Theorem coeidlem 23993

Description: Lemma for coeid 23994. (Contributed by Mario Carneiro, 22-Jul-2014.)

Hypotheses

Ref	Expression
dgrub.1	|- A = (coeff ` F )
dgrub.2	|- N = (deg ` F )
coeid.3	|- ( ph -> F e. (Poly ` S ) )
coeid.4	|- ( ph -> M e. NN0 )
coeid.5	|- ( ph -> B e. ( ( S u. { 0 } ) ^m NN0 ) )
coeid.6	|- ( ph -> ( B " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )
coeid.7	|- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( B ` k ) x. ( z ^ k ) ) ) )

Assertion

Ref	Expression
coeidlem	|- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )

Distinct variable groups:   z, k, A   k, F   ph, k, z   S, k, z   B, k, z   k, M, z   k, N, z

Allowed substitution hint:   F( z)

Proof of Theorem coeidlem

Step	Hyp	Ref	Expression
1		coeid.7	. 2 |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( B ` k ) x. ( z ^ k ) ) ) )
2		dgrub.1	. . . . . . 7 |- A = (coeff ` F )
3		coeid.3	. . . . . . . 8 |- ( ph -> F e. (Poly ` S ) )
4		coeid.4	. . . . . . . 8 |- ( ph -> M e. NN0 )
5		coeid.5	. . . . . . . . . 10 |- ( ph -> B e. ( ( S u. { 0 } ) ^m NN0 ) )
6		plybss 23950	. . . . . . . . . . . . . 14 |- ( F e. (Poly ` S ) -> S C_ CC )
7	3, 6	syl 17	. . . . . . . . . . . . 13 |- ( ph -> S C_ CC )
8		0cnd 10033	. . . . . . . . . . . . . 14 |- ( ph -> 0 e. CC )
9	8	snssd 4340	. . . . . . . . . . . . 13 |- ( ph -> { 0 } C_ CC )
10	7, 9	unssd 3789	. . . . . . . . . . . 12 |- ( ph -> ( S u. { 0 } ) C_ CC )
11		cnex 10017	. . . . . . . . . . . 12 |- CC e. _V
12		ssexg 4804	. . . . . . . . . . . 12 |- ( ( ( S u. { 0 } ) C_ CC /\ CC e. _V ) -> ( S u. { 0 } ) e. _V )
13	10, 11, 12	sylancl 694	. . . . . . . . . . 11 |- ( ph -> ( S u. { 0 } ) e. _V )
14		nn0ex 11298	. . . . . . . . . . 11 |- NN0 e. _V
15		elmapg 7870	. . . . . . . . . . 11 |- ( ( ( S u. { 0 } ) e. _V /\ NN0 e. _V ) -> ( B e. ( ( S u. { 0 } ) ^m NN0 ) <-> B : NN0 --> ( S u. { 0 } ) ) )
16	13, 14, 15	sylancl 694	. . . . . . . . . 10 |- ( ph -> ( B e. ( ( S u. { 0 } ) ^m NN0 ) <-> B : NN0 --> ( S u. { 0 } ) ) )
17	5, 16	mpbid 222	. . . . . . . . 9 |- ( ph -> B : NN0 --> ( S u. { 0 } ) )
18	17, 10	fssd 6057	. . . . . . . 8 |- ( ph -> B : NN0 --> CC )
19		coeid.6	. . . . . . . 8 |- ( ph -> ( B " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )
20	3, 4, 18, 19, 1	coeeq 23983	. . . . . . 7 |- ( ph -> (coeff ` F ) = B )
21	2, 20	syl5req 2669	. . . . . 6 |- ( ph -> B = A )
22	21	adantr 481	. . . . 5 |- ( ( ph /\ z e. CC ) -> B = A )
23		fveq1 6190	. . . . . . 7 |- ( B = A -> ( B ` k ) = ( A ` k ) )
24	23	oveq1d 6665	. . . . . 6 |- ( B = A -> ( ( B ` k ) x. ( z ^ k ) ) = ( ( A ` k ) x. ( z ^ k ) ) )
25	24	sumeq2sdv 14435	. . . . 5 |- ( B = A -> sum_ k e. ( 0 ... M ) ( ( B ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( z ^ k ) ) )
26	22, 25	syl 17	. . . 4 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 0 ... M ) ( ( B ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( z ^ k ) ) )
27	3	adantr 481	. . . . . . . . 9 |- ( ( ph /\ z e. CC ) -> F e. (Poly ` S ) )
28		dgrub.2	. . . . . . . . . 10 |- N = (deg ` F )
29		dgrcl 23989	. . . . . . . . . 10 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
30	28, 29	syl5eqel 2705	. . . . . . . . 9 |- ( F e. (Poly ` S ) -> N e. NN0 )
31	27, 30	syl 17	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> N e. NN0 )
32	31	nn0zd 11480	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> N e. ZZ )
33	4	adantr 481	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> M e. NN0 )
34	33	nn0zd 11480	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> M e. ZZ )
35	22	imaeq1d 5465	. . . . . . . . 9 |- ( ( ph /\ z e. CC ) -> ( B " ( ZZ>= ` ( M + 1 ) ) ) = ( A " ( ZZ>= ` ( M + 1 ) ) ) )
36	19	adantr 481	. . . . . . . . 9 |- ( ( ph /\ z e. CC ) -> ( B " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )
37	35, 36	eqtr3d 2658	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )
38	2, 28	dgrlb 23992	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } ) -> N <_ M )
39	27, 33, 37, 38	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> N <_ M )
40		eluz2 11693	. . . . . . 7 |- ( M e. ( ZZ>= ` N ) <-> ( N e. ZZ /\ M e. ZZ /\ N <_ M ) )
41	32, 34, 39, 40	syl3anbrc 1246	. . . . . 6 |- ( ( ph /\ z e. CC ) -> M e. ( ZZ>= ` N ) )
42		fzss2 12381	. . . . . 6 |- ( M e. ( ZZ>= ` N ) -> ( 0 ... N ) C_ ( 0 ... M ) )
