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Theorem wallispi2 27689
Description: An alternative version of Wallis' formula for π ; this second formula uses factorials and it is later used to proof Stirling's approximation formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypothesis
Ref Expression
wallispi2.1  |-  V  =  ( n  e.  NN  |->  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `
 n ) ^
4 ) )  / 
( ( ! `  ( 2  x.  n
) ) ^ 2 ) )  /  (
( 2  x.  n
)  +  1 ) ) )
Assertion
Ref Expression
wallispi2  |-  V  ~~>  ( pi 
/  2 )

Proof of Theorem wallispi2
Dummy variables  k  m  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . 2  |-  ( k  e.  NN  |->  ( ( ( 2  x.  k
)  /  ( ( 2  x.  k )  -  1 ) )  x.  ( ( 2  x.  k )  / 
( ( 2  x.  k )  +  1 ) ) ) )  =  ( k  e.  NN  |->  ( ( ( 2  x.  k )  /  ( ( 2  x.  k )  - 
1 ) )  x.  ( ( 2  x.  k )  /  (
( 2  x.  k
)  +  1 ) ) ) )
2 ax-1cn 9004 . . . . . . 7  |-  1  e.  CC
32a1i 11 . . . . . 6  |-  ( n  e.  NN  ->  1  e.  CC )
4 2cn 10026 . . . . . . . . 9  |-  2  e.  CC
54a1i 11 . . . . . . . 8  |-  ( n  e.  NN  ->  2  e.  CC )
6 nncn 9964 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  CC )
75, 6mulcld 9064 . . . . . . 7  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  CC )
87, 3addcld 9063 . . . . . 6  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  e.  CC )
9 elnnuz 10478 . . . . . . . 8  |-  ( n  e.  NN  <->  n  e.  ( ZZ>= `  1 )
)
109biimpi 187 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
11 eqidd 2405 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... n )  ->  (
k  e.  NN  |->  ( ( ( 2  x.  k ) ^ 4 )  /  ( ( ( 2  x.  k
)  x.  ( ( 2  x.  k )  -  1 ) ) ^ 2 ) ) )  =  ( k  e.  NN  |->  ( ( ( 2  x.  k
) ^ 4 )  /  ( ( ( 2  x.  k )  x.  ( ( 2  x.  k )  - 
1 ) ) ^
2 ) ) ) )
12 simpr 448 . . . . . . . . . . . . 13  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  k  =  m )
1312oveq2d 6056 . . . . . . . . . . . 12  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( 2  x.  k
)  =  ( 2  x.  m ) )
1413oveq1d 6055 . . . . . . . . . . 11  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k ) ^ 4 )  =  ( ( 2  x.  m ) ^ 4 ) )
1513oveq1d 6055 . . . . . . . . . . . . 13  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k )  -  1 )  =  ( ( 2  x.  m )  -  1 ) )
1613, 15oveq12d 6058 . . . . . . . . . . . 12  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) )  =  ( ( 2  x.  m )  x.  ( ( 2  x.  m )  - 
1 ) ) )
1716oveq1d 6055 . . . . . . . . . . 11  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( ( 2  x.  k )  x.  ( ( 2  x.  k )  -  1 ) ) ^ 2 )  =  ( ( ( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) ) ^ 2 ) )
1814, 17oveq12d 6058 . . . . . . . . . 10  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( ( 2  x.  k ) ^
4 )  /  (
( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) ) ^ 2 ) )  =  ( ( ( 2  x.  m
) ^ 4 )  /  ( ( ( 2  x.  m )  x.  ( ( 2  x.  m )  - 
1 ) ) ^
2 ) ) )
19 elfznn 11036 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... n )  ->  m  e.  NN )
204a1i 11 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  2  e.  CC )
2119nncnd 9972 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  m  e.  CC )
