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Theorem wallispi2 31401
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 2467 . 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 9550 . . . . . . 7  |-  1  e.  CC
32a1i 11 . . . . . 6  |-  ( n  e.  NN  ->  1  e.  CC )
4 2cnd 10608 . . . . . . . 8  |-  ( n  e.  NN  ->  2  e.  CC )
5 nncn 10544 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  CC )
64, 5mulcld 9616 . . . . . . 7  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  CC )
76, 3addcld 9615 . . . . . 6  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  e.  CC )
8 elnnuz 11118 . . . . . . . 8  |-  ( n  e.  NN  <->  n  e.  ( ZZ>= `  1 )
)
98biimpi 194 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
10 eqidd 2468 . . . . . . . . . 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 6300 . . . . . . . . . . . 12  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( 2  x.  k
)  =  ( 2  x.  m ) )
1312oveq1d 6299 . . . . . . . . . . 11  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k ) ^ 4 )  =  ( ( 2  x.  m ) ^ 4 ) )
1412oveq1d 6299 . . . . . . . . . . . . 13  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k )  -  1 )  =  ( ( 2  x.  m )  -  1 ) )
1512, 14oveq12d 6302 . . . . . . . . . . . 12  |-  ( ( m  e.  ( 1 ... n )  /\  k  =  m )  ->  ( ( 2  x.  k )  x.  (
( 2  x.  k
)  -  1 ) )  =  ( ( 2  x.  m )  x.  ( ( 2  x.  m )  - 
1 ) ) )
1615oveq1d 6299 . . . . . . . . . . 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 6302 . . . . . . . . . 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 11714 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... n )  ->  m  e.  NN )
19 2cnd 10608 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  2  e.  CC )
2018nncnd 10552 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  m  e.  CC )
2119, 20mulcld 9616 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  e.  CC )
22 4nn0 10814 . . . . . . . . . . . . 13  |-  4  e.  NN0
2322a1i 11 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  4  e.  NN0 )
2421, 23expcld 12278 . . . . . . . . . . 11  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
) ^ 4 )  e.  CC )
252a1i 11 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  1  e.  CC )
2621, 25subcld 9930 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  -  1 )  e.  CC )
2721, 26mulcld 9616 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) )  e.  CC )
2827sqcld 12276 . . . . . . . . . . 11  |-  ( m  e.  ( 1 ... n )  ->  (
( ( 2  x.  m )  x.  (
( 2  x.  m
)  -  1 ) ) ^ 2 )  e.  CC )
29 2ne0 10628 . . . . . . . . . . . . . . 15  |-  2  =/=  0
3029a1i 11 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  2  =/=  0 )
3118nnne0d 10580 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  m  =/=  0 )
3219, 20, 30, 31mulne0d 10201 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  =/=  0 )
33 1re 9595 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
3433a1i 11 . . . . . . . . . . . . . . 15  |-  ( m  e.  ( 1 ... n )  ->  1  e.  RR )
35 2re 10605 . . . . . . . . . . . . . . . . . 18  |-  2  e.  RR
3635a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  2  e.  RR )
3736, 34remulcld 9624 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  1 )  e.  RR )
3818nnred 10551 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  m  e.  RR )
3936, 38remulcld 9624 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  e.  RR )
40 1lt2 10702 . . . . . . . . . . . . . . . . . 18  |-  1  <  2
4140a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  1  <  2 )
42 2cn 10606 . . . . . . . . . . . . . . . . . 18  |-  2  e.  CC
4342mulid1i 9598 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  1 )  =  2
4441, 43syl6breqr 4487 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  1  <  ( 2  x.  1 ) )
45 0le2 10626 . . . . . . . . . . . . . . . . . 18  |-  0  <_  2
4645a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  0  <_  2 )
47 elfzle1 11689 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 1 ... n )  ->  1  <_  m )
4834, 38, 36, 46, 47lemul2ad 10486 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  1 )  <_  ( 2  x.  m ) )
4934, 37, 39, 44, 48ltletrd 9741 . . . . . . . . . . . . . . 15  |-  ( m  e.  ( 1 ... n )  ->  1  <  ( 2  x.  m
) )
5034, 49gtned 9719 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... n )  ->  (
2  x.  m )  =/=  1 )
5121, 25, 50subne0d 9939 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  -  1 )  =/=  0 )
5221, 26, 32, 51mulne0d 10201 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  (
( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) )  =/=  0 )
53 2z 10896 . . . . . . . . . . . . 13  |-  2  e.  ZZ
5453a1i 11 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... n )  ->  2  e.  ZZ )
5527, 52, 54expne0d 12284 . . . . . . . . . . 11  |-  ( m  e.  ( 1 ... n )  ->  (
( ( 2  x.  m )  x.  (
( 2  x.  m
)  -  1 ) ) ^ 2 )  =/=  0 )
5624, 28, 55divcld 10320 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... n )  ->  (
( ( 2  x.  m ) ^ 4 )  /  ( ( ( 2  x.  m
)  x.  ( ( 2  x.  m )  -  1 ) ) ^ 2 ) )  e.  CC )
5710, 17, 18, 56fvmptd 5955 . . . . . . . . 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 2555 . . . . . . . 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 9576 . . . . . . . 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 12095 . . . . . 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 10693 . . . . . . . . . 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 10583 . . . . . . . 8  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  NN )
6766peano2nnd 10553 . . . . . . 7  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  e.  NN )
6867nnne0d 10580 . . . . . 6  |-  ( n  e.  NN  ->  (
( 2  x.  n
)  +  1 )  =/=  0 )
693, 7, 62, 68div32d 10343 . . . . 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 10320 . . . . . 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 9614 . . . . 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 31400 . . . . . 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 6299 . . . . 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 2512 . . . 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 4529 . . 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 31399 . . . 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 4529 . . 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 2507 . 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 31398 1  |-  V  ~~>  ( pi 
/  2 )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447    |-> cmpt 4505   ` cfv 5588  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    < clt 9628    <_ cle 9629    - cmin 9805    / cdiv 10206   NNcn 10536   2c2 10585   4c4 10587   NN0cn0 10795   ZZcz 10864   ZZ>=cuz 11082   ...cfz 11672    seqcseq 12075   ^cexp 12134   !cfa 12321    ~~> cli 13270   picpi 13664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cc 8815  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-ofr 6525  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-acn 8323  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-sum 13472  df-ef 13665  df-sin 13667  df-cos 13668  df-pi 13670  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-haus 19610  df-cmp 19681  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-xms 20586  df-ms 20587  df-tms 20588  df-cncf 21145  df-ovol 21639  df-vol 21640  df-mbf 21791  df-itg1 21792  df-itg2 21793  df-ibl 21794  df-itg 21795  df-0p 21840  df-limc 22033  df-dv 22034
This theorem is referenced by:  stirlinglem15  31416
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