Theorem knoppndvlem10 32512

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

Hypotheses

Ref	Expression
knoppndvlem10.t	|- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
knoppndvlem10.f	|- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
knoppndvlem10.a	|- A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M )
knoppndvlem10.b	|- B = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( M + 1 ) )
knoppndvlem10.c	|- ( ph -> C e. ( -u 1 (,) 1 ) )
knoppndvlem10.j	|- ( ph -> J e. NN0 )
knoppndvlem10.m	|- ( ph -> M e. ZZ )
knoppndvlem10.n	|- ( ph -> N e. NN )

Assertion

Ref	Expression
knoppndvlem10	|- ( ph -> ( abs ` ( ( ( F ` B ) ` J ) - ( ( F ` A ) ` J ) ) ) = ( ( ( abs ` C ) ^ J ) / 2 ) )

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

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

Proof of Theorem knoppndvlem10

Step	Hyp	Ref	Expression
1		knoppndvlem10.t	. . . . . . 7 |- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
2		knoppndvlem10.f	. . . . . . 7 |- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
3		knoppndvlem10.b	. . . . . . 7 |- B = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( M + 1 ) )
4		knoppndvlem10.c	. . . . . . . 8 |- ( ph -> C e. ( -u 1 (,) 1 ) )
5	4	adantr 481	. . . . . . 7 |- ( ( ph /\ 2 || M ) -> C e. ( -u 1 (,) 1 ) )
6		knoppndvlem10.j	. . . . . . . 8 |- ( ph -> J e. NN0 )
7	6	adantr 481	. . . . . . 7 |- ( ( ph /\ 2 || M ) -> J e. NN0 )
8		knoppndvlem10.m	. . . . . . . . 9 |- ( ph -> M e. ZZ )
9	8	peano2zd 11485	. . . . . . . 8 |- ( ph -> ( M + 1 ) e. ZZ )
10	9	adantr 481	. . . . . . 7 |- ( ( ph /\ 2 || M ) -> ( M + 1 ) e. ZZ )
11		knoppndvlem10.n	. . . . . . . 8 |- ( ph -> N e. NN )
12	11	adantr 481	. . . . . . 7 |- ( ( ph /\ 2 || M ) -> N e. NN )
13		notnot 136	. . . . . . . . 9 |- ( 2 || M -> -. -. 2 || M )
14	13	adantl 482	. . . . . . . 8 |- ( ( ph /\ 2 || M ) -> -. -. 2 || M )
15	8	adantr 481	. . . . . . . . 9 |- ( ( ph /\ 2 || M ) -> M e. ZZ )
16		oddp1even 15068	. . . . . . . . 9 |- ( M e. ZZ -> ( -. 2 || M <-> 2 || ( M + 1 ) ) )
17	15, 16	syl 17	. . . . . . . 8 |- ( ( ph /\ 2 || M ) -> ( -. 2 || M <-> 2 || ( M + 1 ) ) )
18	14, 17	mtbid 314	. . . . . . 7 |- ( ( ph /\ 2 || M ) -> -. 2 || ( M + 1 ) )
19	1, 2, 3, 5, 7, 10, 12, 18	knoppndvlem9 32511	. . . . . 6 |- ( ( ph /\ 2 || M ) -> ( ( F ` B ) ` J ) = ( ( C ^ J ) / 2 ) )
20		knoppndvlem10.a	. . . . . . 7 |- A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M )
21	14	notnotrd 128	. . . . . . 7 |- ( ( ph /\ 2 || M ) -> 2 || M )
