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Theorem wallispi2 37929
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 2450 . 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 1cnd 9656 . . . . . 6  |-  ( n  e.  NN  ->  1  e.  CC )
3 2cnd 10679 . . . . . . . 8  |-  ( n  e.  NN  ->  2  e.  CC )
4 nncn 10614 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  CC )
53, 4mulcld 9660 . . . . . . 7  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  CC )
65, 2addcld 9659 . . . . . 6  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  e.  CC )
7 elnnuz 11192 . . . . . . . 8  |-  ( n  e.  NN  <->  n  e.  ( ZZ>= `  1 )
)
87biimpi 198 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
9 eqidd 2451 . . . . . . . . . 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 ) ) ) )
10 simpr 463 . . . . . . . . . . . . 13  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  k  =  m )
1110oveq2d 6304 . . . . . . . . . . . 12  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( 2  x.  k
)  =  ( 2  x.  m ) )
1211oveq1d 6303 . . . . . . . . . . 11  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k ) ^ 4 )  =  ( ( 2  x.  m ) ^ 4 ) )
1311oveq1d 6303 . . . . . . . . . . . . 13  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k )  -  1 )  =  ( ( 2  x.  m )  -  1 ) )
1411, 13oveq12d 6306 . . . . . . . . . . . 12  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) )  =  ( ( 2  x.  m )  x.  ( ( 2  x.  m )  - 
1 ) ) )
1514oveq1d 6303 . . . . . . . . . . 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 ) )
1612, 15oveq12d 6306 . . . . . . . . . 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 ) ) )
17 elfznn 11825 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... n )  ->  m  e.  NN )
18 2cnd 10679 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  2  e.  CC )
1917nncnd 10622 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  m  e.  CC )
2018, 19mulcld 9660 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  e.  CC )
21 4nn0 10885 . . . . . . . . . . . . 13  |-  4  e.  NN0
2221a1i 11 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  4  e.  NN0 )
2320, 22expcld 12413 . . . . . . . . . . 11  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
) ^ 4 )  e.  CC )
24 1cnd 9656 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  1  e.  CC )
2520, 24subcld 9983 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  -  1 )  e.  CC )
2620, 25mulcld 9660 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) )  e.  CC )
2726sqcld 12411 . . . . . . . . . . 11  |-  ( m  e.  ( 1 ... n )  ->  (
( ( 2  x.  m )  x.  (
( 2  x.  m
)  -  1 ) ) ^ 2 )  e.  CC )
28 2ne0 10699 . . . . . . . . . . . . . . 15  |-  2  =/=  0
2928a1i 11 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  2  =/=  0 )
3017nnne0d 10651 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  m  =/=  0 )
3118, 19, 29, 30mulne0d 10261 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  =/=  0 )
32 1red 9655 . . . . . . . . . . . . . . 15  |-  ( m  e.  ( 1 ... n )  ->  1  e.  RR )
33 2re 10676 . . . . . . . . . . . . . . . . . 18  |-  2  e.  RR
3433a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  2  e.  RR )
3534, 32remulcld 9668 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  1 )  e.  RR )
3617nnred 10621 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  m  e.  RR )
3734, 36remulcld 9668 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  e.  RR )
38 1lt2 10773 . . . . . . . . . . . . . . . . . 18  |-  1  <  2
3938a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  1  <  2 )
40 2t1e2 10755 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  1 )  =  2
4139, 40syl6breqr 4442 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  1  <  ( 2  x.  1 ) )
42 0le2 10697 . . . . . . . . . . . . . . . . . 18  |-  0  <_  2
4342a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  0  <_  2 )
44 elfzle1 11799 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  1  <_  m )
4532, 36, 34, 43, 44lemul2ad 10544 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  1 )  <_  ( 2  x.  m ) )
4632, 35, 37, 41, 45ltletrd 9792 . . . . . . . . . . . . . . 15  |-  ( m  e.  ( 1 ... n )  ->  1  <  ( 2  x.  m
) )
4732, 46gtned 9767 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  =/=  1 )
