Theorem plycjlemc 14996

Description: Lemma for plycj 14997. (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 11013	. . . . 5 |- ( z e. CC -> ( * ` z ) e. CC )
3	2	adantl 277	. . . 4 |- ( ( ph /\ z e. CC ) -> ( * ` z ) e. CC )
4		cjf 11012	. . . . . 6 |- * : CC --> CC
5	4	a1i 9	. . . . 5 |- ( ph -> * : CC --> CC )
6	5	feqmptd 5614	. . . 4 |- ( ph -> * = ( z e. CC |-> ( * ` z ) ) )
7		0zd 9338	. . . . . . . 8 |- ( ph -> 0 e. ZZ )
8		plycjlemc.n	. . . . . . . . 9 |- ( ph -> N e. NN0 )
9	8	nn0zd 9446	. . . . . . . 8 |- ( ph -> N e. ZZ )
10	7, 9	fzfigd 10523	. . . . . . 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 14969	. . . . . . . . . . . . 13 |- ( F e. (Poly ` S ) -> S C_ CC )
15	13, 14	syl 14	. . . . . . . . . . . 12 |- ( ph -> S C_ CC )
16		0cn 8018	. . . . . . . . . . . . 13 |- 0 e. CC
17		snssi 3766	. . . . . . . . . . . . 13 |- ( 0 e. CC -> { 0 } C_ CC )
18	16, 17	mp1i 10	. . . . . . . . . . . 12 |- ( ph -> { 0 } C_ CC )
19	15, 18	unssd 3339	. . . . . . . . . . 11 |- ( ph -> ( S u. { 0 } ) C_ CC )
20	12, 19	fssd 5420	. . . . . . . . . 10 |- ( ph -> A : NN0 --> CC )
21	20	adantr 276	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... N ) ) -> A : NN0 --> CC )
22		elfznn0 10189	. . . . . . . . . 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 5698	. . . . . . . 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 10765	. . . . . . 7 |- ( ( ( ph /\ x e. CC ) /\ k e. ( 0 ... N ) ) -> ( x ^ k ) e. CC )
29	25, 28	mulcld 8047	. . . . . 6 |- ( ( ( ph /\ x e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. ( x ^ k ) ) e. CC )
30	11, 29	fsumcl 11565	. . . . 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 5929	. . . . . . . . 9 |- ( z = x -> ( z ^ k ) = ( x ^ k ) )
33	32	oveq2d 5938	. . . . . . . 8 |- ( z = x -> ( ( A ` k ) x. ( z ^ k ) ) = ( ( A ` k ) x. ( x ^ k ) ) )
34	33	sumeq2sdv 11535	. . . . . . 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 4129	. . . . . 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 2245	. . . . 5 |- ( ph -> F = ( x e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) ) )
37		fveq2 5558	. . . . 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 5728	. . . 4 |- ( ph -> ( * o. F ) = ( x e. CC |-> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) ) ) )
39		oveq1 5929	. . . . . . 7 |- ( x = ( * ` z ) -> ( x ^ k ) = ( ( * ` z ) ^ k ) )
40	39	oveq2d 5938	. . . . . 6 |- ( x = ( * ` z ) -> ( ( A ` k ) x. ( x ^ k ) ) = ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) )
41	40	sumeq2sdv 11535	. . . . 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 5562	. . . 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 5728	. . 3 |- ( ph -> ( ( * o. F ) o. * ) = ( z e. CC |-> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) ) )
44	1, 43	eqtrid 2241	. 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 10765	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( * ` z ) ^ k ) e. CC )
50	46, 49	mulcld 8047	. . . . 5 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) e. CC )
51	45, 50	fsumcj 11639	. . . 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 11131	. . . . . 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 5632	. . . . . . . 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 11123	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( ( * ` z ) ^ k ) ) = ( ( * ` ( * ` z ) ) ^ k ) )
57		cjcj 11048	. . . . . . . . . 10 |- ( z e. CC -> ( * ` ( * ` z ) ) = z )
58	57	ad2antlr 489	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( * ` z ) ) = z )
59	58	oveq1d 5937	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( * ` ( * ` z ) ) ^ k ) = ( z ^ k ) )
60	56, 59	eqtr2d 2230	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( z ^ k ) = ( * ` ( ( * ` z ) ^ k ) ) )
61	55, 60	oveq12d 5940	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) = ( ( * ` ( A ` k ) ) x. ( * ` ( ( * ` z ) ^ k ) ) ) )
62	52, 61	eqtr4d 2232	. . . . 5 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) = ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) )
63	62	sumeq2dv 11533	. . . 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 2229	. . 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 4123	. 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 2229	1 |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   u. cun 3155   C_ wss 3157   {csn 3622   |-> cmpt 4094   o. ccom 4667   -->wf 5254   ` cfv 5258  (class class class)co 5922   Fincfn 6799   CCcc 7877   0cc0 7879   x. cmul 7884   NN0cn0 9249   ...cfz 10083   ^cexp 10630   *ccj 11004   sum_csu 11518  Polycply 14964

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  df-ply 14966

This theorem is referenced by:  plycj  14997