Metamath Proof Explorer

Theorem knoppndvlem9

Description: Lemma for knoppndv . (Contributed by Asger C. Ipsen, 15-Jun-2021) (Revised by Asger C. Ipsen, 5-Jul-2021)

Ref Expression
Hypotheses knoppndvlem9.t 
|- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
knoppndvlem9.f 
|- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
knoppndvlem9.a 
|- A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M )
knoppndvlem9.c 
|- ( ph -> C e. ( -u 1 (,) 1 ) )
knoppndvlem9.j 
|- ( ph -> J e. NN0 )
knoppndvlem9.m 
|- ( ph -> M e. ZZ )
knoppndvlem9.n 
|- ( ph -> N e. NN )
knoppndvlem9.1 
|- ( ph -> -. 2 || M )
Assertion knoppndvlem9 
|- ( ph -> ( ( F ` A ) ` J ) = ( ( C ^ J ) / 2 ) )

Proof

Step Hyp Ref Expression
1 knoppndvlem9.t 
 |-  T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
2 knoppndvlem9.f 
 |-  F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
3 knoppndvlem9.a 
 |-  A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M )
4 knoppndvlem9.c 
 |-  ( ph -> C e. ( -u 1 (,) 1 ) )
5 knoppndvlem9.j 
 |-  ( ph -> J e. NN0 )
6 knoppndvlem9.m 
 |-  ( ph -> M e. ZZ )
7 knoppndvlem9.n 
 |-  ( ph -> N e. NN )
8 knoppndvlem9.1 
 |-  ( ph -> -. 2 || M )
9 1 2 3 5 6 7 knoppndvlem7 
 |-  ( ph -> ( ( F ` A ) ` J ) = ( ( C ^ J ) x. ( T ` ( M / 2 ) ) ) )
10 odd2np1 
 |-  ( M e. ZZ -> ( -. 2 || M <-> E. m e. ZZ ( ( 2 x. m ) + 1 ) = M ) )
11 6 10 syl 
 |-  ( ph -> ( -. 2 || M <-> E. m e. ZZ ( ( 2 x. m ) + 1 ) = M ) )
12 8 11 mpbid 
 |-  ( ph -> E. m e. ZZ ( ( 2 x. m ) + 1 ) = M )
13 eqcom 
 |-  ( ( ( 2 x. m ) + 1 ) = M <-> M = ( ( 2 x. m ) + 1 ) )
14 13 biimpi 
 |-  ( ( ( 2 x. m ) + 1 ) = M -> M = ( ( 2 x. m ) + 1 ) )
15 14 oveq1d 
 |-  ( ( ( 2 x. m ) + 1 ) = M -> ( M / 2 ) = ( ( ( 2 x. m ) + 1 ) / 2 ) )
16 15 adantl 
 |-  ( ( m e. ZZ /\ ( ( 2 x. m ) + 1 ) = M ) -> ( M / 2 ) = ( ( ( 2 x. m ) + 1 ) / 2 ) )
17 16 adantl 
 |-  ( ( ph /\ ( m e. ZZ /\ ( ( 2 x. m ) + 1 ) = M ) ) -> ( M / 2 ) = ( ( ( 2 x. m ) + 1 ) / 2 ) )
18 2cnd 
 |-  ( ( ph /\ m e. ZZ ) -> 2 e. CC )
19 zcn 
 |-  ( m e. ZZ -> m e. CC )
20 19 adantl 
 |-  ( ( ph /\ m e. ZZ ) -> m e. CC )
21 18 20 mulcld 
