Theorem knoppcnlem4 32486

Description: Lemma for knoppcn 32494. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)

Hypotheses

Ref	Expression
knoppcnlem4.t	|- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
knoppcnlem4.f	|- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
knoppcnlem4.n	|- ( ph -> N e. NN )
knoppcnlem4.1	|- ( ph -> C e. RR )
knoppcnlem4.2	|- ( ph -> A e. RR )
knoppcnlem4.3	|- ( ph -> M e. NN0 )

Assertion

Ref	Expression
knoppcnlem4	|- ( ph -> ( abs ` ( ( F ` A ) ` M ) ) <_ ( ( m e. NN0 |-> ( ( abs ` C ) ^ m ) ) ` M ) )

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

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

Proof of Theorem knoppcnlem4

Step	Hyp	Ref	Expression
1		knoppcnlem4.f	. . . 4 |- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
2		knoppcnlem4.2	. . . 4 |- ( ph -> A e. RR )
3		knoppcnlem4.3	. . . 4 |- ( ph -> M e. NN0 )
4	1, 2, 3	knoppcnlem1 32483	. . 3 |- ( ph -> ( ( F ` A ) ` M ) = ( ( C ^ M ) x. ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) )
5	4	fveq2d 6195	. 2 |- ( ph -> ( abs ` ( ( F ` A ) ` M ) ) = ( abs ` ( ( C ^ M ) x. ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) ) )
6		knoppcnlem4.1	. . . . . . . 8 |- ( ph -> C e. RR )
7	6	recnd 10068	. . . . . . 7 |- ( ph -> C e. CC )
8	7, 3	expcld 13008	. . . . . 6 |- ( ph -> ( C ^ M ) e. CC )
9		knoppcnlem4.t	. . . . . . . 8 |- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
10		2re 11090	. . . . . . . . . . . 12 |- 2 e. RR
11	10	a1i 11	. . . . . . . . . . 11 |- ( ph -> 2 e. RR )
12		knoppcnlem4.n	. . . . . . . . . . . 12 |- ( ph -> N e. NN )
13		nnre 11027	. . . . . . . . . . . 12 |- ( N e. NN -> N e. RR )
14	12, 13	syl 17	. . . . . . . . . . 11 |- ( ph -> N e. RR )
15	11, 14	remulcld 10070	. . . . . . . . . 10 |- ( ph -> ( 2 x. N ) e. RR )
16	15, 3	reexpcld 13025	. . . . . . . . 9 |- ( ph -> ( ( 2 x. N ) ^ M ) e. RR )
17	16, 2	remulcld 10070	. . . . . . . 8 |- ( ph -> ( ( ( 2 x. N ) ^ M ) x. A ) e. RR )
18	9, 17	dnicld2 32463	. . . . . . 7 |- ( ph -> ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) e. RR )
19	18	recnd 10068	. . . . . 6 |- ( ph -> ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) e. CC )
20	8, 19	absmuld 14193	. . . . 5 |- ( ph -> ( abs ` ( ( C ^ M ) x. ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) ) = ( ( abs ` ( C ^ M ) ) x. ( abs ` ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) ) )
21	7, 3	absexpd 14191	. . . . . 6 |- ( ph -> ( abs ` ( C ^ M ) ) = ( ( abs ` C ) ^ M ) )
22	21	oveq1d 6665	. . . . 5 |- ( ph -> ( ( abs ` ( C ^ M ) ) x. ( abs ` ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) ) = ( ( ( abs ` C ) ^ M ) x. ( abs ` ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) ) )
23	20, 22	eqtrd 2656	. . . 4 |- ( ph -> ( abs ` ( ( C ^ M ) x. ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) ) = ( ( ( abs ` C ) ^ M ) x. ( abs ` ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) ) )
24	19	abscld 14175	. . . . . 6 |- ( ph -> ( abs ` ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) e. RR )
25		1red 10055	. . . . . 6 |- ( ph -> 1 e. RR )
26	7	abscld 14175	. . . . . . 7 |- ( ph -> ( abs ` C ) e. RR )
27	26, 3	reexpcld 13025	. . . . . 6 |- ( ph -> ( ( abs ` C ) ^ M ) e. RR )
28	7	absge0d 14183	. . . . . . 7 |- ( ph -> 0 <_ ( abs ` C ) )
29	26, 3, 28	expge0d 13026	. . . . . 6 |- ( ph -> 0 <_ ( ( abs ` C ) ^ M ) )
30	9	dnival 32461	. . . . . . . . . 10 |- ( ( ( ( 2 x. N ) ^ M ) x. A ) e. RR -> ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) = ( abs ` ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) ) )
31	17, 30	syl 17	. . . . . . . . 9 |- ( ph -> ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) = ( abs ` ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) ) )
32	31	fveq2d 6195	. . . . . . . 8 |- ( ph -> ( abs ` ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) = ( abs ` ( abs ` ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) ) ) )
33		halfre 11246	. . . . . . . . . . . . . 14 |- ( 1 / 2 ) e. RR
34	33	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> ( 1 / 2 ) e. RR )
