Theorem plycjlemc 15276

Description: Lemma for plycj 15277. (Contributed by Mario Carneiro, 24-Jul-2014.) (Revised by Jim Kingdon, 22-Sep-2025.)

Hypotheses

Ref	Expression
plycjlemc.n	|- ( ph -> N e. NN0 )
plycjlem.2	|- G = ( ( * o. F ) o. * )
plycjlemc.a	|- ( ph -> A : NN0 --> ( S u. { 0 } ) )
plycjlemc.f	|- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )
plycjlemc.p	|- ( ph -> F e. (Poly ` S ) )

Assertion

Ref	Expression
plycjlemc	|- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) ) )

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

Allowed substitution hints:   G( z, k)

Proof of Theorem plycjlemc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		plycjlem.2	. . 3 |- G = ( ( * o. F ) o. * )
2		cjcl 11203	. . . . 5 |- ( z e. CC -> ( * ` z ) e. CC )
3	2	adantl 277	. . . 4 |- ( ( ph /\ z e. CC ) -> ( * ` z ) e. CC )
4		cjf 11202	. . . . . 6 |- * : CC --> CC
5	4	a1i 9	. . . . 5 |- ( ph -> * : CC --> CC )
6	5	feqmptd 5639	. . . 4 |- ( ph -> * = ( z e. CC |-> ( * ` z ) ) )
7		0zd 9391	. . . . . . . 8 |- ( ph -> 0 e. ZZ )
8		plycjlemc.n	. . . . . . . . 9 |- ( ph -> N e. NN0 )
9	8	nn0zd 9500	. . . . . . . 8 |- ( ph -> N e. ZZ )
10	7, 9	fzfigd 10583	. . . . . . 7 |- ( ph -> ( 0 ... N ) e. Fin )
11	10	adantr 276	. . . . . 6 |- ( ( ph /\ x e. CC ) -> ( 0 ... N ) e. Fin )
12		plycjlemc.a	. . . . . . . . . . 11 |- ( ph -> A : NN0 --> ( S u. { 0 } ) )
13		plycjlemc.p	. . . . . . . . . . . . 13 |- ( ph -> F e. (Poly ` S ) )
14		plybss 15249	. . . . . . . . . . . . 13 |- ( F e. (Poly ` S ) -> S C_ CC )
15	13, 14	syl 14	. . . . . . . . . . . 12 |- ( ph -> S C_ CC )
16		0cn 8071	. . . . . . . . . . . . 13 |- 0 e. CC
17		snssi 3779	. . . . . . . . . . . . 13 |- ( 0 e. CC -> { 0 } C_ CC )
18	16, 17	mp1i 10	. . . . . . . . . . . 12 |- ( ph -> { 0 } C_ CC )
19	15, 18	unssd 3350	. . . . . . . . . . 11 |- ( ph -> ( S u. { 0 } ) C_ CC )
20	12, 19	fssd 5444	. . . . . . . . . 10 |- ( ph -> A : NN0 --> CC )
21	20	adantr 276	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... N ) ) -> A : NN0 --> CC )
22		elfznn0 10243	. . . . . . . . . 10 |- ( k e. ( 0 ... N ) -> k e. NN0 )
23	22	adantl 277	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... N ) ) -> k e. NN0 )
24	21, 23	ffvelcdmd 5723	. . . . . . . 8 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
25	24	adantlr 477	. . . . . . 7 |- ( ( ( ph /\ x e. CC ) /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
26		simplr 528	. . . . . . . 8 |- ( ( ( ph /\ x e. CC ) /\ k e. ( 0 ... N ) ) -> x e. CC )
27	22	adantl 277	. . . . . . . 8 |- ( ( ( ph /\ x e. CC ) /\ k e. ( 0 ... N ) ) -> k e. NN0 )
28	26, 27	expcld 10825	. . . . . . 7 |- ( ( ( ph /\ x e. CC ) /\ k e. ( 0 ... N ) ) -> ( x ^ k ) e. CC )
29	25, 28	mulcld 8100	. . . . . 6 |- ( ( ( ph /\ x e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. ( x ^ k ) ) e. CC )
30	11, 29	fsumcl 11755	. . . . 5 |- ( ( ph /\ x e. CC ) -> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) e. CC )
31		plycjlemc.f	. . . . . 6 |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )
32		oveq1 5958	. . . . . . . . 9 |- ( z = x -> ( z ^ k ) = ( x ^ k ) )
33	32	oveq2d 5967	. . . . . . . 8 |- ( z = x -> ( ( A ` k ) x. ( z ^ k ) ) = ( ( A ` k ) x. ( x ^ k ) ) )
34	33	sumeq2sdv 11725	. . . . . . 7 |- ( z = x -> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) )
35	34	cbvmptv 4144	. . . . . 6 |- ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) = ( x e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) )
36	31, 35	eqtrdi 2255	. . . . 5 |- ( ph -> F = ( x e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) ) )
37		fveq2 5583	. . . . 5 |- ( z = sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) -> ( * ` z ) = ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) ) )
38	30, 36, 6, 37	fmptco 5753	. . . 4 |- ( ph -> ( * o. F ) = ( x e. CC |-> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) ) ) )
39		oveq1 5958	. . . . . . 7 |- ( x = ( * ` z ) -> ( x ^ k ) = ( ( * ` z ) ^ k ) )
