |  x  ||  A } )  <-> 
( p  e.  Prime  /\  ( k  e.  NN  /\  ( p ^ k
)  e.  { x  e.  NN  |  x  ||  A } ) ) )
6356, 61, 623bitr4g 281 . . 3  |-  ( A  e.  NN  ->  (
( p  e.  ( ( 1 ... A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( p  pCnt  A ) ) )  <->  ( (
p  e.  Prime  /\  k  e.  NN )  /\  (
p ^ k )  e.  { x  e.  NN  |  x  ||  A } ) ) )
647sselda 3155 . . . . 5  |-  ( ( A  e.  NN  /\  n  e.  { x  e.  NN  |  x  ||  A } )  ->  n  e.  NN )
65 vmacl 20318 . . . . 5  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
6664, 65syl 17 . . . 4  |-  ( ( A  e.  NN  /\  n  e.  { x  e.  NN  |  x  ||  A } )  ->  (Λ `  n )  e.  RR )
6766recnd 8829 . . 3  |-  ( ( A  e.  NN  /\  n  e.  { x  e.  NN  |  x  ||  A } )  ->  (Λ `  n )  e.  CC )
68 simprr 736 . . 3  |-  ( ( A  e.  NN  /\  ( n  e.  { x  e.  NN  |  x  ||  A }  /\  (Λ `  n )  =  0 ) )  ->  (Λ `  n )  =  0 )
691, 5, 7, 10, 63, 67, 68fsumvma 20414 . 2  |-  ( A  e.  NN  ->  sum_ n  e.  { x  e.  NN  |  x  ||  A } 
(Λ `  n )  = 
sum_ p  e.  (
( 1 ... A
)  i^i  Prime ) sum_ k  e.  ( 1 ... ( p  pCnt  A ) ) (Λ `  (
p ^ k ) ) )
7057simprbi 452 . . . . . . 7  |-  ( p  e.  ( ( 1 ... A )  i^i 
Prime )  ->  p  e. 
Prime )
7170ad2antlr 710 . . . . . 6  |-  ( ( ( A  e.  NN  /\  p  e.  ( ( 1 ... A )  i^i  Prime ) )  /\  k  e.  ( 1 ... ( p  pCnt  A ) ) )  ->  p  e.  Prime )
72 elfznn 10785 . . . . . . 7  |-  ( k  e.  ( 1 ... ( p  pCnt  A
) )  ->  k  e.  NN )
7372adantl 454 . . . . . 6  |-  ( ( ( A  e.  NN  /\  p  e.  ( ( 1 ... A )  i^i  Prime ) )  /\  k  e.  ( 1 ... ( p  pCnt  A ) ) )  -> 
k  e.  NN )
74 vmappw 20316 . . . . . 6  |-  ( ( p  e.  Prime  /\  k  e.  NN )  ->  (Λ `  ( p ^ k
) )  =  ( log `  p ) )
7571, 73, 74syl2anc 645 . . . . 5  |-  ( ( ( A  e.  NN  /\  p  e.  ( ( 1 ... A )  i^i  Prime ) )  /\  k  e.  ( 1 ... ( p  pCnt  A ) ) )  -> 
(Λ `  ( p ^
k ) )  =  ( log `  p
) )
7675sumeq2dv 12141 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  sum_ k  e.  ( 1 ... (
p  pCnt  A )
) (Λ `  ( p ^ k ) )  =  sum_ k  e.  ( 1 ... ( p 
pCnt  A ) ) ( log `  p ) )
77 fzfid 11001 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
1 ... ( p  pCnt  A ) )  e.  Fin )
7870, 23syl 17 . . . . . . . . 9  |-  ( p  e.  ( ( 1 ... A )  i^i 
Prime )  ->  p  e.  NN )
7978adantl 454 . . . . . . . 8  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  NN )
8079nnrpd 10356 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  RR+ )
8180relogcld 19936 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  p )  e.  RR )
8281recnd 8829 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  p )  e.  CC )
83 fsumconst 12217 . . . . 5  |-  ( ( ( 1 ... (
p  pCnt  A )
)  e.  Fin  /\  ( log `  p )  e.  CC )  ->  sum_ k  e.  ( 1 ... ( p  pCnt  A ) ) ( log `  p )  =  ( ( # `  (
1 ... ( p  pCnt  A ) ) )  x.  ( log `  p
) ) )
8477, 82, 83syl2anc 645 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  sum_ k  e.  ( 1 ... (
p  pCnt  A )
) ( log `  p
)  =  ( (
# `  ( 1 ... ( p  pCnt  A
) ) )  x.  ( log `  p
) ) )
8570, 12sylan2 462 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p  pCnt  A )  e.  NN0 )
86 hashfz1 11311 . . . . . 6  |-  ( ( p  pCnt  A )  e.  NN0  ->  ( # `  (
1 ... ( p  pCnt  A ) ) )  =  ( p  pCnt  A
) )
