Theorem knoppndvlem6 32508

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

Hypotheses

Ref	Expression
knoppndvlem6.t	|- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
knoppndvlem6.f	|- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
knoppndvlem6.w	|- W = ( w e. RR |-> sum_ i e. NN0 ( ( F ` w ) ` i ) )
knoppndvlem6.a	|- A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M )
knoppndvlem6.c	|- ( ph -> C e. ( -u 1 (,) 1 ) )
knoppndvlem6.j	|- ( ph -> J e. NN0 )
knoppndvlem6.m	|- ( ph -> M e. ZZ )
knoppndvlem6.n	|- ( ph -> N e. NN )

Assertion

Ref	Expression
knoppndvlem6	|- ( ph -> ( W ` A ) = sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) )

Distinct variable groups:   A, i, n, w, y   x, A, i, w   C, n, y   i, F, w   i, J, n, y   n, N, y   x, N   T, n, y   ph, i, n, w, y

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

Proof of Theorem knoppndvlem6

Step	Hyp	Ref	Expression
1		knoppndvlem6.w	. . . . 5 |- W = ( w e. RR |-> sum_ i e. NN0 ( ( F ` w ) ` i ) )
2	1	a1i 11	. . . 4 |- ( ph -> W = ( w e. RR |-> sum_ i e. NN0 ( ( F ` w ) ` i ) ) )
3		fveq2 6191	. . . . . . 7 |- ( w = A -> ( F ` w ) = ( F ` A ) )
4	3	fveq1d 6193	. . . . . 6 |- ( w = A -> ( ( F ` w ) ` i ) = ( ( F ` A ) ` i ) )
5	4	sumeq2sdv 14435	. . . . 5 |- ( w = A -> sum_ i e. NN0 ( ( F ` w ) ` i ) = sum_ i e. NN0 ( ( F ` A ) ` i ) )
6	5	adantl 482	. . . 4 |- ( ( ph /\ w = A ) -> sum_ i e. NN0 ( ( F ` w ) ` i ) = sum_ i e. NN0 ( ( F ` A ) ` i ) )
7		knoppndvlem6.a	. . . . . 6 |- A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M )
8	7	a1i 11	. . . . 5 |- ( ph -> A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) )
9		knoppndvlem6.n	. . . . . 6 |- ( ph -> N e. NN )
10		knoppndvlem6.j	. . . . . . 7 |- ( ph -> J e. NN0 )
11	10	nn0zd 11480	. . . . . 6 |- ( ph -> J e. ZZ )
12		knoppndvlem6.m	. . . . . 6 |- ( ph -> M e. ZZ )
13	9, 11, 12	knoppndvlem1 32503	. . . . 5 |- ( ph -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) e. RR )
14	8, 13	eqeltrd 2701	. . . 4 |- ( ph -> A e. RR )
15		sumex 14418	. . . . 5 |- sum_ i e. NN0 ( ( F ` A ) ` i ) e. _V
16	15	a1i 11	. . . 4 |- ( ph -> sum_ i e. NN0 ( ( F ` A ) ` i ) e. _V )
17	2, 6, 14, 16	fvmptd 6288	. . 3 |- ( ph -> ( W ` A ) = sum_ i e. NN0 ( ( F ` A ) ` i ) )
18		nn0uz 11722	. . . 4 |- NN0 = ( ZZ>= ` 0 )
19		eqid 2622	. . . 4 |- ( ZZ>= ` ( J + 1 ) ) = ( ZZ>= ` ( J + 1 ) )
20		peano2nn0 11333	. . . . 5 |- ( J e. NN0 -> ( J + 1 ) e. NN0 )
21	10, 20	syl 17	. . . 4 |- ( ph -> ( J + 1 ) e. NN0 )
22		eqidd 2623	. . . 4 |- ( ( ph /\ i e. NN0 ) -> ( ( F ` A ) ` i ) = ( ( F ` A ) ` i ) )
23		knoppndvlem6.t	. . . . . 6 |- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
24		knoppndvlem6.f	. . . . . 6 |- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
25	9	adantr 481	. . . . . 6 |- ( ( ph /\ i e. NN0 ) -> N e. NN )
26		knoppndvlem6.c	. . . . . . . . 9 |- ( ph -> C e. ( -u 1 (,) 1 ) )
27	26	knoppndvlem3 32505	. . . . . . . 8 |- ( ph -> ( C e. RR /\ ( abs ` C ) < 1 ) )
28	27	simpld 475	. . . . . . 7 |- ( ph -> C e. RR )
29	28	adantr 481	. . . . . 6 |- ( ( ph /\ i e. NN0 ) -> C e. RR )
30	14	adantr 481	. . . . . 6 |- ( ( ph /\ i e. NN0 ) -> A e. RR )
31		simpr 477	. . . . . 6 |- ( ( ph /\ i e. NN0 ) -> i e. NN0 )
32	23, 24, 25, 29, 30, 31	knoppcnlem3 32485	. . . . 5 |- ( ( ph /\ i e. NN0 ) -> ( ( F ` A ) ` i ) e. RR )
