Theorem knoppndvlem7 32509

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

Hypotheses

Ref	Expression
knoppndvlem7.t	|- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
knoppndvlem7.f	|- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
knoppndvlem7.a	|- A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M )
knoppndvlem7.j	|- ( ph -> J e. NN0 )
knoppndvlem7.m	|- ( ph -> M e. ZZ )
knoppndvlem7.n	|- ( ph -> N e. NN )

Assertion

Ref	Expression
knoppndvlem7	|- ( ph -> ( ( F ` A ) ` J ) = ( ( C ^ J ) x. ( T ` ( M / 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 knoppndvlem7

Step	Hyp	Ref	Expression
1		knoppndvlem7.f	. . 3 |- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
2		knoppndvlem7.a	. . . . 5 |- A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M )
3	2	a1i 11	. . . 4 |- ( ph -> A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) )
4		knoppndvlem7.n	. . . . 5 |- ( ph -> N e. NN )
5		knoppndvlem7.j	. . . . . 6 |- ( ph -> J e. NN0 )
6	5	nn0zd 11480	. . . . 5 |- ( ph -> J e. ZZ )
7		knoppndvlem7.m	. . . . 5 |- ( ph -> M e. ZZ )
8	4, 6, 7	knoppndvlem1 32503	. . . 4 |- ( ph -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) e. RR )
9	3, 8	eqeltrd 2701	. . 3 |- ( ph -> A e. RR )
10	1, 9, 5	knoppcnlem1 32483	. 2 |- ( ph -> ( ( F ` A ) ` J ) = ( ( C ^ J ) x. ( T ` ( ( ( 2 x. N ) ^ J ) x. A ) ) ) )
11	2	oveq2i 6661	. . . . . 6 |- ( ( ( 2 x. N ) ^ J ) x. A ) = ( ( ( 2 x. N ) ^ J ) x. ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) )
12	11	a1i 11	. . . . 5 |- ( ph -> ( ( ( 2 x. N ) ^ J ) x. A ) = ( ( ( 2 x. N ) ^ J ) x. ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) )
13		2cnd 11093	. . . . . . . . . 10 |- ( ph -> 2 e. CC )
14		nnz 11399	. . . . . . . . . . . 12 |- ( N e. NN -> N e. ZZ )
15	4, 14	syl 17	. . . . . . . . . . 11 |- ( ph -> N e. ZZ )
16	15	zcnd 11483	. . . . . . . . . 10 |- ( ph -> N e. CC )
17	13, 16	mulcld 10060	. . . . . . . . 9 |- ( ph -> ( 2 x. N ) e. CC )
18	17, 5	expcld 13008	. . . . . . . 8 |- ( ph -> ( ( 2 x. N ) ^ J ) e. CC )
19		2ne0 11113	. . . . . . . . . . . 12 |- 2 =/= 0
20	19	a1i 11	. . . . . . . . . . 11 |- ( ph -> 2 =/= 0 )
21		0red 10041	. . . . . . . . . . . . 13 |- ( ph -> 0 e. RR )
22		1red 10055	. . . . . . . . . . . . . 14 |- ( ph -> 1 e. RR )
23	15	zred 11482	. . . . . . . . . . . . . 14 |- ( ph -> N e. RR )
24		0lt1 10550	. . . . . . . . . . . . . . 15 |- 0 < 1
25	24	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> 0 < 1 )
26		nnge1 11046	. . . . . . . . . . . . . . 15 |- ( N e. NN -> 1 <_ N )
27	4, 26	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> 1 <_ N )
28	21, 22, 23, 25, 27	ltletrd 10197	. . . . . . . . . . . . 13 |- ( ph -> 0 < N )
29	21, 28	ltned 10173	. . . . . . . . . . . 12 |- ( ph -> 0 =/= N )
30	29	necomd 2849	. . . . . . . . . . 11 |- ( ph -> N =/= 0 )
31	13, 16, 20, 30	mulne0d 10679	. . . . . . . . . 10 |- ( ph -> ( 2 x. N ) =/= 0 )
32	6	znegcld 11484	. . . . . . . . . 10 |- ( ph -> -u J e. ZZ )
33	17, 31, 32	expclzd 13013	. . . . . . . . 9 |- ( ph -> ( ( 2 x. N ) ^ -u J ) e. CC )