8785, 86syl 17 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( # `
 ( 1 ... ( p  pCnt  A
) ) )  =  ( p  pCnt  A
) )
8887oveq1d 5807 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
( # `  ( 1 ... ( p  pCnt  A ) ) )  x.  ( log `  p
) )  =  ( ( p  pCnt  A
)  x.  ( log `  p ) ) )
8976, 84, 883eqtrd 2294 . . 3  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  sum_ k  e.  ( 1 ... (
p  pCnt  A )
) (Λ `  ( p ^ k ) )  =  ( ( p 
pCnt  A )  x.  ( log `  p ) ) )
9089sumeq2dv 12141 . 2  |-  ( A  e.  NN  ->  sum_ p  e.  ( ( 1 ... A )  i^i  Prime )
sum_ k  e.  ( 1 ... ( p 
pCnt  A ) ) (Λ `  ( p ^ k
) )  =  sum_ p  e.  ( ( 1 ... A )  i^i 
Prime ) ( ( p 
pCnt  A )  x.  ( log `  p ) ) )
91 pclogsum 20416 . 2  |-  ( A  e.  NN  ->  sum_ p  e.  ( ( 1 ... A )  i^i  Prime ) ( ( p  pCnt  A )  x.  ( log `  p ) )  =  ( log `  A
) )
9269, 90, 913eqtrd 2294 1  |-  ( A  e.  NN  ->  sum_ n  e.  { x  e.  NN  |  x  ||  A } 
(Λ `  n )  =  ( log `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   {crab 2522    i^i cin 3126    C_ wss 3127   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Fincfn 6831   CCcc 8703   RRcr 8704   0cc0 8705   1c1 8706    x. cmul 8710    <_ cle 8836   NNcn 9714   NN0cn0 9932   ZZcz 9991   ...cfz 10748   ^cexp 11070   #chash 11303   sum_csu 12123    || cdivides 12493   Primecprime 12720    pCnt cpc 12851   logclog 19874  Λcvma 20291
This theorem is referenced by:  logfac2  20418  dchrvmasumlem1  20606  vmalogdivsum2  20649  logsqvma  20653
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-ixp 6786  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-fi 7133  df-sup 7162  df-oi 7193  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9933  df-z 9992  df-dec 10092  df-uz 10198  df-q 10284  df-rp 10322  df-xneg 10419  df-xadd 10420  df-xmul 10421  df-ioo 10626  df-ioc 10627  df-ico 10628  df-icc 10629  df-fz 10749  df-fzo 10837  df-fl 10891  df-mod 10940  df-seq 11013  df-exp 11071  df-fac 11255  df-bc 11282  df-hash 11304  df-shft 11527  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-limsup 11910  df-clim 11927  df-rlim 11928  df-sum 12124  df-ef 12311  df-sin 12313  df-cos 12314  df-pi 12316  df-divides 12494  df-gcd 12648  df-prime 12721  df-pc 12852  df-struct 13112  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-mulr 13184  df-starv 13185  df-sca 13186  df-vsca 13187  df-tset 13189  df-ple 13190  df-ds 13192  df-hom 13194  df-cco 13195  df-rest 13289  df-topn 13290  df-topgen 13306  df-pt 13307  df-prds 13310  df-xrs 13365  df-0g 13366  df-gsum 13367  df-qtop 13372  df-imas 13373  df-xps 13375  df-mre 13450  df-mrc 13451  df-acs 13453  df-mnd 14329  df-submnd 14378  df-mulg 14454  df-cntz 14755  df-cmn 15053  df-xmet 16335  df-met 16336  df-bl 16337  df-mopn 16338  df-cnfld 16340  df-top 16598  df-bases 16600  df-topon 16601  df-topsp 16602  df-cld 16718  df-ntr 16719  df-cls 16720  df-nei 16797  df-lp 16830  df-perf 16831  df-cn 16919  df-cnp 16920  df-haus 17005  df-tx 17219  df-hmeo 17408  df-fbas 17482  df-fg 17483  df-fil 17503  df-fm 17595  df-flim 17596  df-flf 17597  df-xms 17847  df-ms 17848  df-tms 17849  df-cncf 18344  df-limc 19178  df-dv 19179  df-log 19876  df-vma 20297
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