Theorem plycjlemc 15551

Description: Lemma for plycj 15552. (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 11469	. . . . 5 |- ( z e. CC -> ( * ` z ) e. CC )
3	2	adantl 277	. . . 4 |- ( ( ph /\ z e. CC ) -> ( * ` z ) e. CC )
4		cjf 11468	. . . . . 6 |- * : CC --> CC
5	4	a1i 9	. . . . 5 |- ( ph -> * : CC --> CC )
6	5	feqmptd 5708	. . . 4 |- ( ph -> * = ( z e. CC |-> ( * ` z ) ) )
7		0zd 9534	. . . . . . . 8 |- ( ph -> 0 e. ZZ )
8		plycjlemc.n	. . . . . . . . 9 |- ( ph -> N e. NN0 )
9	8	nn0zd 9643	. . . . . . . 8 |- ( ph -> N e. ZZ )
10	7, 9	fzfigd 10737	. . . . . . 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 15524	. . . . . . . . . . . . 13 |- ( F e. (Poly ` S ) -> S C_ CC )
15	13, 14	syl 14	. . . . . . . . . . . 12 |- ( ph -> S C_ CC )
16		0cn 8214	. . . . . . . . . . . . 13 |- 0 e. CC
17		snssi 3822	. . . . . . . . . . . . 13 |- ( 0 e. CC -> { 0 } C_ CC )
18	16, 17	mp1i 10	. . . . . . . . . . . 12 |- ( ph -> { 0 } C_ CC )
19	15, 18	unssd 3385	. . . . . . . . . . 11 |- ( ph -> ( S u. { 0 } ) C_ CC )
20	12, 19	fssd 5502	. . . . . . . . . 10 |- ( ph -> A : NN0 --> CC )
21	20	adantr 276	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... N ) ) -> A : NN0 --> CC )
22		elfznn0 10392	. . . . . . . . . 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 5791	. . . . . . . 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 529	. . . . . . . 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 10979	. . . . . . 7 |- ( ( ( ph /\ x e. CC ) /\ k e. ( 0 ... N ) ) -> ( x ^ k ) e. CC )
29	25, 28	mulcld 8243	. . . . . 6 |- ( ( ( ph /\ x e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. ( x ^ k ) ) e. CC )
30	11, 29	fsumcl 12022	. . . . 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 6035	. . . . . . . . 9 |- ( z = x -> ( z ^ k ) = ( x ^ k ) )
33	32	oveq2d 6044	. . . . . . . 8 |- ( z = x -> ( ( A ` k ) x. ( z ^ k ) ) = ( ( A ` k ) x. ( x ^ k ) ) )
34	33	sumeq2sdv 11991	. . . . . . 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 4190	. . . . . 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 2280	. . . . 5 |- ( ph -> F = ( x e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) ) )
37		fveq2 5648	. . . . 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 5821	. . . 4 |- ( ph -> ( * o. F ) = ( x e. CC |-> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) ) ) )
39		oveq1 6035	. . . . . . 7 |- ( x = ( * ` z ) -> ( x ^ k ) = ( ( * ` z ) ^ k ) )
40	39	oveq2d 6044	. . . . . 6 |- ( x = ( * ` z ) -> ( ( A ` k ) x. ( x ^ k ) ) = ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) )
41	40	sumeq2sdv 11991	. . . . 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 5652	. . . 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 5821	. . 3 |- ( ph -> ( ( * o. F ) o. * ) = ( z e. CC |-> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) ) )
44	1, 43	eqtrid 2276	. 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 10979	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( * ` z ) ^ k ) e. CC )
50	46, 49	mulcld 8243	. . . . 5 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) e. CC )
51	45, 50	fsumcj 12096	. . . 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 11587	. . . . . 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 5726	. . . . . . . 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 11579	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( ( * ` z ) ^ k ) ) = ( ( * ` ( * ` z ) ) ^ k ) )
57		cjcj 11504	. . . . . . . . . 10 |- ( z e. CC -> ( * ` ( * ` z ) ) = z )
58	57	ad2antlr 489	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( * ` z ) ) = z )
59	58	oveq1d 6043	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( * ` ( * ` z ) ) ^ k ) = ( z ^ k ) )
60	56, 59	eqtr2d 2265	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( z ^ k ) = ( * ` ( ( * ` z ) ^ k ) ) )
61	55, 60	oveq12d 6046	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) = ( ( * ` ( A ` k ) ) x. ( * ` ( ( * ` z ) ^ k ) ) ) )
62	52, 61	eqtr4d 2267	. . . . 5 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) = ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) )
63	62	sumeq2dv 11989	. . . 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 2264	. . 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 4184	. 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 2264	1 |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   u. cun 3199   C_ wss 3201   {csn 3673   |-> cmpt 4155   o. ccom 4735   -->wf 5329   ` cfv 5333  (class class class)co 6028   Fincfn 6952   CCcc 8073   0cc0 8075   x. cmul 8080   NN0cn0 9445   ...cfz 10286   ^cexp 10844   *ccj 11460   sum_csu 11974  Polycply 15519

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-n0 9446  df-z 9523  df-uz 9799  df-q 9897  df-rp 9932  df-fz 10287  df-fzo 10421  df-seqfrec 10754  df-exp 10845  df-ihash 11082  df-cj 11463  df-re 11464  df-im 11465  df-rsqrt 11619  df-abs 11620  df-clim 11900  df-sumdc 11975  df-ply 15521

This theorem is referenced by:  plycj  15552