35	17, 34	readdcld 10069	. . . . . . . . . . . 12 |- ( ph -> ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) e. RR )
36		reflcl 12597	. . . . . . . . . . . 12 |- ( ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) e. RR -> ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) e. RR )
37	35, 36	syl 17	. . . . . . . . . . 11 |- ( ph -> ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) e. RR )
38	37, 17	resubcld 10458	. . . . . . . . . 10 |- ( ph -> ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) e. RR )
39	38	recnd 10068	. . . . . . . . 9 |- ( ph -> ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) e. CC )
40		absidm 14063	. . . . . . . . 9 |- ( ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) e. CC -> ( abs ` ( abs ` ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) ) ) = ( abs ` ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) ) )
41	39, 40	syl 17	. . . . . . . 8 |- ( ph -> ( abs ` ( abs ` ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) ) ) = ( abs ` ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) ) )
42	32, 41	eqtrd 2656	. . . . . . 7 |- ( ph -> ( abs ` ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) = ( abs ` ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) ) )
43	31, 18	eqeltrrd 2702	. . . . . . . 8 |- ( ph -> ( abs ` ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) ) e. RR )
44		rddif 14080	. . . . . . . . 9 |- ( ( ( ( 2 x. N ) ^ M ) x. A ) e. RR -> ( abs ` ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) ) <_ ( 1 / 2 ) )
45	17, 44	syl 17	. . . . . . . 8 |- ( ph -> ( abs ` ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) ) <_ ( 1 / 2 ) )
46		halflt1 11250	. . . . . . . . . 10 |- ( 1 / 2 ) < 1
47		1re 10039	. . . . . . . . . . 11 |- 1 e. RR
48	33, 47	ltlei 10159	. . . . . . . . . 10 |- ( ( 1 / 2 ) < 1 -> ( 1 / 2 ) <_ 1 )
49	46, 48	ax-mp 5	. . . . . . . . 9 |- ( 1 / 2 ) <_ 1
50	49	a1i 11	. . . . . . . 8 |- ( ph -> ( 1 / 2 ) <_ 1 )
51	43, 34, 25, 45, 50	letrd 10194	. . . . . . 7 |- ( ph -> ( abs ` ( ( |_ ` ( ( ( ( 2 x. N ) ^ M ) x. A ) + ( 1 / 2 ) ) ) - ( ( ( 2 x. N ) ^ M ) x. A ) ) ) <_ 1 )
52	42, 51	eqbrtrd 4675	. . . . . 6 |- ( ph -> ( abs ` ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) <_ 1 )
53	24, 25, 27, 29, 52	lemul2ad 10964	. . . . 5 |- ( ph -> ( ( ( abs ` C ) ^ M ) x. ( abs ` ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) ) <_ ( ( ( abs ` C ) ^ M ) x. 1 ) )
54		ax-1rid 10006	. . . . . 6 |- ( ( ( abs ` C ) ^ M ) e. RR -> ( ( ( abs ` C ) ^ M ) x. 1 ) = ( ( abs ` C ) ^ M ) )
55	27, 54	syl 17	. . . . 5 |- ( ph -> ( ( ( abs ` C ) ^ M ) x. 1 ) = ( ( abs ` C ) ^ M ) )
56	53, 55	breqtrd 4679	. . . 4 |- ( ph -> ( ( ( abs ` C ) ^ M ) x. ( abs ` ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) ) <_ ( ( abs ` C ) ^ M ) )
57	23, 56	eqbrtrd 4675	. . 3 |- ( ph -> ( abs ` ( ( C ^ M ) x. ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) ) <_ ( ( abs ` C ) ^ M ) )
58		eqidd 2623	. . . . 5 |- ( ph -> ( m e. NN0 |-> ( ( abs ` C ) ^ m ) ) = ( m e. NN0 |-> ( ( abs ` C ) ^ m ) ) )
59		oveq2 6658	. . . . . 6 |- ( m = M -> ( ( abs ` C ) ^ m ) = ( ( abs ` C ) ^ M ) )
60	59	adantl 482	. . . . 5 |- ( ( ph /\ m = M ) -> ( ( abs ` C ) ^ m ) = ( ( abs ` C ) ^ M ) )
61	58, 60, 3, 27	fvmptd 6288	. . . 4 |- ( ph -> ( ( m e. NN0 |-> ( ( abs ` C ) ^ m ) ) ` M ) = ( ( abs ` C ) ^ M ) )
62	61	eqcomd 2628	. . 3 |- ( ph -> ( ( abs ` C ) ^ M ) = ( ( m e. NN0 |-> ( ( abs ` C ) ^ m ) ) ` M ) )
63	57, 62	breqtrd 4679	. 2 |- ( ph -> ( abs ` ( ( C ^ M ) x. ( T ` ( ( ( 2 x. N ) ^ M ) x. A ) ) ) ) <_ ( ( m e. NN0 |-> ( ( abs ` C ) ^ m ) ) ` M ) )
64	5, 63	eqbrtrd 4675	1 |- ( ph -> ( abs ` ( ( F ` A ) ` M ) ) <_ ( ( m e. NN0 |-> ( ( abs ` C ) ^ m ) ) ` M ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   |_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  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-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-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976

This theorem is referenced by:  knoppcnlem6  32488