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Theorem vmasum 20953
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 5687 . . 3  |-  ( n  =  ( p ^
k )  ->  (Λ `  n )  =  (Λ `  ( p ^ k
) ) )
2 fzfid 11267 . . . 4  |-  ( A  e.  NN  ->  (
1 ... A )  e. 
Fin )
3 sgmss 20842 . . . 4  |-  ( A  e.  NN  ->  { x  e.  NN  |  x  ||  A }  C_  ( 1 ... A ) )
4 ssfi 7288 . . . 4  |-  ( ( ( 1 ... A
)  e.  Fin  /\  { x  e.  NN  |  x  ||  A }  C_  ( 1 ... A
) )  ->  { x  e.  NN  |  x  ||  A }  e.  Fin )
52, 3, 4syl2anc 643 . . 3  |-  ( A  e.  NN  ->  { x  e.  NN  |  x  ||  A }  e.  Fin )
6 ssrab2 3388 . . . 4  |-  { x  e.  NN  |  x  ||  A }  C_  NN
76a1i 11 . . 3  |-  ( A  e.  NN  ->  { x  e.  NN  |  x  ||  A }  C_  NN )
8 inss1 3521 . . . 4  |-  ( ( 1 ... A )  i^i  Prime )  C_  (
1 ... A )
9 ssfi 7288 . . . 4  |-  ( ( ( 1 ... A
)  e.  Fin  /\  ( ( 1 ... A )  i^i  Prime ) 
C_  ( 1 ... A ) )  -> 
( ( 1 ... A )  i^i  Prime )  e.  Fin )
102, 8, 9sylancl 644 . . 3  |-  ( A  e.  NN  ->  (
( 1 ... A
)  i^i  Prime )  e. 
Fin )
11 pccl 13178 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
p  pCnt  A )  e.  NN0 )
1211ancoms 440 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  NN0 )
1312nn0zd 10329 . . . . . . . 8  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  ZZ )
14 fznn 11070 . . . . . . . 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 685 . . . . . 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 773 . . . . . . 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 13038 . . . . . . . . . . . . . . 15  |-  ( p  e.  Prime  ->  p  e.  ZZ )
1918adantl 453 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  ZZ )
20 iddvdsexp 12828 . . . . . . . . . . . . . 14  |-  ( ( p  e.  ZZ  /\  k  e.  NN )  ->  p  ||  ( p ^ k ) )
2119, 20sylan 458 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  p  ||  (
p ^ k ) )
2218ad2antlr 708 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  p  e.  ZZ )
23 prmnn 13037 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  Prime  ->  p  e.  NN )
2423adantl 453 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  NN )
25 nnnn0 10184 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  k  e.  NN0 )
26 nnexpcl 11349 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  NN  /\  k  e.  NN0 )  -> 
( p ^ k
)  e.  NN )
2724, 25, 26syl2an 464 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( p ^
k )  e.  NN )
2827nnzd 10330 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( p ^
k )  e.  ZZ )
29 nnz 10259 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  A  e.  ZZ )
3029ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  A  e.  ZZ )
31 dvdstr 12838 . . . . . . . . . . . . . 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 657 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( ( p ^ k )  ||  A  ->  p  ||  A
) )
34 simpll 731 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  A  e.  NN )
35 dvdsle 12850 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  A  e.  NN )  ->  ( p  ||  A  ->  p  <_  A )
)
3622, 34, 35syl2anc 643 . . . . . . . . . . . 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 708 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  p  e.  NN )
39 fznn 11070 . . . . . . . . . . . . 13  |-  ( A  e.  ZZ  ->  (
p  e.  ( 1 ... A )  <->  ( p  e.  NN  /\  p  <_  A ) ) )
4039baibd 876 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  p  e.  NN )  ->  ( p  e.  ( 1 ... A )  <-> 
p  <_  A )
)
4130, 38, 40syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( p  e.  ( 1 ... A
)  <->  p  <_  A ) )
4237, 41sylibrd 226 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( ( p ^ k )  ||  A  ->  p  e.  ( 1 ... A ) ) )
4342pm4.71rd 617 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( ( p ^ k )  ||  A 
<->  ( p  e.  ( 1 ... A )  /\  ( p ^
k )  ||  A
) ) )
44 breq1 4175 . . . . . . . . . . 11  |-  ( x  =  ( p ^
k )  ->  (
x  ||  A  <->  ( p ^ k )  ||  A ) )
4544elrab3 3053 . . . . . . . . . 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 732 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  p  e.  Prime )
