Theorem plycoeid3 14993

Description: Reconstruct a polynomial as an explicit sum of the coefficient function up to an index no smaller than the degree of the polynomial. (Contributed by Jim Kingdon, 17-Oct-2025.)

Hypotheses

Ref	Expression
plycoeid3.d	|- ( ph -> D e. NN0 )
plycoeid3.a	|- ( ph -> A : NN0 --> CC )
plycoeid3.z	|- ( ph -> ( A " ( ZZ>= ` ( D + 1 ) ) ) = { 0 } )
plycoeid3.f	|- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( z ^ k ) ) ) )
plycoeid3.m	|- ( ph -> M e. ( ZZ>= ` D ) )
plycoeid3.x	|- ( ph -> X e. CC )

Assertion

Ref	Expression
plycoeid3	|- ( ph -> ( F ` X ) = sum_ j e. ( 0 ... M ) ( ( A ` j ) x. ( X ^ j ) ) )

Distinct variable groups:   A, j, z   A, k, z   D, k, z   j, M   k, M   j, X, z   k, X

Allowed substitution hints:   ph( z, j, k)   D( j)   F( z, j, k)   M( z)

Proof of Theorem plycoeid3

Dummy variables r q p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plycoeid3.f	. . . . . 6 |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( z ^ k ) ) ) )
2	1	fveq1d 5560	. . . . 5 |- ( ph -> ( F ` X ) = ( ( z e. CC |-> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( z ^ k ) ) ) ` X ) )
3		eqid 2196	. . . . . 6 |- ( z e. CC |-> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( z ^ k ) ) )
4		oveq1 5929	. . . . . . . 8 |- ( z = X -> ( z ^ k ) = ( X ^ k ) )
5	4	oveq2d 5938	. . . . . . 7 |- ( z = X -> ( ( A ` k ) x. ( z ^ k ) ) = ( ( A ` k ) x. ( X ^ k ) ) )
6	5	sumeq2sdv 11535	. . . . . 6 |- ( z = X -> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( X ^ k ) ) )
7		plycoeid3.x	. . . . . 6 |- ( ph -> X e. CC )
8		fveq2 5558	. . . . . . . . 9 |- ( q = k -> ( A ` q ) = ( A ` k ) )
9		oveq2 5930	. . . . . . . . 9 |- ( q = k -> ( X ^ q ) = ( X ^ k ) )
10	8, 9	oveq12d 5940	. . . . . . . 8 |- ( q = k -> ( ( A ` q ) x. ( X ^ q ) ) = ( ( A ` k ) x. ( X ^ k ) ) )
11	10	cbvsumv 11526	. . . . . . 7 |- sum_ q e. ( 0 ... D ) ( ( A ` q ) x. ( X ^ q ) ) = sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( X ^ k ) )
12		0zd 9338	. . . . . . . . 9 |- ( ph -> 0 e. ZZ )
13		plycoeid3.d	. . . . . . . . . 10 |- ( ph -> D e. NN0 )
14	13	nn0zd 9446	. . . . . . . . 9 |- ( ph -> D e. ZZ )
15	12, 14	fzfigd 10523	. . . . . . . 8 |- ( ph -> ( 0 ... D ) e. Fin )
16		plycoeid3.a	. . . . . . . . . . 11 |- ( ph -> A : NN0 --> CC )
17	16	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ q e. ( 0 ... D ) ) -> A : NN0 --> CC )
18		elfznn0 10189	. . . . . . . . . . 11 |- ( q e. ( 0 ... D ) -> q e. NN0 )
19	18	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ q e. ( 0 ... D ) ) -> q e. NN0 )
20	17, 19	ffvelcdmd 5698	. . . . . . . . 9 |- ( ( ph /\ q e. ( 0 ... D ) ) -> ( A ` q ) e. CC )
21	7	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ q e. ( 0 ... D ) ) -> X e. CC )
