Theorem plycjlemc 15617

Description: Lemma for plycj 15618. (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 11529	. . . . 5 |- ( z e. CC -> ( * ` z ) e. CC )
3	2	adantl 277	. . . 4 |- ( ( ph /\ z e. CC ) -> ( * ` z ) e. CC )
4		cjf 11528	. . . . . 6 |- * : CC --> CC
5	4	a1i 9	. . . . 5 |- ( ph -> * : CC --> CC )
6	5	feqmptd 5729	. . . 4 |- ( ph -> * = ( z e. CC |-> ( * ` z ) ) )
7		0zd 9588	. . . . . . . 8 |- ( ph -> 0 e. ZZ )
8		plycjlemc.n	. . . . . . . . 9 |- ( ph -> N e. NN0 )
9	8	nn0zd 9697	. . . . . . . 8 |- ( ph -> N e. ZZ )
10	7, 9	fzfigd 10792	. . . . . . 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 15590	. . . . . . . . . . . . 13 |- ( F e. (Poly ` S ) -> S C_ CC )
15	13, 14	syl 14	. . . . . . . . . . . 12 |- ( ph -> S C_ CC )
16		0cn 8265	. . . . . . . . . . . . 13 |- 0 e. CC
17		snssi 3837	. . . . . . . . . . . . 13 |- ( 0 e. CC -> { 0 } C_ CC )
18	16, 17	mp1i 10	. . . . . . . . . . . 12 |- ( ph -> { 0 } C_ CC )
19	15, 18	unssd 3394	. . . . . . . . . . 11 |- ( ph -> ( S u. { 0 } ) C_ CC )
20	12, 19	fssd 5521	. . . . . . . . . 10 |- ( ph -> A : NN0 --> CC )
21	20	adantr 276	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... N ) ) -> A : NN0 --> CC )
22		elfznn0 10447	. . . . . . . . . 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 5812	. . . . . . . 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 11034	. . . . . . 7 |- ( ( ( ph /\ x e. CC ) /\ k e. ( 0 ... N ) ) -> ( x ^ k ) e. CC )
29	25, 28	mulcld 8293	. . . . . 6 |- ( ( ( ph /\ x e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. ( x ^ k ) ) e. CC )
30	11, 29	fsumcl 12082	. . . . 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 6056	. . . . . . . . 9 |- ( z = x -> ( z ^ k ) = ( x ^ k ) )
33	32	oveq2d 6065	. . . . . . . 8 |- ( z = x -> ( ( A ` k ) x. ( z ^ k ) ) = ( ( A ` k ) x. ( x ^ k ) ) )
34	33	sumeq2sdv 12051	. . . . . . 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 4205	. . . . . 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 2281	. . . . 5 |- ( ph -> F = ( x e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) ) )
37		fveq2 5669	. . . . 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 5842	. . . 4 |- ( ph -> ( * o. F ) = ( x e. CC |-> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) ) ) )
39		oveq1 6056	. . . . . . 7 |- ( x = ( * ` z ) -> ( x ^ k ) = ( ( * ` z ) ^ k ) )
40	39	oveq2d 6065	. . . . . 6 |- ( x = ( * ` z ) -> ( ( A ` k ) x. ( x ^ k ) ) = ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) )
41	40	sumeq2sdv 12051	. . . . 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 5673	. . . 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 5842	. . 3 |- ( ph -> ( ( * o. F ) o. * ) = ( z e. CC |-> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) ) )
44	1, 43	eqtrid 2277	. 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 11034	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( * ` z ) ^ k ) e. CC )
50	46, 49	mulcld 8293	. . . . 5 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) e. CC )
51	45, 50	fsumcj 12156	. . . 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 11647	. . . . . 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 5747	. . . . . . . 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 11639	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( ( * ` z ) ^ k ) ) = ( ( * ` ( * ` z ) ) ^ k ) )
57		cjcj 11564	. . . . . . . . . 10 |- ( z e. CC -> ( * ` ( * ` z ) ) = z )
58	57	ad2antlr 489	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( * ` z ) ) = z )
59	58	oveq1d 6064	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( * ` ( * ` z ) ) ^ k ) = ( z ^ k ) )
60	56, 59	eqtr2d 2266	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( z ^ k ) = ( * ` ( ( * ` z ) ^ k ) ) )
61	55, 60	oveq12d 6067	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) = ( ( * ` ( A ` k ) ) x. ( * ` ( ( * ` z ) ^ k ) ) ) )
62	52, 61	eqtr4d 2268	. . . . 5 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) = ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) )
63	62	sumeq2dv 12049	. . . 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 2265	. . 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 4199	. 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 2265	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 2203   u. cun 3208   C_ wss 3210   {csn 3688   |-> cmpt 4170   o. ccom 4752   -->wf 5347   ` cfv 5351  (class class class)co 6049   Fincfn 6974   CCcc 8124   0cc0 8126   x. cmul 8131   NN0cn0 9495   ...cfz 10341   ^cexp 10899   *ccj 11520   sum_csu 12034  Polycply 15585

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-oadd 6650  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-fz 10342  df-fzo 10476  df-seqfrec 10809  df-exp 10900  df-ihash 11137  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-clim 11960  df-sumdc 12035  df-ply 15587

This theorem is referenced by:  plycj  15618