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

Theorem vmasum 21005
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
Dummy variables  k  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5731 . . 3  |-  ( n  =  ( p ^
k )  ->  (Λ `  n )  =  (Λ `  ( p ^ k
) ) )
2 fzfid 11317 . . . 4  |-  ( A  e.  NN  ->  (
1 ... A )  e. 
Fin )
3 sgmss 20894 . . . 4  |-  ( A  e.  NN  ->  { x  e.  NN  |  x  ||  A }  C_  ( 1 ... A ) )
4 ssfi 7332 . . . 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 3430 . . . 4  |-  { x  e.  NN  |  x  ||  A }  C_  NN
76a1i 11 . . 3  |-  ( A  e.  NN  ->  { x  e.  NN  |  x  ||  A }  C_  NN )
8 inss1 3563 . . . 4  |-  ( ( 1 ... A )  i^i  Prime )  C_  (
1 ... A )
9 ssfi 7332 . . . 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 13228 . . . . . . . . . 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 10378 . . . . . . . 8  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  ZZ )
14 fznn 11120 . . . . . . . 8  |-  ( ( p  pCnt  A )  e.  ZZ  ->  ( k  e.  ( 1 ... (
p  pCnt  A )
)  <->  ( k  e.  NN  /\  k  <_ 
( p  pCnt  A
) ) ) )
1513, 14syl 16 . . . . . . 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 13088 . . . . . . . . . . . . . . 15  |-  ( p  e.  Prime  ->  p  e.  ZZ )
1918adantl 454 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  ZZ )
20 iddvdsexp 12878 . . . . . . . . . . . . . 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 13087 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  Prime  ->  p  e.  NN )
2423adantl 454 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  NN )
25 nnnn0 10233 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  k  e.  NN0 )
26 nnexpcl 11399 . . . . . . . . . . . . . . . 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 10379 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( p ^
k )  e.  ZZ )
29 nnz 10308 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  A  e.  ZZ )
3029ad2antrr 708 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  A  e.  ZZ )
31 dvdstr 12888 . . . . . . . . . . . . . 14  |-  ( ( p  e.  ZZ  /\  ( p ^ k
)  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( p  ||  ( p ^ k
)  /\  ( p ^ k )  ||  A )  ->  p  ||  A ) )
3222, 28, 30, 31syl3anc 1185 . . . . . . . . . . . . 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 12900 . . . . . . . . . . . . 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 43 . . . . . . . . . . 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 11120 . . . . . . . . . . . . 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 4218 . . . . . . . . . . 11  |-  ( x  =  ( p ^
k )  ->  (
x  ||  A  <->  ( p ^ k )  ||  A ) )
4544elrab3 3095 . . . . . . . . . 10  |-  ( ( p ^ k )  e.  NN  ->  (
( p ^ k
)  e.  { x  e.  NN  |  x  ||  A }  <->  ( p ^
k )  ||  A
) )
4627, 45syl 16 . . . . . . . . 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 13247 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  A  e.  ZZ  /\  k  e. 
NN0 )  ->  (
k  <_  ( p  pCnt  A )  <->  ( p ^ k )  ||  A ) )
5047, 30, 48, 49syl3anc 1185 . . . . . . . . . 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 3532 . . . . . 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 3350 . . . . 5  |-  ( ( A  e.  NN  /\  n  e.  { x  e.  NN  |  x  ||  A } )  ->  n  e.  NN )
65 vmacl 20906 . . . . 5  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
6664, 65syl 16 . . . 4  |-  ( ( A  e.  NN  /\  n  e.  { x  e.  NN  |  x  ||  A } )  ->  (Λ `  n )  e.  RR )
6766recnd 9119 . . 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 21002 . 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 11085 . . . . . . 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 20904 . . . . . 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 12502 . . . 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 11317 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
1 ... ( p  pCnt  A ) )  e.  Fin )
7870, 23syl 16 . . . . . . . . 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 10652 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  RR+ )
8180relogcld 20523 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  p )  e.  RR )
8281recnd 9119 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  p )  e.  CC )
83 fsumconst 12578 . . . . 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 11635 . . . . . 6  |-  ( ( p  pCnt  A )  e.  NN0  ->  ( # `  (
1 ... ( p  pCnt  A ) ) )  =  ( p  pCnt  A
) )
8785, 86syl 16 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( # `
 ( 1 ... ( p  pCnt  A
) ) )  =  ( p  pCnt  A
) )
8887oveq1d 6099 . . . 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 2474 . . 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 12502 . 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 21004 . 2  |-  ( A  e.  NN  ->  sum_ p  e.  ( ( 1 ... A )  i^i  Prime ) ( ( p  pCnt  A )  x.  ( log `  p ) )  =  ( log `  A
) )
9269, 90, 913eqtrd 2474 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 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711    i^i cin 3321    C_ wss 3322   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Fincfn 7112   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996    x. cmul 9000    <_ cle 9126   NNcn 10005   NN0cn0 10226   ZZcz 10287   ...cfz 11048   ^cexp 11387   #chash 11623   sum_csu 12484    || cdivides 12857   Primecprime 13084    pCnt cpc 13215   logclog 20457  Λcvma 20879
This theorem is referenced by:  logfac2  21006  dchrvmasumlem1  21194  vmalogdivsum2  21237  logsqvma  21241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ioc 10926  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-fac 11572  df-bc 11599  df-hash 11624  df-shft 11887  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-limsup 12270  df-clim 12287  df-rlim 12288  df-sum 12485  df-ef 12675  df-sin 12677  df-cos 12678  df-pi 12680  df-dvds 12858  df-gcd 13012  df-prm 13085  df-pc 13216  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lp 17205  df-perf 17206  df-cn 17296  df-cnp 17297  df-haus 17384  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cncf 18913  df-limc 19758  df-dv 19759  df-log 20459  df-vma 20885
  Copyright terms: Public domain W3C validator