Theorem plycoeid3 15484

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 5641	. . . . 5 |- ( ph -> ( F ` X ) = ( ( z e. CC |-> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( z ^ k ) ) ) ` X ) )
3		eqid 2231	. . . . . 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 6025	. . . . . . . 8 |- ( z = X -> ( z ^ k ) = ( X ^ k ) )
5	4	oveq2d 6034	. . . . . . 7 |- ( z = X -> ( ( A ` k ) x. ( z ^ k ) ) = ( ( A ` k ) x. ( X ^ k ) ) )
6	5	sumeq2sdv 11932	. . . . . 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 5639	. . . . . . . . 9 |- ( q = k -> ( A ` q ) = ( A ` k ) )
9		oveq2 6026	. . . . . . . . 9 |- ( q = k -> ( X ^ q ) = ( X ^ k ) )
10	8, 9	oveq12d 6036	. . . . . . . 8 |- ( q = k -> ( ( A ` q ) x. ( X ^ q ) ) = ( ( A ` k ) x. ( X ^ k ) ) )
11	10	cbvsumv 11923	. . . . . . 7 |- sum_ q e. ( 0 ... D ) ( ( A ` q ) x. ( X ^ q ) ) = sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( X ^ k ) )
12		0zd 9491	. . . . . . . . 9 |- ( ph -> 0 e. ZZ )
13		plycoeid3.d	. . . . . . . . . 10 |- ( ph -> D e. NN0 )
14	13	nn0zd 9600	. . . . . . . . 9 |- ( ph -> D e. ZZ )
15	12, 14	fzfigd 10694	. . . . . . . 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 10349	. . . . . . . . . . 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 5783	. . . . . . . . 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 10936	. . . . . . . . 9 |- ( ( ph /\ q e. ( 0 ... D ) ) -> ( X ^ q ) e. CC )
23	20, 22	mulcld 8200	. . . . . . . 8 |- ( ( ph /\ q e. ( 0 ... D ) ) -> ( ( A ` q ) x. ( X ^ q ) ) e. CC )
24	15, 23	fsumcl 11963	. . . . . . 7 |- ( ph -> sum_ q e. ( 0 ... D ) ( ( A ` q ) x. ( X ^ q ) ) e. CC )
25	11, 24	eqeltrrid 2319	. . . . . 6 |- ( ph -> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( X ^ k ) ) e. CC )
26	3, 6, 7, 25	fvmptd3 5740	. . . . 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 2264	. . . 4 |- ( ph -> ( F ` X ) = sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( X ^ k ) ) )
28		fveq2 5639	. . . . . 6 |- ( k = r -> ( A ` k ) = ( A ` r ) )
29		oveq2 6026	. . . . . 6 |- ( k = r -> ( X ^ k ) = ( X ^ r ) )
30	28, 29	oveq12d 6036	. . . . 5 |- ( k = r -> ( ( A ` k ) x. ( X ^ k ) ) = ( ( A ` r ) x. ( X ^ r ) ) )
31	30	cbvsumv 11923	. . . 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 2280	. . 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 10299	. . . . 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 10349	. . . . . . 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 5783	. . . . 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 10936	. . . . 5 |- ( ( ph /\ r e. ( 0 ... D ) ) -> ( X ^ r ) e. CC )
42	39, 41	mulcld 8200	. . . 4 |- ( ( ph /\ r e. ( 0 ... D ) ) -> ( ( A ` r ) x. ( X ^ r ) ) e. CC )
43		eldifn 3330	. . . . . . . . . 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 3329	. . . . . . . . . . . . . 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 10349	. . . . . . . . . . . . 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 10371	. . . . . . . . . . . . . 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 2310	. . . . . . . . . . 11 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> r e. ( ( 0 ... D ) u. ( ZZ>= ` ( D + 1 ) ) ) )
53		elun 3348	. . . . . . . . . . 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 736	. . . . . . . . 9 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( r e. ( ZZ>= ` ( D + 1 ) ) \/ r e. ( 0 ... D ) ) )
