Theorem knoppcnlem9 32491

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

Hypotheses

Ref	Expression
knoppcnlem9.t	|- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
knoppcnlem9.f	|- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
knoppcnlem9.w	|- W = ( w e. RR |-> sum_ i e. NN0 ( ( F ` w ) ` i ) )
knoppcnlem9.n	|- ( ph -> N e. NN )
knoppcnlem9.1	|- ( ph -> C e. RR )
knoppcnlem9.2	|- ( ph -> ( abs ` C ) < 1 )

Assertion

Ref	Expression
knoppcnlem9	|- ( ph -> seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) W )

Distinct variable groups:   C, m, n, y   i, F, m, w, z   n, N, y   x, N   T, n, y   ph, i, m, w, z, n, y   x, i, m, w, z

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

Proof of Theorem knoppcnlem9

Dummy variables f k v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		knoppcnlem9.t	. . . 4 |- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
2		knoppcnlem9.f	. . . 4 |- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
3		knoppcnlem9.n	. . . 4 |- ( ph -> N e. NN )
4		knoppcnlem9.1	. . . 4 |- ( ph -> C e. RR )
5		knoppcnlem9.2	. . . 4 |- ( ph -> ( abs ` C ) < 1 )
6	1, 2, 3, 4, 5	knoppcnlem6 32488	. . 3 |- ( ph -> seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) e. dom ( ~~> u ` RR ) )
7		seqex 12803	. . . 4 |- seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) e. _V
8	7	eldm 5321	. . 3 |- ( seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) e. dom ( ~~> u ` RR ) <-> E. f seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f )
9	6, 8	sylib 208	. 2 |- ( ph -> E. f seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f )
10		simpr 477	. . . . 5 |- ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) -> seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f )
11		ulmcl 24135	. . . . . . . 8 |- ( seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f -> f : RR --> CC )
12	11	feqmptd 6249	. . . . . . 7 |- ( seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f -> f = ( w e. RR |-> ( f ` w ) ) )
13	12	adantl 482	. . . . . 6 |- ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) -> f = ( w e. RR |-> ( f ` w ) ) )
14		nn0uz 11722	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
15		0zd 11389	. . . . . . . . 9 |- ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) -> 0 e. ZZ )
16		eqidd 2623	. . . . . . . . 9 |- ( ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) /\ i e. NN0 ) -> ( ( F ` w ) ` i ) = ( ( F ` w ) ` i ) )
17	3	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. RR ) /\ i e. NN0 ) -> N e. NN )
18	4	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. RR ) /\ i e. NN0 ) -> C e. RR )
19		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. RR ) /\ i e. NN0 ) -> w e. RR )
20		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. RR ) /\ i e. NN0 ) -> i e. NN0 )
21	1, 2, 17, 18, 19, 20	knoppcnlem3 32485	. . . . . . . . . . 11 |- ( ( ( ph /\ w e. RR ) /\ i e. NN0 ) -> ( ( F ` w ) ` i ) e. RR )
22	21	adantllr 755	. . . . . . . . . 10 |- ( ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) /\ i e. NN0 ) -> ( ( F ` w ) ` i ) e. RR )
23	22	recnd 10068	. . . . . . . . 9 |- ( ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) /\ i e. NN0 ) -> ( ( F ` w ) ` i ) e. CC )
24	1, 2, 3, 4	knoppcnlem8 32490	. . . . . . . . . . 11 |- ( ph -> seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) : NN0 --> ( CC ^m RR ) )
25	24	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) -> seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) : NN0 --> ( CC ^m RR ) )
26		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) -> w e. RR )
27		seqex 12803	. . . . . . . . . . 11 |- seq 0 ( + , ( F ` w ) ) e. _V
28	27	a1i 11	. . . . . . . . . 10 |- ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) -> seq 0 ( + , ( F ` w ) ) e. _V )
29	3	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w e. RR ) /\ k e. NN0 ) -> N e. NN )
30	4	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w e. RR ) /\ k e. NN0 ) -> C e. RR )
31		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w e. RR ) /\ k e. NN0 ) -> k e. NN0 )
32	1, 2, 29, 30, 31	knoppcnlem7 32489	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. RR ) /\ k e. NN0 ) -> ( seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ` k ) = ( v e. RR |-> ( seq 0 ( + , ( F ` v ) ) ` k ) ) )
