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Metamath Proof Explorer

Theorem wallispilem2

Description: A first set of properties for the sequence I that will be used in the proof of the Wallis product formula. (Contributed by Glauco Siliprandi, 29-Jun-2017)

Ref Expression
Hypothesis wallispilem2.1 
|- I = ( n e. NN0 |-> S. ( 0 (,) _pi ) ( ( sin ` x ) ^ n ) _d x )
Assertion wallispilem2 
|- ( ( I ` 0 ) = _pi /\ ( I ` 1 ) = 2 /\ ( N e. ( ZZ>= ` 2 ) -> ( I ` N ) = ( ( ( N - 1 ) / N ) x. ( I ` ( N - 2 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 wallispilem2.1 
 |-  I = ( n e. NN0 |-> S. ( 0 (,) _pi ) ( ( sin ` x ) ^ n ) _d x )
2 0nn0 
 |-  0 e. NN0
3 oveq2 
 |-  ( n = 0 -> ( ( sin ` x ) ^ n ) = ( ( sin ` x ) ^ 0 ) )
4 3 adantr 
 |-  ( ( n = 0 /\ x e. ( 0 (,) _pi ) ) -> ( ( sin ` x ) ^ n ) = ( ( sin ` x ) ^ 0 ) )
5 ioosscn 
 |-  ( 0 (,) _pi ) C_ CC
6 5 sseli 
 |-  ( x e. ( 0 (,) _pi ) -> x e. CC )
7 6 sincld 
 |-  ( x e. ( 0 (,) _pi ) -> ( sin ` x ) e. CC )
8 7 adantl 
 |-  ( ( n = 0 /\ x e. ( 0 (,) _pi ) ) -> ( sin ` x ) e. CC )
9 8 exp0d 
 |-  ( ( n = 0 /\ x e. ( 0 (,) _pi ) ) -> ( ( sin ` x ) ^ 0 ) = 1 )
10 4 9 eqtrd 
 |-  ( ( n = 0 /\ x e. ( 0 (,) _pi ) ) -> ( ( sin ` x ) ^ n ) = 1 )
11 10 itgeq2dv 
 |-  ( n = 0 -> S. ( 0 (,) _pi ) ( ( sin ` x ) ^ n ) _d x = S. ( 0 (,) _pi ) 1 _d x )
12 ioombl 
 |-  ( 0 (,) _pi ) e. dom vol
13 0re 
 |-  0 e. RR
14 pire 
 |-  _pi e. RR
15 ioovolcl 
 |-  ( ( 0 e. RR /\ _pi e. RR ) -> ( vol ` ( 0 (,) _pi ) ) e. RR )
16 13 14 15 mp2an 
 |-  ( vol ` ( 0 (,) _pi ) ) e. RR
17 ax-1cn 
 |-  1 e. CC
18 itgconst 
 |-  ( ( ( 0 (,) _pi ) e. dom vol /\ ( vol ` ( 0 (,) _pi ) ) e. RR /\ 1 e. CC ) -> S. ( 0 (,) _pi ) 1 _d x = ( 1 x. ( vol ` ( 0 (,) _pi ) ) ) )
19 12 16 17 18 mp3an 
 |-  S. ( 0 (,) _pi ) 1 _d x = ( 1 x. ( vol ` ( 0 (,) _pi ) ) )
20 16 recni 
 |-  ( vol ` ( 0 (,) _pi ) ) e. CC
21 20 mullidi 
 |-  ( 1 x. ( vol ` ( 0 (,) _pi ) ) ) = ( vol ` ( 0 (,) _pi ) )
22 pipos 
 |-  0 < _pi
23 13 14 22 ltleii 
 |-  0 <_ _pi
24 volioo 
 |-  ( ( 0 e. RR /\ _pi e. RR /\ 0 <_ _pi ) -> ( vol ` ( 0 (,) _pi ) ) = ( _pi - 0 ) )
25 13 14 23 24 mp3an 
 |-  ( vol ` ( 0 (,) _pi ) ) = ( _pi - 0 )
26 14 recni 
 |-  _pi e. CC
27 26 subid1i 
 |-  ( _pi - 0 ) = _pi
28 25 27 eqtri 
 |-  ( vol ` ( 0 (,) _pi ) ) = _pi
29 21 28 eqtri 
 |-  ( 1 x. ( vol ` ( 0 (,) _pi ) ) ) = _pi
30 19 29 eqtri 
 |-  S. ( 0 (,) _pi ) 1 _d x = _pi
31 11 30 eqtrdi 
 |-  ( n = 0 -> S. ( 0 (,) _pi ) ( ( sin ` x ) ^ n ) _d x = _pi )
32 14 elexi 
 |-  _pi e. _V
33 31 1 32 fvmpt 
 |-  ( 0 e. NN0 -> ( I ` 0 ) = _pi )
34 2 33 ax-mp 
 |-  ( I ` 0 ) = _pi
35 1nn0 
 |-  1 e. NN0
36 simpl 
 |-  ( ( n = 1 /\ x e. ( 0 (,) _pi ) ) -> n = 1 )
37 36 oveq2d 
 |-  ( ( n = 1 /\ x e. ( 0 (,) _pi ) ) -> ( ( sin ` x ) ^ n ) = ( ( sin ` x ) ^ 1 ) )
38 7 adantl 
 |-  ( ( n = 1 /\ x e. ( 0 (,) _pi ) ) -> ( sin ` x ) e. CC )
39 38 exp1d 
 |-  ( ( n = 1 /\ x e. ( 0 (,) _pi ) ) -> ( ( sin ` x ) ^ 1 ) = ( sin ` x ) )
40 37 39 eqtrd 
 |-  ( ( n = 1 /\ x e. ( 0 (,) _pi ) ) -> ( ( sin ` x ) ^ n ) = ( sin ` x ) )
41 40 itgeq2dv 
 |-  ( n = 1 -> S. ( 0 (,) _pi ) ( ( sin ` x ) ^ n ) _d x = S. ( 0 (,) _pi ) ( sin ` x ) _d x )
42 itgex 
 |-  S. ( 0 (,) _pi ) ( sin ` x ) _d x e. _V
43 41 1 42 fvmpt 
 |-  ( 1 e. NN0 -> ( I ` 1 ) = S. ( 0 (,) _pi ) ( sin ` x ) _d x )
44 35 43 ax-mp 
 |-  ( I ` 1 ) = S. ( 0 (,) _pi ) ( sin ` x ) _d x
45 itgsin0pi 
 |-  S. ( 0 (,) _pi ) ( sin ` x ) _d x = 2
46 44 45 eqtri 
 |-  ( I ` 1 ) = 2
47 id 
 |-  ( N e. ( ZZ>= ` 2 ) -> N e. ( ZZ>= ` 2 ) )
48 1 47 itgsinexp 
 |-  ( N e. ( ZZ>= ` 2 ) -> ( I ` N ) = ( ( ( N - 1 ) / N ) x. ( I ` ( N - 2 ) ) ) )
49 34 46 48 3pm3.2i 
 |-  ( ( I ` 0 ) = _pi /\ ( I ` 1 ) = 2 /\ ( N e. ( ZZ>= ` 2 ) -> ( I ` N ) = ( ( ( N - 1 ) / N ) x. ( I ` ( N - 2 ) ) ) ) )