56	44, 55	ecased 1385	. . . . . . . 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 3281	. . . . . . . . . . 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 5486	. . . . . . . . . . 11 |- ( ph -> Fun A )
61		peano2nn0 9442	. . . . . . . . . . . . . . . 16 |- ( D e. NN0 -> ( D + 1 ) e. NN0 )
62	13, 61	syl 14	. . . . . . . . . . . . . . 15 |- ( ph -> ( D + 1 ) e. NN0 )
63		nn0uz 9791	. . . . . . . . . . . . . . 15 |- NN0 = ( ZZ>= ` 0 )
64	62, 63	eleqtrdi 2324	. . . . . . . . . . . . . 14 |- ( ph -> ( D + 1 ) e. ( ZZ>= ` 0 ) )
65		uzss 9777	. . . . . . . . . . . . . 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 3276	. . . . . . . . . . . 12 |- ( ph -> ( ZZ>= ` ( D + 1 ) ) C_ NN0 )
68	16	fdmd 5489	. . . . . . . . . . . 12 |- ( ph -> dom A = NN0 )
69	67, 68	sseqtrrd 3266	. . . . . . . . . . 11 |- ( ph -> ( ZZ>= ` ( D + 1 ) ) C_ dom A )
70		funimass4 5696	. . . . . . . . . . 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 2620	. . . . . . . 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 3687	. . . . . . 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 6033	. . . . 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 10936	. . . . . 6 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( X ^ r ) e. CC )
80	79	mul02d 8571	. . . . 5 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( 0 x. ( X ^ r ) ) = 0 )
81	77, 80	eqtrd 2264	. . . 4 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( ( A ` r ) x. ( X ^ r ) ) = 0 )
82		elfzelz 10260	. . . . . . 7 |- ( p e. ( 0 ... M ) -> p e. ZZ )
83	82	adantl 277	. . . . . 6 |- ( ( ph /\ p e. ( 0 ... M ) ) -> p e. ZZ )
84		0zd 9491	. . . . . 6 |- ( ( ph /\ p e. ( 0 ... M ) ) -> 0 e. ZZ )
85	14	adantr 276	. . . . . 6 |- ( ( ph /\ p e. ( 0 ... M ) ) -> D e. ZZ )
86		fzdcel 10275	. . . . . 6 |- ( ( p e. ZZ /\ 0 e. ZZ /\ D e. ZZ ) -> DECID p e. ( 0 ... D ) )
87	83, 84, 85, 86	syl3anc 1273	. . . . 5 |- ( ( ph /\ p e. ( 0 ... M ) ) -> DECID p e. ( 0 ... D ) )
88	87	ralrimiva 2605	. . . 4 |- ( ph -> A. p e. ( 0 ... M )DECID p e. ( 0 ... D ) )
89		eluzelz 9765	. . . . . 6 |- ( M e. ( ZZ>= ` D ) -> M e. ZZ )
90	33, 89	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
91	12, 90	fzfigd 10694	. . . 4 |- ( ph -> ( 0 ... M ) e. Fin )
92	35, 42, 81, 88, 91	fisumss 11955	. . 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 2264	. 2 |- ( ph -> ( F ` X ) = sum_ r e. ( 0 ... M ) ( ( A ` r ) x. ( X ^ r ) ) )
94		fveq2 5639	. . . 4 |- ( r = j -> ( A ` r ) = ( A ` j ) )
95		oveq2 6026	. . . 4 |- ( r = j -> ( X ^ r ) = ( X ^ j ) )
96	94, 95	oveq12d 6036	. . 3 |- ( r = j -> ( ( A ` r ) x. ( X ^ r ) ) = ( ( A ` j ) x. ( X ^ j ) ) )
97	96	cbvsumv 11923	. 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 2280	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 715  DECID wdc 841   = wceq 1397   e. wcel 2202   A.wral 2510   \ cdif 3197   u. cun 3198   C_ wss 3200   {csn 3669   |-> cmpt 4150   dom cdm 4725   "cima 4728   Fun wfun 5320   -->wf 5322   ` cfv 5326  (class class class)co 6018   CCcc 8030   0cc0 8032   1c1 8033   + caddc 8035   x. cmul 8037   NN0cn0 9402   ZZcz 9479   ZZ>=cuz 9755   ...cfz 10243   ^cexp 10801   sum_csu 11915

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-seqfrec 10711  df-exp 10802  df-ihash 11039  df-cj 11404  df-re 11405  df-im 11406  df-rsqrt 11560  df-abs 11561  df-clim 11841  df-sumdc 11916

This theorem is referenced by:  dvply2g  15493