Theorem knoppndvlem9 32511

Description: Lemma for knoppndv 32525. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)

Hypotheses

Ref	Expression
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

Ref	Expression
knoppndvlem9	|- ( ph -> ( ( F ` A ) ` J ) = ( ( C ^ J ) / 2 ) )

Distinct variable groups:   A, n, y   C, n, y   n, J   n, N, y   T, n, y   ph, n, y

Allowed substitution hints:   ph( x)   A( x)   C( x)   T( x)   F( x, y, n)   J( x, y)   M( x, y, n)   N( x)

Proof of Theorem knoppndvlem9

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		knoppndvlem9.t	. . 3 |- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
2		knoppndvlem9.f	. . 3 |- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
3		knoppndvlem9.a	. . 3 |- A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M )
4		knoppndvlem9.j	. . 3 |- ( ph -> J e. NN0 )
5		knoppndvlem9.m	. . 3 |- ( ph -> M e. ZZ )
6		knoppndvlem9.n	. . 3 |- ( ph -> N e. NN )
7	1, 2, 3, 4, 5, 6	knoppndvlem7 32509	. 2 |- ( ph -> ( ( F ` A ) ` J ) = ( ( C ^ J ) x. ( T ` ( M / 2 ) ) ) )
8		knoppndvlem9.1	. . . . 5 |- ( ph -> -. 2 || M )
9		odd2np1 15065	. . . . . 6 |- ( M e. ZZ -> ( -. 2 || M <-> E. m e. ZZ ( ( 2 x. m ) + 1 ) = M ) )
10	5, 9	syl 17	. . . . 5 |- ( ph -> ( -. 2 || M <-> E. m e. ZZ ( ( 2 x. m ) + 1 ) = M ) )
11	8, 10	mpbid 222	. . . 4 |- ( ph -> E. m e. ZZ ( ( 2 x. m ) + 1 ) = M )
12		eqcom 2629	. . . . . . . . . . 11 |- ( ( ( 2 x. m ) + 1 ) = M <-> M = ( ( 2 x. m ) + 1 ) )
13	12	biimpi 206	. . . . . . . . . 10 |- ( ( ( 2 x. m ) + 1 ) = M -> M = ( ( 2 x. m ) + 1 ) )
14	13	oveq1d 6665	. . . . . . . . 9 |- ( ( ( 2 x. m ) + 1 ) = M -> ( M / 2 ) = ( ( ( 2 x. m ) + 1 ) / 2 ) )
15	14	adantl 482	. . . . . . . 8 |- ( ( m e. ZZ /\ ( ( 2 x. m ) + 1 ) = M ) -> ( M / 2 ) = ( ( ( 2 x. m ) + 1 ) / 2 ) )
16	15	adantl 482	. . . . . . 7 |- ( ( ph /\ ( m e. ZZ /\ ( ( 2 x. m ) + 1 ) = M ) ) -> ( M / 2 ) = ( ( ( 2 x. m ) + 1 ) / 2 ) )
17		2cnd 11093	. . . . . . . . . . 11 |- ( ( ph /\ m e. ZZ ) -> 2 e. CC )
18		zcn 11382	. . . . . . . . . . . 12 |- ( m e. ZZ -> m e. CC )
19	18	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ m e. ZZ ) -> m e. CC )
20	17, 19	mulcld 10060	. . . . . . . . . 10 |- ( ( ph /\ m e. ZZ ) -> ( 2 x. m ) e. CC )
21		1cnd 10056	. . . . . . . . . 10 |- ( ( ph /\ m e. ZZ ) -> 1 e. CC )
22		2ne0 11113	. . . . . . . . . . 11 |- 2 =/= 0
23	22	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ m e. ZZ ) -> 2 =/= 0 )
24	20, 21, 17, 23	divdird 10839	. . . . . . . . 9 |- ( ( ph /\ m e. ZZ ) -> ( ( ( 2 x. m ) + 1 ) / 2 ) = ( ( ( 2 x. m ) / 2 ) + ( 1 / 2 ) ) )