2220, 21mulcld 9064 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  e.  CC )
23 4nn0 10196 . . . . . . . . . . . . 13  |-  4  e.  NN0
2423a1i 11 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  4  e.  NN0 )
2522, 24expcld 11478 . . . . . . . . . . 11  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
) ^ 4 )  e.  CC )
262a1i 11 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  1  e.  CC )
2722, 26subcld 9367 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  -  1 )  e.  CC )
2822, 27mulcld 9064 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) )  e.  CC )
2928sqcld 11476 . . . . . . . . . . 11  |-  ( m  e.  ( 1 ... n )  ->  (
( ( 2  x.  m )  x.  (
( 2  x.  m
)  -  1 ) ) ^ 2 )  e.  CC )
30 2ne0 10039 . . . . . . . . . . . . . . 15  |-  2  =/=  0
3130a1i 11 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  2  =/=  0 )
3219nnne0d 10000 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  m  =/=  0 )
3320, 21, 31, 32mulne0d 9630 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  =/=  0 )
34 1re 9046 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
3534a1i 11 . . . . . . . . . . . . . . 15  |-  ( m  e.  ( 1 ... n )  ->  1  e.  RR )
36 2re 10025 . . . . . . . . . . . . . . . . . 18  |-  2  e.  RR
3736a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  2  e.  RR )
3837, 35remulcld 9072 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  1 )  e.  RR )
3919nnred 9971 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  m  e.  RR )
4037, 39remulcld 9072 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  e.  RR )
41 1lt2 10098 . . . . . . . . . . . . . . . . . 18  |-  1  <  2
4241a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  1  <  2 )
434mulid1i 9048 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  1 )  =  2
4442, 43syl6breqr 4212 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  1  <  ( 2  x.  1 ) )
45 0re 9047 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR
46 2pos 10038 . . . . . . . . . . . . . . . . . . 19  |-  0  <  2
4745, 36, 46ltleii 9152 . . . . . . . . . . . . . . . . . 18  |-  0  <_  2
4847a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  0  <_  2 )
49 elfzle1 11016 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  1  <_  m )
5035, 39, 37, 48, 49lemul2ad 9907 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  1 )  <_  ( 2  x.  m ) )
5135, 38, 40, 44, 50ltletrd 9186 . . . . . . . . . . . . . . 15  |-  ( m  e.  ( 1 ... n )  ->  1  <  ( 2  x.  m
) )
5235, 51gtned 9164 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  =/=  1 )
5322, 26, 52subne0d 9376 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  -  1 )  =/=  0 )
5422, 27, 33, 53mulne0d 9630 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) )  =/=  0 )
55 2z 10268 . . . . . . . . . . . . 13  |-  2  e.  ZZ
5655a1i 11 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  2  e.  ZZ )
5728, 54, 56expne0d 11484 . . . . . . . . . . 11  |-  ( m  e.  ( 1 ... n )  ->  (
( ( 2  x.  m )  x.  (
( 2  x.  m
)  -  1 ) ) ^ 2 )  =/=  0 )
5825, 29, 57divcld 9746 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... n )  ->  (
