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

Theorem vmasum 20451
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
Dummy variables  k  p are mutually distinct and distinct from all other variables.

Proof of Theorem vmasum
StepHypRef Expression
1 fveq2 5487 . . 3  |-  ( n  =  ( p ^
k )  ->  (Λ `  n )  =  (Λ `  ( p ^ k
) ) )
2 fzfid 11031 . . . 4  |-  ( A  e.  NN  ->  (
1 ... A )  e. 
Fin )
3 sgmss 20340 . . . 4  |-  ( A  e.  NN  ->  { x  e.  NN  |  x  ||  A }  C_  ( 1 ... A ) )
4 ssfi 7080 . . . 4  |-  ( ( ( 1 ... A
)  e.  Fin  /\  { x  e.  NN  |  x  ||  A }  C_  ( 1 ... A
) )  ->  { x  e.  NN  |  x  ||  A }  e.  Fin )
52, 3, 4syl2anc 644 . . 3  |-  ( A  e.  NN  ->  { x  e.  NN  |  x  ||  A }  e.  Fin )
6 ssrab2 3261 . . . 4  |-  { x  e.  NN  |  x  ||  A }  C_  NN
76a1i 12 . . 3  |-  ( A  e.  NN  ->  { x  e.  NN  |  x  ||  A }  C_  NN )
8 inss1 3392 . . . 4  |-  ( ( 1 ... A )  i^i  Prime )  C_  (
1 ... A )
9 ssfi 7080 . . . 4  |-  ( ( ( 1 ... A
)  e.  Fin  /\  ( ( 1 ... A )  i^i  Prime ) 
C_  ( 1 ... A ) )  -> 
( ( 1 ... A )  i^i  Prime )  e.  Fin )
102, 8, 9sylancl 645 . . 3  |-  ( A  e.  NN  ->  (
( 1 ... A
)  i^i  Prime )  e. 
Fin )
11 pccl 12898 . . . . . . . . . 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 10112 . . . . . . . 8  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  ZZ )
14 fznn 10848 . . . . . . . 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 686 . . . . . 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 774 . . . . . . 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 12758 . . . . . . . . . . . . . . 15  |-  ( p  e.  Prime  ->  p  e.  ZZ )
1918adantl 454 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  ZZ )
20 iddvdsexp 12548 . . . . . . . . . . . . . 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 709 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  p  e.  ZZ )
23 prmnn 12757 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  Prime  ->  p  e.  NN )
2423adantl 454 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  NN )
25 nnnn0 9969 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  k  e.  NN0 )
26 nnexpcl 11112 . . . . . . . . . . . . . . . 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 10113 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( p ^
k )  e.  ZZ )
29 nnz 10042 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  A  e.  ZZ )
3029ad2antrr 708 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  A  e.  ZZ )
31 dvdstr 12558 . . . . . . . . . . . . . 14  |-  ( ( p  e.  ZZ  /\  ( p ^ k
)  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( p  ||  ( p ^ k
)  /\  ( p ^ k )  ||  A )  ->  p  ||  A ) )
3222, 28, 30, 31syl3anc 1184 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( ( p 
||  ( p ^
k )  /\  (
p ^ k ) 
||  A )  ->  p  ||  A ) )
3321, 32mpand 658 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( ( p ^ k )  ||  A  ->  p  ||  A
) )
34 simpll 732 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  A  e.  NN )
35 dvdsle 12570 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  A  e.  NN )  ->  ( p  ||  A  ->  p  <_  A )
)
3622, 34, 35syl2anc 644 . . . . . . . . . . . 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 709 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  p  e.  NN )
39 fznn 10848 . . . . . . . . . . . . 13  |-  ( A  e.  ZZ  ->  (
p  e.  ( 1 ... A )  <->  ( p  e.  NN  /\  p  <_  A ) ) )
4039baibd 877 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  p  e.  NN )  ->  ( p  e.  ( 1 ... A )  <-> 
p  <_  A )
)
4130, 38, 40syl2anc 644 . . . . . . . . . . 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 618 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( ( p ^ k )  ||  A 
<->  ( p  e.  ( 1 ... A )  /\  ( p ^
k )  ||  A
) ) )
44 breq1 4029 . . . . . . . . . . 11  |-  ( x  =  ( p ^
k )  ->  (
x  ||  A  <->  ( p ^ k )  ||  A ) )
4544elrab3 2927 . . . . . . . . . 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 733 . . . . . . . . . . 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 12917 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  A  e.  ZZ  /\  k  e. 