4820, 24, 47subne0d 9992 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  -  1 )  =/=  0 )
4920, 25, 31, 48mulne0d 10261 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) )  =/=  0 )
50 2z 10966 . . . . . . . . . . . . 13  |-  2  e.  ZZ
5150a1i 11 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  2  e.  ZZ )
5226, 49, 51expne0d 12419 . . . . . . . . . . 11  |-  ( m  e.  ( 1 ... n )  ->  (
( ( 2  x.  m )  x.  (
( 2  x.  m
)  -  1 ) ) ^ 2 )  =/=  0 )
5323, 27, 52divcld 10380 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... n )  ->  (
( ( 2  x.  m ) ^ 4 )  /  ( ( ( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) ) ^ 2 ) )  e.  CC )
549, 16, 17, 53fvmptd 5952 . . . . . . . . 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 ) ) )
5554, 53eqeltrd 2528 . . . . . . . 8  |-  ( m  e.  ( 1 ... n )  ->  (
( k  e.  NN  |->  ( ( ( 2  x.  k ) ^
4 )  /  (
( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) ) ^ 2 ) ) ) `  m
)  e.  CC )
5655adantl 468 . . . . . . 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 )
57 mulcl 9620 . . . . . . . 8  |-  ( ( m  e.  CC  /\  w  e.  CC )  ->  ( m  x.  w
)  e.  CC )
5857adantl 468 . . . . . . 7  |-  ( ( n  e.  NN  /\  ( m  e.  CC  /\  w  e.  CC ) )  ->  ( m  x.  w )  e.  CC )
598, 56, 58seqcl 12230 . . . . . 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 )
60 2nn 10764 . . . . . . . . . 10  |-  2  e.  NN
6160a1i 11 . . . . . . . . 9  |-  ( n  e.  NN  ->  2  e.  NN )
62 id 22 . . . . . . . . 9  |-  ( n  e.  NN  ->  n  e.  NN )
6361, 62nnmulcld 10654 . . . . . . . 8  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  NN )
6463peano2nnd 10623 . . . . . . 7  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  e.  NN )
6564nnne0d 10651 . . . . . 6  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  =/=  0 )
662, 6, 59, 65div32d 10403 . . . . 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 ) ) ) )
6759, 6, 65divcld 10380 . . . . . 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 )
6867mulid2d 9658 . . . . 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 ) ) )
69 wallispi2lem2 37928 . . . . . 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 ) ) )
7069oveq1d 6303 . . . . 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 ) ) )
7166, 68, 703eqtrd 2488 . . . 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 ) ) )
7271mpteq2ia 4484 . . 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 ) ) )
73 wallispi2lem1 37927 . . . 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 ) ) )
7473mpteq2ia 4484 . . 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 ) ) )
75 wallispi2.1 . . 3  |-  V  =  ( n  e.  NN  |->  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `
 n ) ^
4 ) )  / 
( ( ! `  ( 2  x.  n
) ) ^ 2 ) )  /  (
( 2  x.  n
)  +  1 ) ) )
7672, 74, 753eqtr4ri 2483 . 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 ) )
771, 76wallispi 37926 1  |-  V  ~~>  ( pi 
/  2 )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621   class class class wbr 4401    |-> cmpt 4460   ` cfv 5581  (class class class)co 6288   CCcc 9534   RRcr 9535   0cc0 9536   1c1 9537    + caddc 9539    x. cmul 9541    < clt 9672    <_ cle 9673    - cmin 9857    / cdiv 10266   NNcn 10606   2c2 10656   4c4 10658   NN0cn0 10866   ZZcz 10934   ZZ>=cuz 11156   ...cfz 11781    seqcseq 12210   ^cexp 12269   !cfa 12456    ~~> cli 13541   picpi 14112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cc 8862  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-disj 4373  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-ofr 6529  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-omul 7184  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-acn 8373  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ioc 11637  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13123  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-nei 20107  df-lp 20145  df-perf 20146  df-cn 20236  df-cnp 20237  df-haus 20324  df-cmp 20395  df-tx 20570  df-hmeo 20763  df-fil 20854  df-fm 20946  df-flim 20947  df-flf 20948  df-xms 21328  df-ms 21329  df-tms 21330  df-cncf 21903  df-ovol 22409  df-vol 22411  df-mbf 22570  df-itg1 22571  df-itg2 22572  df-ibl 22573  df-itg 22574  df-0p 22621  df-limc 22814  df-dv 22815
This theorem is referenced by:  stirlinglem15  37944
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