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Theorem wallispi2 29866
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 2442 . 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 9339 . . . . . . 7  |-  1  e.  CC
32a1i 11 . . . . . 6  |-  ( n  e.  NN  ->  1  e.  CC )
4 2cnd 10393 . . . . . . . 8  |-  ( n  e.  NN  ->  2  e.  CC )
5 nncn 10329 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  CC )
64, 5mulcld 9405 . . . . . . 7  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  CC )
76, 3addcld 9404 . . . . . 6  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  e.  CC )
8 elnnuz 10896 . . . . . . . 8  |-  ( n  e.  NN  <->  n  e.  ( ZZ>= `  1 )
)
98biimpi 194 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
10 eqidd 2443 . . . . . . . . . 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 ) ) ) )
11 simpr 461 . . . . . . . . . . . . 13  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  k  =  m )
1211oveq2d 6106 . . . . . . . . . . . 12  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( 2  x.  k
)  =  ( 2  x.  m ) )
1312oveq1d 6105 . . . . . . . . . . 11  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k ) ^ 4 )  =  ( ( 2  x.  m ) ^ 4 ) )
1412oveq1d 6105 . . . . . . . . . . . . 13  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k )  -  1 )  =  ( ( 2  x.  m )  -  1 ) )
1512, 14oveq12d 6108 . . . . . . . . . . . 12  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) )  =  ( ( 2  x.  m )  x.  ( ( 2  x.  m )  - 
1 ) ) )
1615oveq1d 6105 . . . . . . . . . . 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 ) )
1713, 16oveq12d 6108 . . . . . . . . . 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 ) ) )
18 elfznn 11477 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... n )  ->  m  e.  NN )
19 2cnd 10393 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  2  e.  CC )
2018nncnd 10337 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  m  e.  CC )
2119, 20mulcld 9405 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  e.  CC )
22 4nn0 10597 . . . . . . . . . . . . 13  |-  4  e.  NN0
2322a1i 11 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  4  e.  NN0 )
2421, 23expcld 12007 . . . . . . . . . . 11  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
) ^ 4 )  e.  CC )
252a1i 11 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  1  e.  CC )
2621, 25subcld 9718 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  -  1 )  e.  CC )
2721, 26mulcld 9405 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) )  e.  CC )
2827sqcld 12005 . . . . . . . . . . 11  |-  ( m  e.  ( 1 ... n )  ->  (
( ( 2  x.  m )  x.  (
( 2  x.  m
)  -  1 ) ) ^ 2 )  e.  CC )
29 2ne0 10413 . . . . . . . . . . . . . . 15  |-  2  =/=  0
3029a1i 11 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  2  =/=  0 )
3118nnne0d 10365 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  m  =/=  0 )
3219, 20, 30, 31mulne0d 9987 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  =/=  0 )
33 1re 9384 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
3433a1i 11 . . . . . . . . . . . . . . 15  |-  ( m  e.  ( 1 ... n )  ->  1  e.  RR )
35 2re 10390 . . . . . . . . . . . . . . . . . 18  |-  2  e.  RR
3635a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  2  e.  RR )
3736, 34remulcld 9413 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  1 )  e.  RR )
3818nnred 10336 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  m  e.  RR )
3936, 38remulcld 9413 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  e.  RR )
40 1lt2 10487 . . . . . . . . . . . . . . . . . 18  |-  1  <  2
4140a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  1  <  2 )
42 2cn 10391 . . . . . . . . . . . . . . . . . 18  |-  2  e.  CC
4342mulid1i 9387 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  1 )  =  2
4441, 43syl6breqr 4331 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  1  <  ( 2  x.  1 ) )
45 0le2 10411 . . . . . . . . . . . . . . . . . 18  |-  0  <_  2
4645a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  0  <_  2 )
47 elfzle1 11453 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  1  <_  m )
4834, 38, 36, 46, 47lemul2ad 10272 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  1 )  <_  ( 2  x.  m ) )
4934, 37, 39, 44, 48ltletrd 9530 . . . . . . . . . . . . . . 15  |-  ( m  e.  ( 1 ... n )  ->  1  <  ( 2  x.  m
) )
5034, 49gtned 9508 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  =/=  1 )
5121, 25, 50subne0d 9727 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  -  1 )  =/=  0 )
5221, 26, 32, 51mulne0d 9987 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) )  =/=  0 )
53 2z 10677 . . . . . . . . . . . . 13  |-  2  e.  ZZ
5453a1i 11 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  2  e.  ZZ )
5527, 52, 54expne0d 12013 . . . . . . . . . . 11  |-  ( m  e.  ( 1 ... n )  ->  (
( ( 2  x.  m )  x.  (
( 2  x.  m
)  -  1 ) ) ^ 2 )  =/=  0 )