NN0 )  ->  (
k  <_  ( p  pCnt  A )  <->  ( p ^ k )  ||  A ) )
5047, 30, 48, 49syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( k  <_ 
( p  pCnt  A
)  <->  ( p ^
k )  ||  A
) )
5150anbi2d 686 . . . . . . . . 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 624 . . . . . . 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 624 . . . 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 3361 . . . . . 6  |-  ( p  e.  ( ( 1 ... A )  i^i 
Prime )  <->  ( p  e.  ( 1 ... A
)  /\  p  e.  Prime ) )
5857anbi1i 678 . . . . 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 632 . . . . 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 774 . . . . 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 632 . . . 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 3183 . . . . 5  |-  ( ( A  e.  NN  /\  n  e.  { x  e.  NN  |  x  ||  A } )  ->  n  e.  NN )
65 vmacl 20352 . . . . 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 8858 . . 3  |-  ( ( A  e.  NN  /\  n  e.  { x  e.  NN  |  x  ||  A } )  ->  (Λ `  n )  e.  CC )
68 simprr 735 . . 3  |-  ( ( A  e.  NN  /\  ( n  e.  { x  e.  NN  |  x  ||  A }  /\  (Λ `  n )  =  0 ) )  ->  (Λ `  n )  =  0 )
691, 5, 7, 10, 63, 67, 68fsumvma 20448 . 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 709 . . . . . 6  |-  ( ( ( A  e.  NN  /\  p  e.  ( ( 1 ... A )  i^i  Prime ) )  /\  k  e.  ( 1 ... ( p  pCnt  A ) ) )  ->  p  e.  Prime )
72 elfznn 10815 . . . . . . 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 20350 . . . . . 6  |-  ( ( p  e.  Prime  /\  k  e.  NN )  ->  (Λ `  ( p ^ k
) )  =  ( log `  p ) )
7571, 73, 74syl2anc 644 . . . . 5  |-  ( ( ( A  e.  NN  /\  p  e.  ( ( 1 ... A )  i^i  Prime ) )  /\  k  e.  ( 1 ... ( p  pCnt  A ) ) )  -> 
(Λ `  ( p ^
k ) )  =  ( log `  p
) )
7675sumeq2dv 12172 . . . 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 11031 . . . . 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 10386 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  RR+ )
8180relogcld 19970 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  p )  e.  RR )
8281recnd 8858 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  p )  e.  CC )
83 fsumconst 12248 . . . . 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 644 . . . 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 11341 . . . . . 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 5836 . . . 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 2322 . . 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 12172 . 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 20450 . 2  |-  ( A  e.  NN  ->  sum_ p  e.  ( ( 1 ... A )  i^i  Prime ) ( ( p  pCnt  A )  x.  ( log `  p ) )  =  ( log `  A
) )
9269, 90, 913eqtrd 2322 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 1625    e. wcel 1687   {crab 2550    i^i cin 3154    C_ wss 3155   class class class wbr 4026   ` cfv 5223  (class class class)co 5821   Fincfn 6860   CCcc 8732   RRcr 8733   0cc0 8734   1c1 8735    x. cmul 8739    <_ cle 8865   NNcn 9743   NN0cn0 9962   ZZcz 10021   ...cfz 10778   ^cexp 11100   #chash 11333   sum_csu 12154    || cdivides 12527   Primecprime 12754    pCnt cpc 12885   logclog 19908  Λcvma 20325
This theorem is referenced by:  logfac2  20452  dchrvmasumlem1  20640  vmalogdivsum2  20683  logsqvma  20687
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-inf2 7339  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-pre-sup 8812  ax-addf 8813  ax-mulf 8814
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-int 3866  df-iun 3910  df-iin 3911  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-se 4354  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-isom 5232  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-of 6041  df-1st 6085  df-2nd 6086  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-1o 6476  df-2o 6477  df-oadd 6480  df-er 6657  df-map 6771  df-pm 6772  df-ixp 6815  df-en 6861  df-dom 6862  df-sdom 6863  df-fin 6864  df-fi 7162  df-sup 7191  df-oi 7222  df-card 7569  df-cda 7791  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-div 9421  df-nn 9744  df-2 9801  df-3 9802  df-4 9803  df-5 9804  df-6 9805  df-7 9806  df-8 9807  df-9 9808  df-10 9809  df-n0 9963  df-z 10022  df-dec 10122  df-uz 10228  df-q 10314  df-rp 10352  df-xneg 10449  df-xadd 10450  df-xmul 10451  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-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-sum 12155  df-ef 12345  df-sin 12347  df-cos 12348  df-pi 12350  df-dvds 12528  df-gcd 12682  df-prm 12755  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-xps 13409  df-mre 13484  df-mrc 13485  df-acs 13487  df-mnd 14363  df-submnd 14412  df-mulg 14488  df-cntz 14789  df-cmn 15087  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  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-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-limc 19212  df-dv 19213  df-log 19910  df-vma 20331
  Copyright terms: Public domain W3C validator