22	21, 19	expcld 10765	. . . . . . . . 9 |- ( ( ph /\ q e. ( 0 ... D ) ) -> ( X ^ q ) e. CC )
23	20, 22	mulcld 8047	. . . . . . . 8 |- ( ( ph /\ q e. ( 0 ... D ) ) -> ( ( A ` q ) x. ( X ^ q ) ) e. CC )
24	15, 23	fsumcl 11565	. . . . . . 7 |- ( ph -> sum_ q e. ( 0 ... D ) ( ( A ` q ) x. ( X ^ q ) ) e. CC )
25	11, 24	eqeltrrid 2284	. . . . . 6 |- ( ph -> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( X ^ k ) ) e. CC )
26	3, 6, 7, 25	fvmptd3 5655	. . . . 5 |- ( ph -> ( ( z e. CC |-> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( z ^ k ) ) ) ` X ) = sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( X ^ k ) ) )
27	2, 26	eqtrd 2229	. . . 4 |- ( ph -> ( F ` X ) = sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( X ^ k ) ) )
28		fveq2 5558	. . . . . 6 |- ( k = r -> ( A ` k ) = ( A ` r ) )
29		oveq2 5930	. . . . . 6 |- ( k = r -> ( X ^ k ) = ( X ^ r ) )
30	28, 29	oveq12d 5940	. . . . 5 |- ( k = r -> ( ( A ` k ) x. ( X ^ k ) ) = ( ( A ` r ) x. ( X ^ r ) ) )
31	30	cbvsumv 11526	. . . 4 |- sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( X ^ k ) ) = sum_ r e. ( 0 ... D ) ( ( A ` r ) x. ( X ^ r ) )
32	27, 31	eqtrdi 2245	. . 3 |- ( ph -> ( F ` X ) = sum_ r e. ( 0 ... D ) ( ( A ` r ) x. ( X ^ r ) ) )
33		plycoeid3.m	. . . . 5 |- ( ph -> M e. ( ZZ>= ` D ) )
34		fzss2 10139	. . . . 5 |- ( M e. ( ZZ>= ` D ) -> ( 0 ... D ) C_ ( 0 ... M ) )
35	33, 34	syl 14	. . . 4 |- ( ph -> ( 0 ... D ) C_ ( 0 ... M ) )
36	16	adantr 276	. . . . . 6 |- ( ( ph /\ r e. ( 0 ... D ) ) -> A : NN0 --> CC )
37		elfznn0 10189	. . . . . . 7 |- ( r e. ( 0 ... D ) -> r e. NN0 )
38	37	adantl 277	. . . . . 6 |- ( ( ph /\ r e. ( 0 ... D ) ) -> r e. NN0 )
39	36, 38	ffvelcdmd 5698	. . . . 5 |- ( ( ph /\ r e. ( 0 ... D ) ) -> ( A ` r ) e. CC )
40	7	adantr 276	. . . . . 6 |- ( ( ph /\ r e. ( 0 ... D ) ) -> X e. CC )
41	40, 38	expcld 10765	. . . . 5 |- ( ( ph /\ r e. ( 0 ... D ) ) -> ( X ^ r ) e. CC )
42	39, 41	mulcld 8047	. . . 4 |- ( ( ph /\ r e. ( 0 ... D ) ) -> ( ( A ` r ) x. ( X ^ r ) ) e. CC )
43		eldifn 3286	. . . . . . . . . 10 |- ( r e. ( ( 0 ... M ) \ ( 0 ... D ) ) -> -. r e. ( 0 ... D ) )
44	43	adantl 277	. . . . . . . . 9 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> -. r e. ( 0 ... D ) )
45		eldifi 3285	. . . . . . . . . . . . . 14 |- ( r e. ( ( 0 ... M ) \ ( 0 ... D ) ) -> r e. ( 0 ... M ) )
46	45	adantl 277	. . . . . . . . . . . . 13 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> r e. ( 0 ... M ) )
47		elfznn0 10189	. . . . . . . . . . . . 13 |- ( r e. ( 0 ... M ) -> r e. NN0 )
48	46, 47	syl 14	. . . . . . . . . . . 12 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> r e. NN0 )