34	33, 13, 20	divcld 10801	. . . . . . . 8 |- ( ph -> ( ( ( 2 x. N ) ^ -u J ) / 2 ) e. CC )
35	7	zcnd 11483	. . . . . . . 8 |- ( ph -> M e. CC )
36	18, 34, 35	mulassd 10063	. . . . . . 7 |- ( ph -> ( ( ( ( 2 x. N ) ^ J ) x. ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) x. M ) = ( ( ( 2 x. N ) ^ J ) x. ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) )
37	36	eqcomd 2628	. . . . . 6 |- ( ph -> ( ( ( 2 x. N ) ^ J ) x. ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) = ( ( ( ( 2 x. N ) ^ J ) x. ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) x. M ) )
38	18, 33, 13, 20	divassd 10836	. . . . . . . . 9 |- ( ph -> ( ( ( ( 2 x. N ) ^ J ) x. ( ( 2 x. N ) ^ -u J ) ) / 2 ) = ( ( ( 2 x. N ) ^ J ) x. ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) )
39	38	eqcomd 2628	. . . . . . . 8 |- ( ph -> ( ( ( 2 x. N ) ^ J ) x. ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) = ( ( ( ( 2 x. N ) ^ J ) x. ( ( 2 x. N ) ^ -u J ) ) / 2 ) )
40	17, 31, 6	expnegd 13015	. . . . . . . . . . 11 |- ( ph -> ( ( 2 x. N ) ^ -u J ) = ( 1 / ( ( 2 x. N ) ^ J ) ) )
41	40	oveq2d 6666	. . . . . . . . . 10 |- ( ph -> ( ( ( 2 x. N ) ^ J ) x. ( ( 2 x. N ) ^ -u J ) ) = ( ( ( 2 x. N ) ^ J ) x. ( 1 / ( ( 2 x. N ) ^ J ) ) ) )
42	17, 31, 6	expne0d 13014	. . . . . . . . . . 11 |- ( ph -> ( ( 2 x. N ) ^ J ) =/= 0 )
43	18, 42	recidd 10796	. . . . . . . . . 10 |- ( ph -> ( ( ( 2 x. N ) ^ J ) x. ( 1 / ( ( 2 x. N ) ^ J ) ) ) = 1 )
44	41, 43	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( ( ( 2 x. N ) ^ J ) x. ( ( 2 x. N ) ^ -u J ) ) = 1 )
45	44	oveq1d 6665	. . . . . . . 8 |- ( ph -> ( ( ( ( 2 x. N ) ^ J ) x. ( ( 2 x. N ) ^ -u J ) ) / 2 ) = ( 1 / 2 ) )
46	39, 45	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( ( 2 x. N ) ^ J ) x. ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) = ( 1 / 2 ) )
47	46	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( ( ( 2 x. N ) ^ J ) x. ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) x. M ) = ( ( 1 / 2 ) x. M ) )
48	35, 13, 20	divrec2d 10805	. . . . . . 7 |- ( ph -> ( M / 2 ) = ( ( 1 / 2 ) x. M ) )
49	48	eqcomd 2628	. . . . . 6 |- ( ph -> ( ( 1 / 2 ) x. M ) = ( M / 2 ) )
50	37, 47, 49	3eqtrd 2660	. . . . 5 |- ( ph -> ( ( ( 2 x. N ) ^ J ) x. ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) = ( M / 2 ) )
51	12, 50	eqtrd 2656	. . . 4 |- ( ph -> ( ( ( 2 x. N ) ^ J ) x. A ) = ( M / 2 ) )
52	51	fveq2d 6195	. . 3 |- ( ph -> ( T ` ( ( ( 2 x. N ) ^ J ) x. A ) ) = ( T ` ( M / 2 ) ) )
53	52	oveq2d 6666	. 2 |- ( ph -> ( ( C ^ J ) x. ( T ` ( ( ( 2 x. N ) ^ J ) x. A ) ) ) = ( ( C ^ J ) x. ( T ` ( M / 2 ) ) ) )
54	10, 53	eqtrd 2656	1 |- ( ph -> ( ( F ` A ) ` J ) = ( ( C ^ J ) x. ( T ` ( M / 2 ) ) ) )

Syntax hints:   -> wi 4   = 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   |_cfl 12591   ^cexp 12860   abscabs 13974

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

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-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861

This theorem is referenced by:  knoppndvlem8  32510  knoppndvlem9  32511