( ( 2  x.  m ) ^ 4 )  /  ( ( ( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) ) ^ 2 ) )  e.  CC )
5911, 18, 19, 58fvmptd 5769 . . . . . . . . 9  |-  ( m  e.  ( 1 ... n )  ->  (
( k  e.  NN  |->  ( ( ( 2  x.  k ) ^
4 )  /  (
( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) ) ^ 2 ) ) ) `  m
)  =  ( ( ( 2  x.  m
) ^ 4 )  /  ( ( ( 2  x.  m )  x.  ( ( 2  x.  m )  - 
1 ) ) ^
2 ) ) )
6059, 58eqeltrd 2478 . . . . . . . 8  |-  ( m  e.  ( 1 ... n )  ->  (
( k  e.  NN  |->  ( ( ( 2  x.  k ) ^
4 )  /  (
( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) ) ^ 2 ) ) ) `  m
)  e.  CC )
6160adantl 453 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  ( 1 ... n ) )  ->  ( ( k  e.  NN  |->  ( ( ( 2  x.  k
) ^ 4 )  /  ( ( ( 2  x.  k )  x.  ( ( 2  x.  k )  - 
1 ) ) ^
2 ) ) ) `
 m )  e.  CC )
62 mulcl 9030 . . . . . . . 8  |-  ( ( m  e.  CC  /\  w  e.  CC )  ->  ( m  x.  w
)  e.  CC )
6362adantl 453 . . . . . . 7  |-  ( ( n  e.  NN  /\  ( m  e.  CC  /\  w  e.  CC ) )  ->  ( m  x.  w )  e.  CC )
6410, 61, 63seqcl 11298 . . . . . 6  |-  ( n  e.  NN  ->  (  seq  1 (  x.  , 
( k  e.  NN  |->  ( ( ( 2  x.  k ) ^
4 )  /  (
( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) ) ^ 2 ) ) ) ) `  n )  e.  CC )
65 2nn 10089 . . . . . . . . . 10  |-  2  e.  NN
6665a1i 11 . . . . . . . . 9  |-  ( n  e.  NN  ->  2  e.  NN )
67 id 20 . . . . . . . . 9  |-  ( n  e.  NN  ->  n  e.  NN )
6866, 67nnmulcld 10003 . . . . . . . 8  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  NN )
6968peano2nnd 9973 . . . . . . 7  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  e.  NN )
7069nnne0d 10000 . . . . . 6  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  =/=  0 )
713, 8, 64, 70div32d 9769 . . . . 5  |-  ( n  e.  NN  ->  (
( 1  /  (
( 2  x.  n
)  +  1 ) )  x.  (  seq  1 (  x.  , 
( k  e.  NN  |->  ( ( ( 2  x.  k ) ^
4 )  /  (
( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) ) ^ 2 ) ) ) ) `  n ) )  =  ( 1  x.  (
(  seq  1 (  x.  ,  ( k  e.  NN  |->  ( ( ( 2  x.  k
) ^ 4 )  /  ( ( ( 2  x.  k )  x.  ( ( 2  x.  k )  - 
1 ) ) ^
2 ) ) ) ) `  n )  /  ( ( 2  x.  n )  +  1 ) ) ) )
7264, 8, 70divcld 9746 . . . . . 6  |-  ( n  e.  NN  ->  (
(  seq  1 (  x.  ,  ( k  e.  NN  |->  ( ( ( 2  x.  k
) ^ 4 )  /  ( ( ( 2  x.  k )  x.  ( ( 2  x.  k )  - 
1 ) ) ^
2 ) ) ) ) `  n )  /  ( ( 2  x.  n )  +  1 ) )  e.  CC )
7372mulid2d 9062 . . . . 5  |-  ( n  e.  NN  ->  (
1  x.  ( (  seq  1 (  x.  ,  ( k  e.  NN  |->  ( ( ( 2  x.  k ) ^ 4 )  / 
( ( ( 2  x.  k )  x.  ( ( 2  x.  k )  -  1 ) ) ^ 2 ) ) ) ) `
 n )  / 
( ( 2  x.  n )  +  1 ) ) )  =  ( (  seq  1
(  x.  ,  ( k  e.  NN  |->  ( ( ( 2  x.  k ) ^ 4 )  /  ( ( ( 2  x.  k
)  x.  ( ( 2  x.  k )  -  1 ) ) ^ 2 ) ) ) ) `  n
)  /  ( ( 2  x.  n )  +  1 ) ) )
74 wallispi2lem2 27688 . . . . . 6  |-  ( n  e.  NN  ->  (  seq  1 (  x.  , 
( k  e.  NN  |->  ( ( ( 2  x.  k ) ^
4 )  /  (
( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) ) ^ 2 ) ) ) ) `  n )  =  ( ( ( 2 ^ ( 4  x.  n
) )  x.  (
( ! `  n
) ^ 4 ) )  /  ( ( ! `  ( 2  x.  n ) ) ^ 2 ) ) )
7574oveq1d 6055 . . . . 5  |-  ( n  e.  NN  ->  (
(  seq  1 (  x.  ,  ( k  e.  NN  |->  ( ( ( 2  x.  k