 |-  ( ( ph /\ m e. ZZ ) -> ( 2 x. m ) e. CC )
22 1cnd 
 |-  ( ( ph /\ m e. ZZ ) -> 1 e. CC )
23 2ne0 
 |-  2 =/= 0
24 23 a1i 
 |-  ( ( ph /\ m e. ZZ ) -> 2 =/= 0 )
25 21 22 18 24 divdird 
 |-  ( ( ph /\ m e. ZZ ) -> ( ( ( 2 x. m ) + 1 ) / 2 ) = ( ( ( 2 x. m ) / 2 ) + ( 1 / 2 ) ) )
26 20 18 24 divcan3d 
 |-  ( ( ph /\ m e. ZZ ) -> ( ( 2 x. m ) / 2 ) = m )
27 26 oveq1d 
 |-  ( ( ph /\ m e. ZZ ) -> ( ( ( 2 x. m ) / 2 ) + ( 1 / 2 ) ) = ( m + ( 1 / 2 ) ) )
28 25 27 eqtrd 
 |-  ( ( ph /\ m e. ZZ ) -> ( ( ( 2 x. m ) + 1 ) / 2 ) = ( m + ( 1 / 2 ) ) )
29 28 adantrr 
 |-  ( ( ph /\ ( m e. ZZ /\ ( ( 2 x. m ) + 1 ) = M ) ) -> ( ( ( 2 x. m ) + 1 ) / 2 ) = ( m + ( 1 / 2 ) ) )
30 17 29 eqtrd 
 |-  ( ( ph /\ ( m e. ZZ /\ ( ( 2 x. m ) + 1 ) = M ) ) -> ( M / 2 ) = ( m + ( 1 / 2 ) ) )
31 30 fveq2d 
 |-  ( ( ph /\ ( m e. ZZ /\ ( ( 2 x. m ) + 1 ) = M ) ) -> ( T ` ( M / 2 ) ) = ( T ` ( m + ( 1 / 2 ) ) ) )
32 id 
 |-  ( m e. ZZ -> m e. ZZ )
33 1 32 dnizphlfeqhlf 
 |-  ( m e. ZZ -> ( T ` ( m + ( 1 / 2 ) ) ) = ( 1 / 2 ) )
34 33 adantl 
 |-  ( ( ph /\ m e. ZZ ) -> ( T ` ( m + ( 1 / 2 ) ) ) = ( 1 / 2 ) )
35 34 adantrr 
 |-  ( ( ph /\ ( m e. ZZ /\ ( ( 2 x. m ) + 1 ) = M ) ) -> ( T ` ( m + ( 1 / 2 ) ) ) = ( 1 / 2 ) )
36 31 35 eqtrd 
 |-  ( ( ph /\ ( m e. ZZ /\ ( ( 2 x. m ) + 1 ) = M ) ) -> ( T ` ( M / 2 ) ) = ( 1 / 2 ) )
37 12 36 rexlimddv 
 |-  ( ph -> ( T ` ( M / 2 ) ) = ( 1 / 2 ) )
38 37 oveq2d 
 |-  ( ph -> ( ( C ^ J ) x. ( T ` ( M / 2 ) ) ) = ( ( C ^ J ) x. ( 1 / 2 ) ) )
39 4 knoppndvlem3 
 |-  ( ph -> ( C e. RR /\ ( abs ` C ) < 1 ) )
40 39 simpld 
 |-  ( ph -> C e. RR )
41 40 recnd 
 |-  ( ph -> C e. CC )
42 41 5 expcld 
 |-  ( ph -> ( C ^ J ) e. CC )
43 1cnd 
 |-  ( ph -> 1 e. CC )
44 2cnd 
 |-  ( ph -> 2 e. CC )
45 23 a1i 
 |-  ( ph -> 2 =/= 0 )
46 42 43 44 45 div12d 
 |-  ( ph -> ( ( C ^ J ) x. ( 1 / 2 ) ) = ( 1 x. ( ( C ^ J ) / 2 ) ) )
47 42 44 45 divcld 
 |-  ( ph -> ( ( C ^ J ) / 2 ) e. CC )
48 47 mulid2d 
 |-  ( ph -> ( 1 x. ( ( C ^ J ) / 2 ) ) = ( ( C ^ J ) / 2 ) )
49 46 48 eqtrd 
 |-  ( ph -> ( ( C ^ J ) x. ( 1 / 2 ) ) = ( ( C ^ J ) / 2 ) )
50 9 38 49 3eqtrd 
 |-  ( ph -> ( ( F ` A ) ` J ) = ( ( C ^ J ) / 2 ) )