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Theorem wallispi2 37504
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 2429 . 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 9658 . . . . . 6  |-  ( n  e.  NN  ->  1  e.  CC )
3 2cnd 10682 . . . . . . . 8  |-  ( n  e.  NN  ->  2  e.  CC )
4 nncn 10617 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  CC )
53, 4mulcld 9662 . . . . . . 7  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  CC )
65, 2addcld 9661 . . . . . 6  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  e.  CC )
7 elnnuz 11195 . . . . . . . 8  |-  ( n  e.  NN  <->  n  e.  ( ZZ>= `  1 )
)
87biimpi 197 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
9 eqidd 2430 . . . . . . . . . 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 462 . . . . . . . . . . . . 13  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  k  =  m )
1110oveq2d 6321 . . . . . . . . . . . 12  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( 2  x.  k
)  =  ( 2  x.  m ) )
1211oveq1d 6320 . . . . . . . . . . 11  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k ) ^ 4 )  =  ( ( 2  x.  m ) ^ 4 ) )
1311oveq1d 6320 . . . . . . . . . . . . 13  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k )  -  1 )  =  ( ( 2  x.  m )  -  1 ) )
1411, 13oveq12d 6323 . . . . . . . . . . . 12  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) )  =  ( ( 2  x.  m )  x.  ( ( 2  x.  m )  - 
1 ) ) )
1514oveq1d 6320 . . . . . . . . . . 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 6323 . . . . . . . . . 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 11826 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... n )  ->  m  e.  NN )
18 2cnd 10682 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  2  e.  CC )
1917nncnd 10625 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  m  e.  CC )
2018, 19mulcld 9662 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  e.  CC )
21 4nn0 10888 . . . . . . . . . . . . 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 9658 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  1  e.  CC )
2520, 24subcld 9985 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  -  1 )  e.  CC )
2620, 25mulcld 9662 . . . . . . . . . . . 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 10702 . . . . . . . . . . . . . . 15  |-  2  =/=  0
2928a1i 11 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  2  =/=  0 )
3017nnne0d 10654 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  m  =/=  0 )
3118, 19, 29, 30mulne0d 10263 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  =/=  0 )
32 1red 9657 . . . . . . . . . . . . . . 15  |-  ( m  e.  ( 1 ... n )  ->  1  e.  RR )
33 2re 10679 . . . . . . . . . . . . . . . . . 18  |-  2  e.  RR
3433a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  2  e.  RR )
3534, 32remulcld 9670 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  1 )  e.  RR )
3617nnred 10624 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  m  e.  RR )
3734, 36remulcld 9670 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  e.  RR )
38 1lt2 10776 . . . . . . . . . . . . . . . . . 18  |-  1  <  2
3938a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  1  <  2 )
40 2t1e2 10758 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  1 )  =  2
4139, 40syl6breqr 4466 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  1  <  ( 2  x.  1 ) )
42 0le2 10700 . . . . . . . . . . . . . . . . . 18  |-  0  <_  2
4342a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  0  <_  2 )
44 elfzle1 11800 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  1  <_  m )
4532, 36, 34, 43, 44lemul2ad 10547 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  1 )  <_  ( 2  x.  m ) )
4632, 35, 37, 41, 45ltletrd 9794 . . . . . . . . . . . . . . 15  |-  ( m  e.  ( 1 ... n )  ->  1  <  ( 2  x.  m
) )
4732, 46gtned 9769 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  =/=  1 )
4820, 24, 47subne0d 9994 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  -  1 )  =/=  0 )
4920, 25, 31, 48mulne0d 10263 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) )  =/=  0 )
50 2z 10969 . . . . . . . . . . . . 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 10382 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... n )  ->  (
( ( 2  x.  m ) ^ 4 )  /  ( ( ( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) ) ^ 2 ) )  e.  CC )
549, 16, 17, 53fvmptd 5970 . . . . . . . . 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 2517 . . . . . . . 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 467 . . . . . . 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 9622 . . . . . . . 8  |-  ( ( m  e.  CC  /\  w  e.  CC )  ->  ( m  x.  w
)  e.  CC )
5857adantl 467 . . . . . . 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 10767 . . . . . . . . . 10  |-  2  e.  NN
6160a1i 11 . . . . . . . . 9  |-  ( n  e.  NN  ->  2  e.  NN )
62 id 23 . . . . . . . . 9  |-  ( n  e.  NN  ->  n  e.  NN )
6361, 62nnmulcld 10657 . . . . . . . 8  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  NN )
6463peano2nnd 10626 . . . . . . 7  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  e.  NN )
6564nnne0d 10654 . . . . . 6  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  =/=  0 )
662, 6, 59, 65div32d 10405 . . . . 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 10382 . . . . . 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 9660 . . . . 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 37503 . . . . . 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 6320 . . . . 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 2474 . . . 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 4508 . . 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 37502 . . . 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 4508 . . 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 2469 . 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 37501 1  |-  V  ~~>  ( pi 
/  2 )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426    |-> cmpt 4484   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541    x. cmul 9543    < clt 9674    <_ cle 9675    - cmin 9859    / cdiv 10268   NNcn 10609   2c2 10659   4c4 10661   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11782    seqcseq 12210   ^cexp 12269   !cfa 12456    ~~> cli 13526   picpi 14097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cc 8863  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-ofr 6546  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-omul 7195  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-acn 8375  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-sum 13731  df-ef 14099  df-sin 14101  df-cos 14102  df-pi 14104  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lp 20083  df-perf 20084  df-cn 20174  df-cnp 20175  df-haus 20262  df-cmp 20333  df-tx 20508  df-hmeo 20701  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-tms 21268  df-cncf 21806  df-ovol 22296  df-vol 22297  df-mbf 22454  df-itg1 22455  df-itg2 22456  df-ibl 22457  df-itg 22458  df-0p 22505  df-limc 22698  df-dv 22699
This theorem is referenced by:  stirlinglem15  37519
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