49		nn0split 10211	. . . . . . . . . . . . . 14 |- ( D e. NN0 -> NN0 = ( ( 0 ... D ) u. ( ZZ>= ` ( D + 1 ) ) ) )
50	13, 49	syl 14	. . . . . . . . . . . . 13 |- ( ph -> NN0 = ( ( 0 ... D ) u. ( ZZ>= ` ( D + 1 ) ) ) )
51	50	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> NN0 = ( ( 0 ... D ) u. ( ZZ>= ` ( D + 1 ) ) ) )
52	48, 51	eleqtrd 2275	. . . . . . . . . . 11 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> r e. ( ( 0 ... D ) u. ( ZZ>= ` ( D + 1 ) ) ) )
53		elun 3304	. . . . . . . . . . 11 |- ( r e. ( ( 0 ... D ) u. ( ZZ>= ` ( D + 1 ) ) ) <-> ( r e. ( 0 ... D ) \/ r e. ( ZZ>= ` ( D + 1 ) ) ) )
54	52, 53	sylib 122	. . . . . . . . . 10 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( r e. ( 0 ... D ) \/ r e. ( ZZ>= ` ( D + 1 ) ) ) )
55	54	orcomd 730	. . . . . . . . 9 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( r e. ( ZZ>= ` ( D + 1 ) ) \/ r e. ( 0 ... D ) ) )
56	44, 55	ecased 1360	. . . . . . . 8 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> r e. ( ZZ>= ` ( D + 1 ) ) )
57		plycoeid3.z	. . . . . . . . . . 11 |- ( ph -> ( A " ( ZZ>= ` ( D + 1 ) ) ) = { 0 } )
58		eqimss 3237	. . . . . . . . . . 11 |- ( ( A " ( ZZ>= ` ( D + 1 ) ) ) = { 0 } -> ( A " ( ZZ>= ` ( D + 1 ) ) ) C_ { 0 } )
59	57, 58	syl 14	. . . . . . . . . 10 |- ( ph -> ( A " ( ZZ>= ` ( D + 1 ) ) ) C_ { 0 } )
60	16	ffund 5411	. . . . . . . . . . 11 |- ( ph -> Fun A )
61		peano2nn0 9289	. . . . . . . . . . . . . . . 16 |- ( D e. NN0 -> ( D + 1 ) e. NN0 )
62	13, 61	syl 14	. . . . . . . . . . . . . . 15 |- ( ph -> ( D + 1 ) e. NN0 )
63		nn0uz 9636	. . . . . . . . . . . . . . 15 |- NN0 = ( ZZ>= ` 0 )
64	62, 63	eleqtrdi 2289	. . . . . . . . . . . . . 14 |- ( ph -> ( D + 1 ) e. ( ZZ>= ` 0 ) )
65		uzss 9622	. . . . . . . . . . . . . 14 |- ( ( D + 1 ) e. ( ZZ>= ` 0 ) -> ( ZZ>= ` ( D + 1 ) ) C_ ( ZZ>= ` 0 ) )
66	64, 65	syl 14	. . . . . . . . . . . . 13 |- ( ph -> ( ZZ>= ` ( D + 1 ) ) C_ ( ZZ>= ` 0 ) )
67	66, 63	sseqtrrdi 3232	. . . . . . . . . . . 12 |- ( ph -> ( ZZ>= ` ( D + 1 ) ) C_ NN0 )
68	16	fdmd 5414	. . . . . . . . . . . 12 |- ( ph -> dom A = NN0 )
69	67, 68	sseqtrrd 3222	. . . . . . . . . . 11 |- ( ph -> ( ZZ>= ` ( D + 1 ) ) C_ dom A )
70		funimass4 5611	. . . . . . . . . . 11 |- ( ( Fun A /\ ( ZZ>= ` ( D + 1 ) ) C_ dom A ) -> ( ( A " ( ZZ>= ` ( D + 1 ) ) ) C_ { 0 } <-> A. r e. ( ZZ>= ` ( D + 1 ) ) ( A ` r ) e. { 0 } ) )
71	60, 69, 70	syl2anc 411	. . . . . . . . . 10 |- ( ph -> ( ( A " ( ZZ>= ` ( D + 1 ) ) ) C_ { 0 } <-> A. r e. ( ZZ>= ` ( D + 1 ) ) ( A ` r ) e. { 0 } ) )
72	59, 71	mpbid 147	. . . . . . . . 9 |- ( ph -> A. r e. ( ZZ>= ` ( D + 1 ) ) ( A ` r ) e. { 0 } )
73	72	r19.21bi 2585	. . . . . . . 8 |- ( ( ph /\ r e. ( ZZ>= ` ( D + 1 ) ) ) -> ( A ` r ) e. { 0 } )
