Theorem plycoeid3 15753

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 5678	. . . . 5 |- ( ph -> ( F ` X ) = ( ( z e. CC |-> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( z ^ k ) ) ) ` X ) )
3		eqid 2234	. . . . . 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 6066	. . . . . . . 8 |- ( z = X -> ( z ^ k ) = ( X ^ k ) )
5	4	oveq2d 6075	. . . . . . 7 |- ( z = X -> ( ( A ` k ) x. ( z ^ k ) ) = ( ( A ` k ) x. ( X ^ k ) ) )
6	5	sumeq2sdv 12085	. . . . . 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 5676	. . . . . . . . 9 |- ( q = k -> ( A ` q ) = ( A ` k ) )
9		oveq2 6067	. . . . . . . . 9 |- ( q = k -> ( X ^ q ) = ( X ^ k ) )
10	8, 9	oveq12d 6077	. . . . . . . 8 |- ( q = k -> ( ( A ` q ) x. ( X ^ q ) ) = ( ( A ` k ) x. ( X ^ k ) ) )
11	10	cbvsumv 12076	. . . . . . 7 |- sum_ q e. ( 0 ... D ) ( ( A ` q ) x. ( X ^ q ) ) = sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( X ^ k ) )
12		0zd 9610	. . . . . . . . 9 |- ( ph -> 0 e. ZZ )
13		plycoeid3.d	. . . . . . . . . 10 |- ( ph -> D e. NN0 )
14	13	nn0zd 9720	. . . . . . . . 9 |- ( ph -> D e. ZZ )
15	12, 14	fzfigd 10821	. . . . . . . 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 10474	. . . . . . . . . . 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 5819	. . . . . . . . 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 11064	. . . . . . . . 9 |- ( ( ph /\ q e. ( 0 ... D ) ) -> ( X ^ q ) e. CC )
23	20, 22	mulcld 8311	. . . . . . . 8 |- ( ( ph /\ q e. ( 0 ... D ) ) -> ( ( A ` q ) x. ( X ^ q ) ) e. CC )
24	15, 23	fsumcl 12116	. . . . . . 7 |- ( ph -> sum_ q e. ( 0 ... D ) ( ( A ` q ) x. ( X ^ q ) ) e. CC )
25	11, 24	eqeltrrid 2322	. . . . . 6 |- ( ph -> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( X ^ k ) ) e. CC )
26	3, 6, 7, 25	fvmptd3 5777	. . . . 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 2267	. . . 4 |- ( ph -> ( F ` X ) = sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( X ^ k ) ) )
28		fveq2 5676	. . . . . 6 |- ( k = r -> ( A ` k ) = ( A ` r ) )
29		oveq2 6067	. . . . . 6 |- ( k = r -> ( X ^ k ) = ( X ^ r ) )
30	28, 29	oveq12d 6077	. . . . 5 |- ( k = r -> ( ( A ` k ) x. ( X ^ k ) ) = ( ( A ` r ) x. ( X ^ r ) ) )
31	30	cbvsumv 12076	. . . 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 2283	. . 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 10423	. . . . 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 10474	. . . . . . 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 5819	. . . . 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 11064	. . . . 5 |- ( ( ph /\ r e. ( 0 ... D ) ) -> ( X ^ r ) e. CC )
42	39, 41	mulcld 8311	. . . 4 |- ( ( ph /\ r e. ( 0 ... D ) ) -> ( ( A ` r ) x. ( X ^ r ) ) e. CC )
43		eldifn 3346	. . . . . . . . . 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 3345	. . . . . . . . . . . . . 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 10474	. . . . . . . . . . . . 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 10496	. . . . . . . . . . . . . 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 2313	. . . . . . . . . . 11 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> r e. ( ( 0 ... D ) u. ( ZZ>= ` ( D + 1 ) ) ) )
53		elun 3364	. . . . . . . . . . 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 737	. . . . . . . . 9 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( r e. ( ZZ>= ` ( D + 1 ) ) \/ r e. ( 0 ... D ) ) )
56	44, 55	ecased 1386	. . . . . . . 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 3296	. . . . . . . . . . 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 5518	. . . . . . . . . . 11 |- ( ph -> Fun A )
61		peano2nn0 9557	. . . . . . . . . . . . . . . 16 |- ( D e. NN0 -> ( D + 1 ) e. NN0 )
62	13, 61	syl 14	. . . . . . . . . . . . . . 15 |- ( ph -> ( D + 1 ) e. NN0 )
63		nn0uz 9911	. . . . . . . . . . . . . . 15 |- NN0 = ( ZZ>= ` 0 )
64	62, 63	eleqtrdi 2327	. . . . . . . . . . . . . 14 |- ( ph -> ( D + 1 ) e. ( ZZ>= ` 0 ) )
65		uzss 9897	. . . . . . . . . . . . . 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 3291	. . . . . . . . . . . 12 |- ( ph -> ( ZZ>= ` ( D + 1 ) ) C_ NN0 )
68	16	fdmd 5521	. . . . . . . . . . . 12 |- ( ph -> dom A = NN0 )
69	67, 68	sseqtrrd 3281	. . . . . . . . . . 11 |- ( ph -> ( ZZ>= ` ( D + 1 ) ) C_ dom A )
70		funimass4 5733	. . . . . . . . . . 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 2632	. . . . . . . 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 3713	. . . . . . 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 6074	. . . . 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 11064	. . . . . 6 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( X ^ r ) e. CC )
80	79	mul02d 8684	. . . . 5 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( 0 x. ( X ^ r ) ) = 0 )
81	77, 80	eqtrd 2267	. . . 4 |- ( ( ph /\ r e. ( ( 0 ... M ) \ ( 0 ... D ) ) ) -> ( ( A ` r ) x. ( X ^ r ) ) = 0 )
82		elfzelz 10382	. . . . . . 7 |- ( p e. ( 0 ... M ) -> p e. ZZ )
83	82	adantl 277	. . . . . 6 |- ( ( ph /\ p e. ( 0 ... M ) ) -> p e. ZZ )
84		0zd 9610	. . . . . 6 |- ( ( ph /\ p e. ( 0 ... M ) ) -> 0 e. ZZ )
85	14	adantr 276	. . . . . 6 |- ( ( ph /\ p e. ( 0 ... M ) ) -> D e. ZZ )
86		fzdcel 10398	. . . . . 6 |- ( ( p e. ZZ /\ 0 e. ZZ /\ D e. ZZ ) -> DECID p e. ( 0 ... D ) )
87	83, 84, 85, 86	syl3anc 1274	. . . . 5 |- ( ( ph /\ p e. ( 0 ... M ) ) -> DECID p e. ( 0 ... D ) )
88	87	ralrimiva 2617	. . . 4 |- ( ph -> A. p e. ( 0 ... M )DECID p e. ( 0 ... D ) )
89		eluzelz 9885	. . . . . 6 |- ( M e. ( ZZ>= ` D ) -> M e. ZZ )
90	33, 89	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
91	12, 90	fzfigd 10821	. . . 4 |- ( ph -> ( 0 ... M ) e. Fin )
92	35, 42, 81, 88, 91	fisumss 12108	. . 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 2267	. 2 |- ( ph -> ( F ` X ) = sum_ r e. ( 0 ... M ) ( ( A ` r ) x. ( X ^ r ) ) )
94		fveq2 5676	. . . 4 |- ( r = j -> ( A ` r ) = ( A ` j ) )
95		oveq2 6067	. . . 4 |- ( r = j -> ( X ^ r ) = ( X ^ j ) )
96	94, 95	oveq12d 6077	. . 3 |- ( r = j -> ( ( A ` r ) x. ( X ^ r ) ) = ( ( A ` j ) x. ( X ^ j ) ) )
97	96	cbvsumv 12076	. 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 2283	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 716  DECID wdc 842   = wceq 1398   e. wcel 2205   A.wral 2522   \ cdif 3211   u. cun 3212   C_ wss 3214   {csn 3695   |-> cmpt 4177   dom cdm 4755   "cima 4758   Fun wfun 5352   -->wf 5354   ` cfv 5358  (class class class)co 6059   CCcc 8142   0cc0 8144   1c1 8145   + caddc 8147   x. cmul 8149   NN0cn0 9517   ZZcz 9598   ZZ>=cuz 9875   ...cfz 10365   ^cexp 10928   sum_csu 12068