4825adantl 453 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  k  e.  NN0 )
49 pcdvdsb 13197 . . . . . . . . . . 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 685 . . . . . . . . 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 278 . . . . . . . 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 623 . . . . . . 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 249 . . . . . 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 245 . . . . 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 623 . . . 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 3490 . . . . . 6  |-  ( p  e.  ( ( 1 ... A )  i^i 
Prime )  <->  ( p  e.  ( 1 ... A
)  /\  p  e.  Prime ) )
5857anbi1i 677 . . . . 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 631 . . . . 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 773 . . . . 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 263 . . . 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 631 . . . 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 280 . . 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 3308 . . . . 5  |-  ( ( A  e.  NN  /\  n  e.  { x  e.  NN  |  x  ||  A } )  ->  n  e.  NN )
65 vmacl 20854 . . . . 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 9070 . . 3  |-  ( ( A  e.  NN  /\  n  e.  { x  e.  NN  |  x  ||  A } )  ->  (Λ `  n )  e.  CC )
68 simprr 734 . . 3  |-  ( ( A  e.  NN  /\  ( n  e.  { x  e.  NN  |  x  ||  A }  /\  (Λ `  n )  =  0 ) )  ->  (Λ `  n )  =  0 )
691, 5, 7, 10, 63, 67, 68fsumvma 20950 . 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 451 . . . . . . 7  |-  ( p  e.  ( ( 1 ... A )  i^i 
Prime )  ->  p  e. 
Prime )
7170ad2antlr 708 . . . . . 6  |-  ( ( ( A  e.  NN  /\  p  e.  ( ( 1 ... A )  i^i  Prime ) )  /\  k  e.  ( 1 ... ( p  pCnt  A ) ) )  ->  p  e.  Prime )
72 elfznn 11036 . . . . . . 7  |-  ( k  e.  ( 1 ... ( p  pCnt  A
) )  ->  k  e.  NN )
7372adantl 453 . . . . . 6  |-  ( ( ( A  e.  NN  /\  p  e.  ( ( 1 ... A )  i^i  Prime ) )  /\  k  e.  ( 1 ... ( p  pCnt  A ) ) )  -> 
k  e.  NN )
74 vmappw 20852 . . . . . 6  |-  ( ( p  e.  Prime  /\  k  e.  NN )  ->  (Λ `  ( p ^ k
) )  =  ( log `  p ) )
7571, 73, 74syl2anc 643 . . . . 5  |-  ( ( ( A  e.  NN  /\  p  e.  ( ( 1 ... A )  i^i  Prime ) )  /\  k  e.  ( 1 ... ( p  pCnt  A ) ) )  -> 
(Λ `  ( p ^
k ) )  =  ( log `  p
) )
7675sumeq2dv 12452 . . . 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 11267 . . . . 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 453 . . . . . . . 8  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  NN )
8079nnrpd 10603 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  RR+ )
8180relogcld 20471 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  p )  e.  RR )
8281recnd 9070 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  p )  e.  CC )
83 fsumconst 12528 . . . . 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 643 . . . 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 461 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p  pCnt  A )  e.  NN0 )
86 hashfz1 11585 . . . . . 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 6055 . . . 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 2440 . . 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 12452 . 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 20952 . 2  |-  ( A  e.  NN  ->  sum_ p  e.  ( ( 1 ... A )  i^i  Prime ) ( ( p  pCnt  A )  x.  ( log `  p ) )  =  ( log `  A
) )
9269, 90, 913eqtrd 2440 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2670    i^i cin 3279    C_ wss 3280   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    <_ cle 9077   NNcn 9956   NN0cn0 10177   ZZcz 10238   ...cfz 10999   ^cexp 11337   #chash 11573   sum_csu 12434    || cdivides 12807   Primecprime 13034    pCnt cpc 13165   logclog 20405  Λcvma 20827
This theorem is referenced by:  logfac2  20954  dchrvmasumlem1  21142  vmalogdivsum2  21185  logsqvma  21189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-vma 20833
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