74	56, 73	syldan 282	. . . . . . 7 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( A ` r ) e. { 0 } )
75		elsni 3640	. . . . . . 7 |- ( ( A ` r ) e. { 0 } -> ( A ` r ) = 0 )
76	74, 75	syl 14	. . . . . 6 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( A ` r ) = 0 )
77	76	oveq1d 5937	. . . . 5 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( ( A ` r ) x. ( X ^ r ) ) = ( 0 x. ( X ^ r ) ) )
78	7	adantr 276	. . . . . . 7 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> X e. CC )
79	78, 48	expcld 10765	. . . . . 6 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( X ^ r ) e. CC )
80	79	mul02d 8418	. . . . 5 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( 0 x. ( X ^ r ) ) = 0 )
81	77, 80	eqtrd 2229	. . . 4 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( ( A ` r ) x. ( X ^ r ) ) = 0 )
82		elfzelz 10100	. . . . . . 7 |- ( p e. ( 0 ... M ) -> p e. ZZ )
83	82	adantl 277	. . . . . 6 |- ( ( ph /\ p e. ( 0 ... M ) ) -> p e. ZZ )
84		0zd 9338	. . . . . 6 |- ( ( ph /\ p e. ( 0 ... M ) ) -> 0 e. ZZ )
85	14	adantr 276	. . . . . 6 |- ( ( ph /\ p e. ( 0 ... M ) ) -> D e. ZZ )
86		fzdcel 10115	. . . . . 6 |- ( ( p e. ZZ /\ 0 e. ZZ /\ D e. ZZ ) -> DECID p e. ( 0 ... D ) )
87	83, 84, 85, 86	syl3anc 1249	. . . . 5 |- ( ( ph /\ p e. ( 0 ... M ) ) -> DECID p e. ( 0 ... D ) )
88	87	ralrimiva 2570	. . . 4 |- ( ph -> A. p e. ( 0 ... M )DECID p e. ( 0 ... D ) )
89		eluzelz 9610	. . . . . 6 |- ( M e. ( ZZ>= ` D ) -> M e. ZZ )
90	33, 89	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
91	12, 90	fzfigd 10523	. . . 4 |- ( ph -> ( 0 ... M ) e. Fin )
92	35, 42, 81, 88, 91	fisumss 11557	. . 3 |- ( ph -> sum_ r e. ( 0 ... D ) ( ( A ` r ) x. ( X ^ r ) ) = sum_ r e. ( 0 ... M ) ( ( A ` r ) x. ( X ^ r ) ) )
93	32, 92	eqtrd 2229	. 2 |- ( ph -> ( F ` X ) = sum_ r e. ( 0 ... M ) ( ( A ` r ) x. ( X ^ r ) ) )
94		fveq2 5558	. . . 4 |- ( r = j -> ( A ` r ) = ( A ` j ) )
95		oveq2 5930	. . . 4 |- ( r = j -> ( X ^ r ) = ( X ^ j ) )
96	94, 95	oveq12d 5940	. . 3 |- ( r = j -> ( ( A ` r ) x. ( X ^ r ) ) = ( ( A ` j ) x. ( X ^ j ) ) )
97	96	cbvsumv 11526	. 2 |- sum_ r e. ( 0 ... M ) ( ( A ` r ) x. ( X ^ r ) ) = sum_ j e. ( 0 ... M ) ( ( A ` j ) x. ( X ^ j ) )
98	93, 97	eqtrdi 2245	1 |- ( ph -> ( F ` X ) = sum_ j e. ( 0 ... M ) ( ( A ` j ) x. ( X ^ j ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2167   A.wral 2475   \ cdif 3154   u. cun 3155   C_ wss 3157   {csn 3622   |-> cmpt 4094   dom cdm 4663   "cima 4666   Fun wfun 5252   -->wf 5254   ` cfv 5258  (class class class)co 5922   CCcc 7877   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083   ^cexp 10630   sum_csu 11518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-sumdc 11519

This theorem is referenced by:  dvply2g  15002