22	1, 2, 20, 5, 7, 15, 12, 21	knoppndvlem8 32510	. . . . . 6 |- ( ( ph /\ 2 || M ) -> ( ( F ` A ) ` J ) = 0 )
23	19, 22	oveq12d 6668	. . . . 5 |- ( ( ph /\ 2 || M ) -> ( ( ( F ` B ) ` J ) - ( ( F ` A ) ` J ) ) = ( ( ( C ^ J ) / 2 ) - 0 ) )
24	4	knoppndvlem3 32505	. . . . . . . . . . 11 |- ( ph -> ( C e. RR /\ ( abs ` C ) < 1 ) )
25	24	simpld 475	. . . . . . . . . 10 |- ( ph -> C e. RR )
26	25	recnd 10068	. . . . . . . . 9 |- ( ph -> C e. CC )
27	26, 6	expcld 13008	. . . . . . . 8 |- ( ph -> ( C ^ J ) e. CC )
28		2cnd 11093	. . . . . . . 8 |- ( ph -> 2 e. CC )
29		2ne0 11113	. . . . . . . . 9 |- 2 =/= 0
30	29	a1i 11	. . . . . . . 8 |- ( ph -> 2 =/= 0 )
31	27, 28, 30	divcld 10801	. . . . . . 7 |- ( ph -> ( ( C ^ J ) / 2 ) e. CC )
32	31	subid1d 10381	. . . . . 6 |- ( ph -> ( ( ( C ^ J ) / 2 ) - 0 ) = ( ( C ^ J ) / 2 ) )
33	32	adantr 481	. . . . 5 |- ( ( ph /\ 2 || M ) -> ( ( ( C ^ J ) / 2 ) - 0 ) = ( ( C ^ J ) / 2 ) )
34	23, 33	eqtrd 2656	. . . 4 |- ( ( ph /\ 2 || M ) -> ( ( ( F ` B ) ` J ) - ( ( F ` A ) ` J ) ) = ( ( C ^ J ) / 2 ) )
35	34	fveq2d 6195	. . 3 |- ( ( ph /\ 2 || M ) -> ( abs ` ( ( ( F ` B ) ` J ) - ( ( F ` A ) ` J ) ) ) = ( abs ` ( ( C ^ J ) / 2 ) ) )
36	3	a1i 11	. . . . . . . . 9 |- ( ph -> B = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( M + 1 ) ) )
37	6	nn0zd 11480	. . . . . . . . . 10 |- ( ph -> J e. ZZ )
38	11, 37, 9	knoppndvlem1 32503	. . . . . . . . 9 |- ( ph -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( M + 1 ) ) e. RR )
39	36, 38	eqeltrd 2701	. . . . . . . 8 |- ( ph -> B e. RR )
40	1, 2, 11, 25, 39, 6	knoppcnlem3 32485	. . . . . . 7 |- ( ph -> ( ( F ` B ) ` J ) e. RR )
41	40	recnd 10068	. . . . . 6 |- ( ph -> ( ( F ` B ) ` J ) e. CC )
42	20	a1i 11	. . . . . . . . 9 |- ( ph -> A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) )
43	11, 37, 8	knoppndvlem1 32503	. . . . . . . . 9 |- ( ph -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) e. RR )
44	42, 43	eqeltrd 2701	. . . . . . . 8 |- ( ph -> A e. RR )
45	1, 2, 11, 25, 44, 6	knoppcnlem3 32485	. . . . . . 7 |- ( ph -> ( ( F ` A ) ` J ) e. RR )
46	45	recnd 10068	. . . . . 6 |- ( ph -> ( ( F ` A ) ` J ) e. CC )
47	41, 46	abssubd 14192	. . . . 5 |- ( ph -> ( abs ` ( ( ( F ` B ) ` J ) - ( ( F ` A ) ` J ) ) ) = ( abs ` ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) )
48	47	adantr 481	. . . 4 |- ( ( ph /\ -. 2 || M ) -> ( abs ` ( ( ( F ` B ) ` J ) - ( ( F ` A ) ` J ) ) ) = ( abs ` ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) )
49	4	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. 2 || M ) -> C e. ( -u 1 (,) 1 ) )