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-iinf 4716  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-mulrcl 8243  ax-addcom 8244  ax-mulcom 8245  ax-addass 8246  ax-mulass 8247  ax-distr 8248  ax-i2m1 8249  ax-0lt1 8250  ax-1rid 8251  ax-0id 8252  ax-rnegex 8253  ax-precex 8254  ax-cnre 8255  ax-pre-ltirr 8256  ax-pre-ltwlin 8257  ax-pre-lttrn 8258  ax-pre-apti 8259  ax-pre-ltadd 8260  ax-pre-mulgt0 8261  ax-pre-mulext 8262  ax-arch 8263  ax-caucvg 8264

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4719  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1st 6348  df-2nd 6349  df-recs 6550  df-irdg 6615  df-frec 6636  df-1o 6661  df-oadd 6665  df-er 6781  df-en 6990  df-dom 6991  df-fin 6992  df-pnf 8327  df-mnf 8328  df-xr 8329  df-ltxr 8330  df-le 8331  df-sub 8464  df-neg 8465  df-reap 8868  df-ap 8875  df-div 8968  df-inn 9259  df-2 9317  df-3 9318  df-4 9319  df-n0 9518  df-z 9599  df-uz 9876  df-q 9974  df-rp 10009  df-fz 10366  df-fzo 10503  df-seqfrec 10838  df-exp 10929  df-ihash 11168  df-cj 11556  df-re 11557  df-im 11558  df-rsqrt 11713  df-abs 11714  df-clim 11994  df-sumdc 12069

This theorem is referenced by:  dvply2g  15762