40	39	oveq2d 5967	. . . . . 6 |- ( x = ( * ` z ) -> ( ( A ` k ) x. ( x ^ k ) ) = ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) )
41	40	sumeq2sdv 11725	. . . . 5 |- ( x = ( * ` z ) -> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) = sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) )
42	41	fveq2d 5587	. . . 4 |- ( x = ( * ` z ) -> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) ) = ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) )
43	3, 6, 38, 42	fmptco 5753	. . 3 |- ( ph -> ( ( * o. F ) o. * ) = ( z e. CC |-> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) ) )
44	1, 43	eqtrid 2251	. 2 |- ( ph -> G = ( z e. CC |-> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) ) )
45	10	adantr 276	. . . . 5 |- ( ( ph /\ z e. CC ) -> ( 0 ... N ) e. Fin )
46	24	adantlr 477	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
47	2	ad2antlr 489	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` z ) e. CC )
48	22	adantl 277	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> k e. NN0 )
49	47, 48	expcld 10825	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( * ` z ) ^ k ) e. CC )
50	46, 49	mulcld 8100	. . . . 5 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) e. CC )
51	45, 50	fsumcj 11829	. . . 4 |- ( ( ph /\ z e. CC ) -> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) = sum_ k e. ( 0 ... N ) ( * ` ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) )
52	46, 49	cjmuld 11321	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) = ( ( * ` ( A ` k ) ) x. ( * ` ( ( * ` z ) ^ k ) ) ) )
53	21	adantlr 477	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> A : NN0 --> CC )
54		fvco3 5657	. . . . . . . 8 |- ( ( A : NN0 --> CC /\ k e. NN0 ) -> ( ( * o. A ) ` k ) = ( * ` ( A ` k ) ) )
55	53, 48, 54	syl2anc 411	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( * o. A ) ` k ) = ( * ` ( A ` k ) ) )
56	47, 48	cjexpd 11313	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( ( * ` z ) ^ k ) ) = ( ( * ` ( * ` z ) ) ^ k ) )
57		cjcj 11238	. . . . . . . . . 10 |- ( z e. CC -> ( * ` ( * ` z ) ) = z )
58	57	ad2antlr 489	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( * ` z ) ) = z )
59	58	oveq1d 5966	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( * ` ( * ` z ) ) ^ k ) = ( z ^ k ) )
60	56, 59	eqtr2d 2240	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( z ^ k ) = ( * ` ( ( * ` z ) ^ k ) ) )
61	55, 60	oveq12d 5969	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) = ( ( * ` ( A ` k ) ) x. ( * ` ( ( * ` z ) ^ k ) ) ) )
62	52, 61	eqtr4d 2242	. . . . 5 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) = ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) )
63	62	sumeq2dv 11723	. . . 4 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 0 ... N ) ( * ` ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) = sum_ k e. ( 0 ... N ) ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) )
64	51, 63	eqtrd 2239	. . 3 |- ( ( ph /\ z e. CC ) -> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) = sum_ k e. ( 0 ... N ) ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) )
65	64	mpteq2dva 4138	. 2 |- ( ph -> ( z e. CC |-> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) ) )
66	44, 65	eqtrd 2239	1 |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   u. cun 3165   C_ wss 3167   {csn 3634   |-> cmpt 4109   o. ccom 4683   -->wf 5272   ` cfv 5276  (class class class)co 5951   Fincfn 6834   CCcc 7930   0cc0 7932   x. cmul 7937   NN0cn0 9302   ...cfz 10137   ^cexp 10690   *ccj 11194   sum_csu 11708  Polycply 15244

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-frec 6484  df-1o 6509  df-oadd 6513  df-er 6627  df-en 6835  df-dom 6836  df-fin 6837  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-fz 10138  df-fzo 10272  df-seqfrec 10600  df-exp 10691  df-ihash 10928  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-clim 11634  df-sumdc 11709  df-ply 15246

This theorem is referenced by:  plycj  15277