33	32	recnd 10068	. . . 4 |- ( ( ph /\ i e. NN0 ) -> ( ( F ` A ) ` i ) e. CC )
34	23, 24, 1, 14, 26, 9	knoppndvlem4 32506	. . . . 5 |- ( ph -> seq 0 ( + , ( F ` A ) ) ~~> ( W ` A ) )
35		seqex 12803	. . . . . 6 |- seq 0 ( + , ( F ` A ) ) e. _V
36		fvex 6201	. . . . . 6 |- ( W ` A ) e. _V
37	35, 36	breldm 5329	. . . . 5 |- ( seq 0 ( + , ( F ` A ) ) ~~> ( W ` A ) -> seq 0 ( + , ( F ` A ) ) e. dom ~~> )
38	34, 37	syl 17	. . . 4 |- ( ph -> seq 0 ( + , ( F ` A ) ) e. dom ~~> )
39	18, 19, 21, 22, 33, 38	isumsplit 14572	. . 3 |- ( ph -> sum_ i e. NN0 ( ( F ` A ) ` i ) = ( sum_ i e. ( 0 ... ( ( J + 1 ) - 1 ) ) ( ( F ` A ) ` i ) + sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) ) )
40	10	nn0cnd 11353	. . . . . . 7 |- ( ph -> J e. CC )
41		1cnd 10056	. . . . . . 7 |- ( ph -> 1 e. CC )
42	40, 41	pncand 10393	. . . . . 6 |- ( ph -> ( ( J + 1 ) - 1 ) = J )
43	42	oveq2d 6666	. . . . 5 |- ( ph -> ( 0 ... ( ( J + 1 ) - 1 ) ) = ( 0 ... J ) )
44	43	sumeq1d 14431	. . . 4 |- ( ph -> sum_ i e. ( 0 ... ( ( J + 1 ) - 1 ) ) ( ( F ` A ) ` i ) = sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) )
45	44	oveq1d 6665	. . 3 |- ( ph -> ( sum_ i e. ( 0 ... ( ( J + 1 ) - 1 ) ) ( ( F ` A ) ` i ) + sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) ) = ( sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) + sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) ) )
46	17, 39, 45	3eqtrd 2660	. 2 |- ( ph -> ( W ` A ) = ( sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) + sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) ) )
47	14	adantr 481	. . . . . . . 8 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> A e. RR )
48		eluznn0 11757	. . . . . . . . 9 |- ( ( ( J + 1 ) e. NN0 /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> i e. NN0 )
49	21, 48	sylan 488	. . . . . . . 8 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> i e. NN0 )
50	24, 47, 49	knoppcnlem1 32483	. . . . . . 7 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( ( F ` A ) ` i ) = ( ( C ^ i ) x. ( T ` ( ( ( 2 x. N ) ^ i ) x. A ) ) ) )
51	7	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) )
52	51	oveq2d 6666	. . . . . . . . . 10 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( ( ( 2 x. N ) ^ i ) x. A ) = ( ( ( 2 x. N ) ^ i ) x. ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) )
53	9	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> N e. NN )
54	49	nn0zd 11480	. . . . . . . . . . 11 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> i e. ZZ )
55	11	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> J e. ZZ )
56	12	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> M e. ZZ )
57		eluzle 11700	. . . . . . . . . . . . 13 |- ( i e. ( ZZ>= ` ( J + 1 ) ) -> ( J + 1 ) <_ i )
58	57	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( J + 1 ) <_ i )
59	55, 54	jca 554	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( J e. ZZ /\ i e. ZZ ) )
60		zltp1le 11427	. . . . . . . . . . . . 13 |- ( ( J e. ZZ /\ i e. ZZ ) -> ( J < i <-> ( J + 1 ) <_ i ) )
61	59, 60	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( J < i <-> ( J + 1 ) <_ i ) )
62	58, 61	mpbird 247	. . . . . . . . . . 11 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> J < i )