25	19, 17, 23	divcan3d 10806	. . . . . . . . . 10 |- ( ( ph /\ m e. ZZ ) -> ( ( 2 x. m ) / 2 ) = m )
26	25	oveq1d 6665	. . . . . . . . 9 |- ( ( ph /\ m e. ZZ ) -> ( ( ( 2 x. m ) / 2 ) + ( 1 / 2 ) ) = ( m + ( 1 / 2 ) ) )
27	24, 26	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ m e. ZZ ) -> ( ( ( 2 x. m ) + 1 ) / 2 ) = ( m + ( 1 / 2 ) ) )
28	27	adantrr 753	. . . . . . 7 |- ( ( ph /\ ( m e. ZZ /\ ( ( 2 x. m ) + 1 ) = M ) ) -> ( ( ( 2 x. m ) + 1 ) / 2 ) = ( m + ( 1 / 2 ) ) )
29	16, 28	eqtrd 2656	. . . . . 6 |- ( ( ph /\ ( m e. ZZ /\ ( ( 2 x. m ) + 1 ) = M ) ) -> ( M / 2 ) = ( m + ( 1 / 2 ) ) )
30	29	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( m e. ZZ /\ ( ( 2 x. m ) + 1 ) = M ) ) -> ( T ` ( M / 2 ) ) = ( T ` ( m + ( 1 / 2 ) ) ) )
31		id 22	. . . . . . . 8 |- ( m e. ZZ -> m e. ZZ )
32	1, 31	dnizphlfeqhlf 32466	. . . . . . 7 |- ( m e. ZZ -> ( T ` ( m + ( 1 / 2 ) ) ) = ( 1 / 2 ) )
33	32	adantl 482	. . . . . 6 |- ( ( ph /\ m e. ZZ ) -> ( T ` ( m + ( 1 / 2 ) ) ) = ( 1 / 2 ) )
34	33	adantrr 753	. . . . 5 |- ( ( ph /\ ( m e. ZZ /\ ( ( 2 x. m ) + 1 ) = M ) ) -> ( T ` ( m + ( 1 / 2 ) ) ) = ( 1 / 2 ) )
35	30, 34	eqtrd 2656	. . . 4 |- ( ( ph /\ ( m e. ZZ /\ ( ( 2 x. m ) + 1 ) = M ) ) -> ( T ` ( M / 2 ) ) = ( 1 / 2 ) )
36	11, 35	rexlimddv 3035	. . 3 |- ( ph -> ( T ` ( M / 2 ) ) = ( 1 / 2 ) )
37	36	oveq2d 6666	. 2 |- ( ph -> ( ( C ^ J ) x. ( T ` ( M / 2 ) ) ) = ( ( C ^ J ) x. ( 1 / 2 ) ) )
38		knoppndvlem9.c	. . . . . . . 8 |- ( ph -> C e. ( -u 1 (,) 1 ) )
39	38	knoppndvlem3 32505	. . . . . . 7 |- ( ph -> ( C e. RR /\ ( abs ` C ) < 1 ) )
40	39	simpld 475	. . . . . 6 |- ( ph -> C e. RR )
41	40	recnd 10068	. . . . 5 |- ( ph -> C e. CC )
42	41, 4	expcld 13008	. . . 4 |- ( ph -> ( C ^ J ) e. CC )
43		1cnd 10056	. . . 4 |- ( ph -> 1 e. CC )
44		2cnd 11093	. . . 4 |- ( ph -> 2 e. CC )
45	22	a1i 11	. . . 4 |- ( ph -> 2 =/= 0 )
46	42, 43, 44, 45	div12d 10837	. . 3 |- ( ph -> ( ( C ^ J ) x. ( 1 / 2 ) ) = ( 1 x. ( ( C ^ J ) / 2 ) ) )
47	42, 44, 45	divcld 10801	. . . 4 |- ( ph -> ( ( C ^ J ) / 2 ) e. CC )
48	47	mulid2d 10058	. . 3 |- ( ph -> ( 1 x. ( ( C ^ J ) / 2 ) ) = ( ( C ^ J ) / 2 ) )
49	46, 48	eqtrd 2656	. 2 |- ( ph -> ( ( C ^ J ) x. ( 1 / 2 ) ) = ( ( C ^ J ) / 2 ) )
50	7, 37, 49	3eqtrd 2660	1 |- ( ph -> ( ( F ` A ) ` J ) = ( ( C ^ J ) / 2 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   (,)cioo 12175   |_cfl 12591   ^cexp 12860   abscabs 13974   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ioo 12179  df-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984

This theorem is referenced by:  knoppndvlem10  32512