50	6	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. 2 || M ) -> J e. NN0 )
51	8	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. 2 || M ) -> M e. ZZ )
52	11	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. 2 || M ) -> N e. NN )
53		simpr 477	. . . . . . . 8 |- ( ( ph /\ -. 2 || M ) -> -. 2 || M )
54	1, 2, 20, 49, 50, 51, 52, 53	knoppndvlem9 32511	. . . . . . 7 |- ( ( ph /\ -. 2 || M ) -> ( ( F ` A ) ` J ) = ( ( C ^ J ) / 2 ) )
55	9	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. 2 || M ) -> ( M + 1 ) e. ZZ )
56	51, 16	syl 17	. . . . . . . . 9 |- ( ( ph /\ -. 2 || M ) -> ( -. 2 || M <-> 2 || ( M + 1 ) ) )
57	53, 56	mpbid 222	. . . . . . . 8 |- ( ( ph /\ -. 2 || M ) -> 2 || ( M + 1 ) )
58	1, 2, 3, 49, 50, 55, 52, 57	knoppndvlem8 32510	. . . . . . 7 |- ( ( ph /\ -. 2 || M ) -> ( ( F ` B ) ` J ) = 0 )
59	54, 58	oveq12d 6668	. . . . . 6 |- ( ( ph /\ -. 2 || M ) -> ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) = ( ( ( C ^ J ) / 2 ) - 0 ) )
60	32	adantr 481	. . . . . 6 |- ( ( ph /\ -. 2 || M ) -> ( ( ( C ^ J ) / 2 ) - 0 ) = ( ( C ^ J ) / 2 ) )
61	59, 60	eqtrd 2656	. . . . 5 |- ( ( ph /\ -. 2 || M ) -> ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) = ( ( C ^ J ) / 2 ) )
62	61	fveq2d 6195	. . . 4 |- ( ( ph /\ -. 2 || M ) -> ( abs ` ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) = ( abs ` ( ( C ^ J ) / 2 ) ) )
63	48, 62	eqtrd 2656	. . 3 |- ( ( ph /\ -. 2 || M ) -> ( abs ` ( ( ( F ` B ) ` J ) - ( ( F ` A ) ` J ) ) ) = ( abs ` ( ( C ^ J ) / 2 ) ) )
64	35, 63	pm2.61dan 832	. 2 |- ( ph -> ( abs ` ( ( ( F ` B ) ` J ) - ( ( F ` A ) ` J ) ) ) = ( abs ` ( ( C ^ J ) / 2 ) ) )
65	27, 28, 30	absdivd 14194	. . 3 |- ( ph -> ( abs ` ( ( C ^ J ) / 2 ) ) = ( ( abs ` ( C ^ J ) ) / ( abs ` 2 ) ) )
66	26, 6	absexpd 14191	. . . 4 |- ( ph -> ( abs ` ( C ^ J ) ) = ( ( abs ` C ) ^ J ) )
67		0le2 11111	. . . . . 6 |- 0 <_ 2
68		2re 11090	. . . . . . 7 |- 2 e. RR
69	68	absidi 14117	. . . . . 6 |- ( 0 <_ 2 -> ( abs ` 2 ) = 2 )
70	67, 69	ax-mp 5	. . . . 5 |- ( abs ` 2 ) = 2
71	70	a1i 11	. . . 4 |- ( ph -> ( abs ` 2 ) = 2 )
72	66, 71	oveq12d 6668	. . 3 |- ( ph -> ( ( abs ` ( C ^ J ) ) / ( abs ` 2 ) ) = ( ( ( abs ` C ) ^ J ) / 2 ) )
73	65, 72	eqtrd 2656	. 2 |- ( ph -> ( abs ` ( ( C ^ J ) / 2 ) ) = ( ( ( abs ` C ) ^ J ) / 2 ) )
74	64, 73	eqtrd 2656	1 |- ( ph -> ( abs ` ( ( ( F ` B ) ` J ) - ( ( F ` A ) ` J ) ) ) = ( ( ( abs ` C ) ^ J ) / 2 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - 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-ico 12181  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:  knoppndvlem15  32517