43	41, 42	syl 17	. . . . 5 |- ( ( ph /\ z e. CC ) -> ( 0 ... N ) C_ ( 0 ... M ) )
44		elfznn0 12433	. . . . . 6 |- ( k e. ( 0 ... N ) -> k e. NN0 )
45		plyssc 23956	. . . . . . . . . . 11 |- (Poly ` S ) C_ (Poly ` CC )
46	45, 3	sseldi 3601	. . . . . . . . . 10 |- ( ph -> F e. (Poly ` CC ) )
47	2	coef3 23988	. . . . . . . . . 10 |- ( F e. (Poly ` CC ) -> A : NN0 --> CC )
48	46, 47	syl 17	. . . . . . . . 9 |- ( ph -> A : NN0 --> CC )
49	48	adantr 481	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> A : NN0 --> CC )
50	49	ffvelrnda 6359	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. NN0 ) -> ( A ` k ) e. CC )
51		expcl 12878	. . . . . . . 8 |- ( ( z e. CC /\ k e. NN0 ) -> ( z ^ k ) e. CC )
52	51	adantll 750	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. NN0 ) -> ( z ^ k ) e. CC )
53	50, 52	mulcld 10060	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. NN0 ) -> ( ( A ` k ) x. ( z ^ k ) ) e. CC )
54	44, 53	sylan2 491	. . . . 5 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. ( z ^ k ) ) e. CC )
55		eldifn 3733	. . . . . . . . 9 |- ( k e. ( ( 0 ... M ) \ ( 0 ... N ) ) -> -. k e. ( 0 ... N ) )
56	55	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... M ) \ ( 0 ... N ) ) ) -> -. k e. ( 0 ... N ) )
57		eldifi 3732	. . . . . . . . . . . 12 |- ( k e. ( ( 0 ... M ) \ ( 0 ... N ) ) -> k e. ( 0 ... M ) )
58		elfznn0 12433	. . . . . . . . . . . 12 |- ( k e. ( 0 ... M ) -> k e. NN0 )
59	57, 58	syl 17	. . . . . . . . . . 11 |- ( k e. ( ( 0 ... M ) \ ( 0 ... N ) ) -> k e. NN0 )
60	2, 28	dgrub 23990	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` S ) /\ k e. NN0 /\ ( A ` k ) =/= 0 ) -> k <_ N )
61	60	3expia 1267	. . . . . . . . . . 11 |- ( ( F e. (Poly ` S ) /\ k e. NN0 ) -> ( ( A ` k ) =/= 0 -> k <_ N ) )
62	27, 59, 61	syl2an 494	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... M ) \ ( 0 ... N ) ) ) -> ( ( A ` k ) =/= 0 -> k <_ N ) )
63		elfzuz 12338	. . . . . . . . . . . 12 |- ( k e. ( 0 ... M ) -> k e. ( ZZ>= ` 0 ) )
64	57, 63	syl 17	. . . . . . . . . . 11 |- ( k e. ( ( 0 ... M ) \ ( 0 ... N ) ) -> k e. ( ZZ>= ` 0 ) )
65		elfz5 12334	. . . . . . . . . . 11 |- ( ( k e. ( ZZ>= ` 0 ) /\ N e. ZZ ) -> ( k e. ( 0 ... N ) <-> k <_ N ) )
66	64, 32, 65	syl2anr 495	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... M ) \ ( 0 ... N ) ) ) -> ( k e. ( 0 ... N ) <-> k <_ N ) )
67	62, 66	sylibrd 249	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... M ) \ ( 0 ... N ) ) ) -> ( ( A ` k ) =/= 0 -> k e. ( 0 ... N ) ) )
68	67	necon1bd 2812	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... M ) \ ( 0 ... N ) ) ) -> ( -. k e. ( 0 ... N ) -> ( A ` k ) = 0 ) )
69	56, 68	mpd 15	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... M ) \ ( 0 ... N ) ) ) -> ( A ` k ) = 0 )
70	69	oveq1d 6665	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... M ) \ ( 0 ... N ) ) ) -> ( ( A ` k ) x. ( z ^ k ) ) = ( 0 x. ( z ^ k ) ) )
71		simpr 477	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> z e. CC )
72	71, 59, 51	syl2an 494	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... M ) \ ( 0 ... N ) ) ) -> ( z ^ k ) e. CC )
73	72	mul02d 10234	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... M ) \ ( 0 ... N ) ) ) -> ( 0 x. ( z ^ k ) ) = 0 )
74	70, 73	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... M ) \ ( 0 ... N ) ) ) -> ( ( A ` k ) x. ( z ^ k ) ) = 0 )
75		fzfid 12772	. . . . 5 |- ( ( ph /\ z e. CC ) -> ( 0 ... M ) e. Fin )
76	43, 54, 74, 75	fsumss 14456	. . . 4 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( z ^ k ) ) )
77	26, 76	eqtr4d 2659	. . 3 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 0 ... M ) ( ( B ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) )
78	77	mpteq2dva 4744	. 2 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( B ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )
79	1, 78	eqtrd 2656	1 |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   ^cexp 12860   sum_csu 14416  Polycply 23940  coeffccoe 23942  degcdgr 23943

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

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-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-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947

This theorem is referenced by:  coeid  23994