63	53, 54, 55, 56, 62	knoppndvlem2 32504	. . . . . . . . . 10 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( ( ( 2 x. N ) ^ i ) x. ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) e. ZZ )
64	52, 63	eqeltrd 2701	. . . . . . . . 9 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( ( ( 2 x. N ) ^ i ) x. A ) e. ZZ )
65	23, 64	dnizeq0 32465	. . . . . . . 8 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( T ` ( ( ( 2 x. N ) ^ i ) x. A ) ) = 0 )
66	65	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( ( C ^ i ) x. ( T ` ( ( ( 2 x. N ) ^ i ) x. A ) ) ) = ( ( C ^ i ) x. 0 ) )
67	28	recnd 10068	. . . . . . . . . 10 |- ( ph -> C e. CC )
68	67	adantr 481	. . . . . . . . 9 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> C e. CC )
69	68, 49	expcld 13008	. . . . . . . 8 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( C ^ i ) e. CC )
70	69	mul01d 10235	. . . . . . 7 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( ( C ^ i ) x. 0 ) = 0 )
71	50, 66, 70	3eqtrd 2660	. . . . . 6 |- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( ( F ` A ) ` i ) = 0 )
72	71	sumeq2dv 14433	. . . . 5 |- ( ph -> sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) = sum_ i e. ( ZZ>= ` ( J + 1 ) ) 0 )
73		ssid 3624	. . . . . . . 8 |- ( ZZ>= ` ( J + 1 ) ) C_ ( ZZ>= ` ( J + 1 ) )
74	73	a1i 11	. . . . . . 7 |- ( ph -> ( ZZ>= ` ( J + 1 ) ) C_ ( ZZ>= ` ( J + 1 ) ) )
75	74	orcd 407	. . . . . 6 |- ( ph -> ( ( ZZ>= ` ( J + 1 ) ) C_ ( ZZ>= ` ( J + 1 ) ) \/ ( ZZ>= ` ( J + 1 ) ) e. Fin ) )
76		sumz 14453	. . . . . 6 |- ( ( ( ZZ>= ` ( J + 1 ) ) C_ ( ZZ>= ` ( J + 1 ) ) \/ ( ZZ>= ` ( J + 1 ) ) e. Fin ) -> sum_ i e. ( ZZ>= ` ( J + 1 ) ) 0 = 0 )
77	75, 76	syl 17	. . . . 5 |- ( ph -> sum_ i e. ( ZZ>= ` ( J + 1 ) ) 0 = 0 )
78	72, 77	eqtrd 2656	. . . 4 |- ( ph -> sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) = 0 )
79	78	oveq2d 6666	. . 3 |- ( ph -> ( sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) + sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) ) = ( sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) + 0 ) )
80	23, 24, 14, 28, 9	knoppndvlem5 32507	. . . . 5 |- ( ph -> sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) e. RR )
81	80	recnd 10068	. . . 4 |- ( ph -> sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) e. CC )
82	81	addid1d 10236	. . 3 |- ( ph -> ( sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) + 0 ) = sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) )
83	79, 82	eqtrd 2656	. 2 |- ( ph -> ( sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) + sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) ) = sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) )
84	46, 83	eqtrd 2656	1 |- ( ph -> ( W ` A ) = sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   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   ZZ>=cuz 11687   (,)cioo 12175   ...cfz 12326   |_cfl 12591   seqcseq 12801   ^cexp 12860   abscabs 13974   ~~> cli 14215   sum_csu 14416

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-inf2 8538  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  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-int 4476  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  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-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ulm 24131

This theorem is referenced by:  knoppndvlem15  32517