MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vmasum Unicode version

Theorem vmasum 20287
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 5377 . . 3  |-  ( n  =  ( p ^
k )  ->  (Λ `  n )  =  (Λ `  ( p ^ k
) ) )
2 fzfid 10913 . . . 4  |-  ( A  e.  NN  ->  (
1 ... A )  e. 
Fin )
3 sgmss 20176 . . . 4  |-  ( A  e.  NN  ->  { x  e.  NN  |  x  ||  A }  C_  ( 1 ... A ) )
4 ssfi 6968 . . . 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 3179 . . . 4  |-  { x  e.  NN  |  x  ||  A }  C_  NN
76a1i 12 . . 3  |-  ( A  e.  NN  ->  { x  e.  NN  |  x  ||  A }  C_  NN )
8 inss1 3296 . . . 4  |-  ( ( 1 ... A )  i^i  Prime )  C_  (
1 ... A )
9 ssfi 6968 . . . 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 12776 . . . . . . . . . 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 9994 . . . . . . . 8  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  ZZ )
14 fznn 10730 . . . . . . . 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 12635 . . . . . . . . . . . . . . 15  |-  ( p  e.  Prime  ->  p  e.  ZZ )
1918adantl 454 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  ZZ )
20 iddvdsexp 12426 . . . . . . . . . . . . . 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 12636 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  Prime  ->  p  e.  NN )
2423adantl 454 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  NN )
25 nnnn0 9851 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  k  e.  NN0 )
26 nnexpcl 10994 . . . . . . . . . . . . . . . 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 9995 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( p ^
k )  e.  ZZ )
29 nnz 9924 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  A  e.  ZZ )
3029ad2antrr 709 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  A  e.  ZZ )
31 dvdstr 12436 . . . . . . . . . . . . . 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 12448 . . . . . . . . . . . . 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 10730 . . . . . . . . . . . . 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 3923 . . . . . . . . . . 11  |-  ( x  =  ( p ^
k )  ->  (
x  ||  A  <->  ( p ^ k )  ||  A ) )
4544elrab3 2861 . . . . . . . . . 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 12795 . . . . . . . . . . 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 3266 . . . . . 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 3103 . . . . 5  |-  ( ( A  e.  NN  /\  n  e.  { x  e.  NN  |  x  ||  A } )  ->  n  e.  NN )
65 vmacl 20188 . . . . 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 8741 . . 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 20284 . 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 10697 . . . . . . 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 20186 . . . . . 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 12053 . . . 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 10913 . . . . 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 10268 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  RR+ )
8180relogcld 19806 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  p )  e.  RR )
8281recnd 8741 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  p )  e.  CC )
83 fsumconst 12129 . . . . 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 11223 . . . . . 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 5725 . . . 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 2289 . . 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 12053 . 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 20286 . 2  |-  ( A  e.  NN  ->  sum_ p  e.  ( ( 1 ... A )  i^i  Prime ) ( ( p  pCnt  A )  x.  ( log `  p ) )  =  ( log `  A
) )
9269, 90, 913eqtrd 2289 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 2512    i^i cin 3077    C_ wss 3078   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Fincfn 6749   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618    x. cmul 8622    <_ cle 8748   NNcn 9626   NN0cn0 9844   ZZcz 9903   ...cfz 10660   ^cexp 10982   #chash 11215   sum_csu 12035    || cdivides 12405   Primecprime 12632    pCnt cpc 12763   logclog 19744  Λcvma 20161
This theorem is referenced by:  logfac2  20288  dchrvmasumlem1  20476  vmalogdivsum2  20519  logsqvma  20523
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-addf 8696  ax-mulf 8697
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-er 6546  df-map 6660  df-pm 6661  df-ixp 6704  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-fi 7049  df-sup 7078  df-oi 7109  df-card 7456  df-cda 7678  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-7 9689  df-8 9690  df-9 9691  df-10 9692  df-n0 9845  df-z 9904  df-dec 10004  df-uz 10110  df-q 10196  df-rp 10234  df-xneg 10331  df-xadd 10332  df-xmul 10333  df-ioo 10538  df-ioc 10539  df-ico 10540  df-icc 10541  df-fz 10661  df-fzo 10749  df-fl 10803  df-mod 10852  df-seq 10925  df-exp 10983  df-fac 11167  df-bc 11194  df-hash 11216  df-shft 11439  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-limsup 11822  df-clim 11839  df-rlim 11840  df-sum 12036  df-ef 12223  df-sin 12225  df-cos 12226  df-pi 12228  df-divides 12406  df-gcd 12560  df-prime 12633  df-pc 12764  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-starv 13097  df-sca 13098  df-vsca 13099  df-tset 13101  df-ple 13102  df-ds 13104  df-hom 13106  df-cco 13107  df-rest 13201  df-topn 13202  df-topgen 13218  df-pt 13219  df-prds 13222  df-xrs 13277  df-0g 13278  df-gsum 13279  df-qtop 13284  df-imas 13285  df-xps 13287  df-mre 13361  df-mrc 13362  df-acs 13363  df-mnd 14202  df-submnd 14251  df-mulg 14327  df-cntz 14628  df-cmn 14926  df-xmet 16205  df-met 16206  df-bl 16207  df-mopn 16208  df-cnfld 16210  df-top 16468  df-bases 16470  df-topon 16471  df-topsp 16472  df-cld 16588  df-ntr 16589  df-cls 16590  df-nei 16667  df-lp 16700  df-perf 16701  df-cn 16789  df-cnp 16790  df-haus 16875  df-tx 17089  df-hmeo 17278  df-fbas 17352  df-fg 17353  df-fil 17373  df-fm 17465  df-flim 17466  df-flf 17467  df-xms 17717  df-ms 17718  df-tms 17719  df-cncf 18214  df-limc 19048  df-dv 19049  df-log 19746  df-vma 20167
  Copyright terms: Public domain W3C validator