5624, 28, 55divcld 10106 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... n )  ->  (
( ( 2  x.  m ) ^ 4 )  /  ( ( ( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) ) ^ 2 ) )  e.  CC )
5710, 17, 18, 56fvmptd 5778 . . . . . . . . 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 ) ) )
5857, 56eqeltrd 2516 . . . . . . . 8  |-  ( m  e.  ( 1 ... n )  ->  (
( k  e.  NN  |->  ( ( ( 2  x.  k ) ^
4 )  /  (
( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) ) ^ 2 ) ) ) `  m
)  e.  CC )
5958adantl 466 . . . . . . 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 )
60 mulcl 9365 . . . . . . . 8  |-  ( ( m  e.  CC  /\  w  e.  CC )  ->  ( m  x.  w
)  e.  CC )
6160adantl 466 . . . . . . 7  |-  ( ( n  e.  NN  /\  ( m  e.  CC  /\  w  e.  CC ) )  ->  ( m  x.  w )  e.  CC )
629, 59, 61seqcl 11825 . . . . . 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 )
63 2nn 10478 . . . . . . . . . 10  |-  2  e.  NN
6463a1i 11 . . . . . . . . 9  |-  ( n  e.  NN  ->  2  e.  NN )
65 id 22 . . . . . . . . 9  |-  ( n  e.  NN  ->  n  e.  NN )
6664, 65nnmulcld 10368 . . . . . . . 8  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  NN )
6766peano2nnd 10338 . . . . . . 7  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  e.  NN )
6867nnne0d 10365 . . . . . 6  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  =/=  0 )
693, 7, 62, 68div32d 10129 . . . . 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 ) ) ) )
7062, 7, 68divcld 10106 . . . . . 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 )
7170mulid2d 9403 . . . . 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 ) ) )
72 wallispi2lem2 29865 . . . . . 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 ) ) )
7372oveq1d 6105 . . . . 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 ) ) )
7469, 71, 733eqtrd 2478 . . . 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 ) ) )
7574mpteq2ia 4373 . . 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 ) ) )
76 wallispi2lem1 29864 . . . 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 ) ) )
7776mpteq2ia 4373 . . 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 ) ) )
78 wallispi2.1 . . 3  |-  V  =  ( n  e.  NN  |->  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `
 n ) ^
4 ) )  / 
( ( ! `  ( 2  x.  n
) ) ^ 2 ) )  /  (
( 2  x.  n
)  +  1 ) ) )
7975, 77, 783eqtr4ri 2473 . 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 ) )
801, 79wallispi 29863 1  |-  V  ~~>  ( pi 
/  2 )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605   class class class wbr 4291    e. cmpt 4349   ` cfv 5417  (class class class)co 6090   CCcc 9279   RRcr 9280   0cc0 9281   1c1 9282    + caddc 9284    x. cmul 9286    < clt 9417    <_ cle 9418    - cmin 9594    / cdiv 9992   NNcn 10321   2c2 10370   4c4 10372   NN0cn0 10578   ZZcz 10645   ZZ>=cuz 10860   ...cfz 11436    seqcseq 11805   ^cexp 11864   !cfa 12050    ~~> cli 12961   picpi 13351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cc 8603  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-disj 4262  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-ofr 6320  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-omul 6924  df-er 7100  df-map 7215  df-pm 7216  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-fi 7660  df-sup 7690  df-oi 7723  df-card 8108  df-acn 8111  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-q 10953  df-rp 10991  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-ioo 11303  df-ioc 11304  df-ico 11305  df-icc 11306  df-fz 11437  df-fzo 11548  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-fac 12051  df-bc 12078  df-hash 12103  df-shft 12555  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-limsup 12948  df-clim 12965  df-rlim 12966  df-sum 13163  df-ef 13352  df-sin 13354  df-cos 13355  df-pi 13357  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-sca 14253  df-vsca 14254  df-ip 14255  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-hom 14261  df-cco 14262  df-rest 14360  df-topn 14361  df-0g 14379  df-gsum 14380  df-topgen 14381  df-pt 14382  df-prds 14385  df-xrs 14439  df-qtop 14444  df-imas 14445  df-xps 14447  df-mre 14523  df-mrc 14524  df-acs 14526  df-mnd 15414  df-submnd 15464  df-mulg 15547  df-cntz 15834  df-cmn 16278  df-psmet 17808  df-xmet 17809  df-met 17810  df-bl 17811  df-mopn 17812  df-fbas 17813  df-fg 17814  df-cnfld 17818  df-top 18502  df-bases 18504  df-topon 18505  df-topsp 18506  df-cld 18622  df-ntr 18623  df-cls 18624  df-nei 18701  df-lp 18739  df-perf 18740  df-cn 18830  df-cnp 18831  df-haus 18918  df-cmp 18989  df-tx 19134  df-hmeo 19327  df-fil 19418  df-fm 19510  df-flim 19511  df-flf 19512  df-xms 19894  df-ms 19895  df-tms 19896  df-cncf 20453  df-ovol 20947  df-vol 20948  df-mbf 21098  df-itg1 21099  df-itg2 21100  df-ibl 21101  df-itg 21102  df-0p 21147  df-limc 21340  df-dv 21341
This theorem is referenced by:  stirlinglem15  29881
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