33	32	adantllr 755	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) /\ k e. NN0 ) -> ( seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ` k ) = ( v e. RR |-> ( seq 0 ( + , ( F ` v ) ) ` k ) ) )
34	33	fveq1d 6193	. . . . . . . . . . 11 |- ( ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) /\ k e. NN0 ) -> ( ( seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ` k ) ` w ) = ( ( v e. RR |-> ( seq 0 ( + , ( F ` v ) ) ` k ) ) ` w ) )
35		eqid 2622	. . . . . . . . . . . . 13 |- ( v e. RR |-> ( seq 0 ( + , ( F ` v ) ) ` k ) ) = ( v e. RR |-> ( seq 0 ( + , ( F ` v ) ) ` k ) )
36	35	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) /\ k e. NN0 ) -> ( v e. RR |-> ( seq 0 ( + , ( F ` v ) ) ` k ) ) = ( v e. RR |-> ( seq 0 ( + , ( F ` v ) ) ` k ) ) )
37		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( v = w -> ( F ` v ) = ( F ` w ) )
38	37	seqeq3d 12809	. . . . . . . . . . . . . 14 |- ( v = w -> seq 0 ( + , ( F ` v ) ) = seq 0 ( + , ( F ` w ) ) )
39	38	fveq1d 6193	. . . . . . . . . . . . 13 |- ( v = w -> ( seq 0 ( + , ( F ` v ) ) ` k ) = ( seq 0 ( + , ( F ` w ) ) ` k ) )
40	39	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) /\ k e. NN0 ) /\ v = w ) -> ( seq 0 ( + , ( F ` v ) ) ` k ) = ( seq 0 ( + , ( F ` w ) ) ` k ) )
41	26	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) /\ k e. NN0 ) -> w e. RR )
42		fvexd 6203	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) /\ k e. NN0 ) -> ( seq 0 ( + , ( F ` w ) ) ` k ) e. _V )
43	36, 40, 41, 42	fvmptd 6288	. . . . . . . . . . 11 |- ( ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) /\ k e. NN0 ) -> ( ( v e. RR |-> ( seq 0 ( + , ( F ` v ) ) ` k ) ) ` w ) = ( seq 0 ( + , ( F ` w ) ) ` k ) )
44	34, 43	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) /\ k e. NN0 ) -> ( ( seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ` k ) ` w ) = ( seq 0 ( + , ( F ` w ) ) ` k ) )
45		simplr 792	. . . . . . . . . 10 |- ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) -> seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f )
46	14, 15, 25, 26, 28, 44, 45	ulmclm 24141	. . . . . . . . 9 |- ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) -> seq 0 ( + , ( F ` w ) ) ~~> ( f ` w ) )
47	14, 15, 16, 23, 46	isumclim 14488	. . . . . . . 8 |- ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) -> sum_ i e. NN0 ( ( F ` w ) ` i ) = ( f ` w ) )
48	47	eqcomd 2628	. . . . . . 7 |- ( ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) /\ w e. RR ) -> ( f ` w ) = sum_ i e. NN0 ( ( F ` w ) ` i ) )
49	48	mpteq2dva 4744	. . . . . 6 |- ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) -> ( w e. RR |-> ( f ` w ) ) = ( w e. RR |-> sum_ i e. NN0 ( ( F ` w ) ` i ) ) )
50		knoppcnlem9.w	. . . . . . . 8 |- W = ( w e. RR |-> sum_ i e. NN0 ( ( F ` w ) ` i ) )
51	50	a1i 11	. . . . . . 7 |- ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) -> W = ( w e. RR |-> sum_ i e. NN0 ( ( F ` w ) ` i ) ) )
52	51	eqcomd 2628	. . . . . 6 |- ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) -> ( w e. RR |-> sum_ i e. NN0 ( ( F ` w ) ` i ) ) = W )
53	13, 49, 52	3eqtrd 2660	. . . . 5 |- ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) -> f = W )
54	10, 53	breqtrd 4679	. . . 4 |- ( ( ph /\ seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f ) -> seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) W )
55	54	ex 450	. . 3 |- ( ph -> ( seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f -> seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) W ) )
56	55	exlimdv 1861	. 2 |- ( ph -> ( E. f seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) f -> seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) W ) )
57	9, 56	mpd 15	1 |- ( ph -> seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ( ~~> u ` RR ) W )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   ^m cmap 7857   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   |_cfl 12591   seqcseq 12801   ^cexp 12860   abscabs 13974   sum_csu 14416   ~~> uculm 24130

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-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:  knoppcn  32494  knoppndvlem4  32506