) ^ 4 )  /  ( ( ( 2  x.  k )  x.  ( ( 2  x.  k )  - 
1 ) ) ^
2 ) ) ) ) `  n )  /  ( ( 2  x.  n )  +  1 ) )  =  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `
 n ) ^
4 ) )  / 
( ( ! `  ( 2  x.  n
) ) ^ 2 ) )  /  (
( 2  x.  n
)  +  1 ) ) )
7671, 73, 753eqtrd 2440 . . . 4  |-  ( n  e.  NN  ->  (
( 1  /  (
( 2  x.  n
)  +  1 ) )  x.  (  seq  1 (  x.  , 
( k  e.  NN  |->  ( ( ( 2  x.  k ) ^
4 )  /  (
( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) ) ^ 2 ) ) ) ) `  n ) )  =  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `
 n ) ^
4 ) )  / 
( ( ! `  ( 2  x.  n
) ) ^ 2 ) )  /  (
( 2  x.  n
)  +  1 ) ) )
7776mpteq2ia 4251 . . 3  |-  ( n  e.  NN  |->  ( ( 1  /  ( ( 2  x.  n )  +  1 ) )  x.  (  seq  1
(  x.  ,  ( k  e.  NN  |->  ( ( ( 2  x.  k ) ^ 4 )  /  ( ( ( 2  x.  k
)  x.  ( ( 2  x.  k )  -  1 ) ) ^ 2 ) ) ) ) `  n
) ) )  =  ( n  e.  NN  |->  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `
 n ) ^
4 ) )  / 
( ( ! `  ( 2  x.  n
) ) ^ 2 ) )  /  (
( 2  x.  n
)  +  1 ) ) )
78 wallispi2lem1 27687 . . . 4  |-  ( n  e.  NN  ->  (  seq  1 (  x.  , 
( k  e.  NN  |->  ( ( ( 2  x.  k )  / 
( ( 2  x.  k )  -  1 ) )  x.  (
( 2  x.  k
)  /  ( ( 2  x.  k )  +  1 ) ) ) ) ) `  n )  =  ( ( 1  /  (
( 2  x.  n
)  +  1 ) )  x.  (  seq  1 (  x.  , 
( k  e.  NN  |->  ( ( ( 2  x.  k ) ^
4 )  /  (
( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) ) ^ 2 ) ) ) ) `  n ) ) )
7978mpteq2ia 4251 . . 3  |-  ( n  e.  NN  |->  (  seq  1 (  x.  , 
( k  e.  NN  |->  ( ( ( 2  x.  k )  / 
( ( 2  x.  k )  -  1 ) )  x.  (
( 2  x.  k
)  /  ( ( 2  x.  k )  +  1 ) ) ) ) ) `  n ) )  =  ( n  e.  NN  |->  ( ( 1  / 
( ( 2  x.  n )  +  1 ) )  x.  (  seq  1 (  x.  , 
( k  e.  NN  |->  ( ( ( 2  x.  k ) ^
4 )  /  (
( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) ) ^ 2 ) ) ) ) `  n ) ) )
80 wallispi2.1 . . 3  |-  V  =  ( n  e.  NN  |->  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `
 n ) ^
4 ) )  / 
( ( ! `  ( 2  x.  n
) ) ^ 2 ) )  /  (
( 2  x.  n
)  +  1 ) ) )
8177, 79, 803eqtr4ri 2435 . 2  |-  V  =  ( n  e.  NN  |->  (  seq  1 (  x.  ,  ( k  e.  NN  |->  ( ( ( 2  x.  k )  /  ( ( 2  x.  k )  - 
1 ) )  x.  ( ( 2  x.  k )  /  (
( 2  x.  k
)  +  1 ) ) ) ) ) `
 n ) )
821, 81wallispi 27686 1  |-  V  ~~>  ( pi 
/  2 )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   NNcn 9956   2c2 10005   4c4 10007   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278   ^cexp 11337   !cfa 11521    ~~> cli 12233   picpi 12624
This theorem is referenced by:  stirlinglem15  27704
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-cc 8271  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-disj 4143  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-ofr 6265  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-omul 6688  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-acn 7785  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-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-cmp 17404  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-ovol 19314  df-vol 19315  df-mbf 19465  df-itg1 19466  df-itg2 19467  df-ibl 19468  df-itg 19469  df-0p 19515  df-limc 19706  df-dv 19707
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