Theorem plycjlemc 15844

Description: Lemma for plycj 15845. (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 11596	. . . . 5 |- ( z e. CC -> ( * ` z ) e. CC )
3	2	adantl 277	. . . 4 |- ( ( ph /\ z e. CC ) -> ( * ` z ) e. CC )
4		cjf 11595	. . . . . 6 |- * : CC --> CC
5	4	a1i 9	. . . . 5 |- ( ph -> * : CC --> CC )
6	5	feqmptd 5753	. . . 4 |- ( ph -> * = ( z e. CC |-> ( * ` z ) ) )
7		0zd 9639	. . . . . . . 8 |- ( ph -> 0 e. ZZ )
8		plycjlemc.n	. . . . . . . . 9 |- ( ph -> N e. NN0 )
9	8	nn0zd 9749	. . . . . . . 8 |- ( ph -> N e. ZZ )
10	7, 9	fzfigd 10851	. . . . . . 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 15817	. . . . . . . . . . . . 13 |- ( F e. (Poly ` S ) -> S C_ CC )
15	13, 14	syl 14	. . . . . . . . . . . 12 |- ( ph -> S C_ CC )
16		0cn 8312	. . . . . . . . . . . . 13 |- 0 e. CC
17		snssi 3857	. . . . . . . . . . . . 13 |- ( 0 e. CC -> { 0 } C_ CC )
18	16, 17	mp1i 10	. . . . . . . . . . . 12 |- ( ph -> { 0 } C_ CC )
19	15, 18	unssd 3405	. . . . . . . . . . 11 |- ( ph -> ( S u. { 0 } ) C_ CC )
20	12, 19	fssd 5545	. . . . . . . . . 10 |- ( ph -> A : NN0 --> CC )
21	20	adantr 276	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... N ) ) -> A : NN0 --> CC )
22		elfznn0 10504	. . . . . . . . . 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 5838	. . . . . . . 8 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
25	24	adantlr 481	. . . . . . 7 |- ( ( ( ph /\ x e. CC ) /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
26		simplr 533	. . . . . . . 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 11094	. . . . . . 7 |- ( ( ( ph /\ x e. CC ) /\ k e. ( 0 ... N ) ) -> ( x ^ k ) e. CC )
29	25, 28	mulcld 8340	. . . . . 6 |- ( ( ( ph /\ x e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. ( x ^ k ) ) e. CC )
30	11, 29	fsumcl 12150	. . . . 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 6086	. . . . . . . . 9 |- ( z = x -> ( z ^ k ) = ( x ^ k ) )
33	32	oveq2d 6095	. . . . . . . 8 |- ( z = x -> ( ( A ` k ) x. ( z ^ k ) ) = ( ( A ` k ) x. ( x ^ k ) ) )
34	33	sumeq2sdv 12119	. . . . . . 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 4225	. . . . . 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 2287	. . . . 5 |- ( ph -> F = ( x e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) ) )
37		fveq2 5693	. . . . 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 5868	. . . 4 |- ( ph -> ( * o. F ) = ( x e. CC |-> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) ) ) )
39		oveq1 6086	. . . . . . 7 |- ( x = ( * ` z ) -> ( x ^ k ) = ( ( * ` z ) ^ k ) )
40	39	oveq2d 6095	. . . . . 6 |- ( x = ( * ` z ) -> ( ( A ` k ) x. ( x ^ k ) ) = ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) )
41	40	sumeq2sdv 12119	. . . . 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 5697	. . . 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 5868	. . 3 |- ( ph -> ( ( * o. F ) o. * ) = ( z e. CC |-> ( * ` sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) ) )
44	1, 43	eqtrid 2283	. 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 481	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
47	2	ad2antlr 493	. . . . . . 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 11094	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( * ` z ) ^ k ) e. CC )
50	46, 49	mulcld 8340	. . . . 5 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) e. CC )
51	45, 50	fsumcj 12224	. . . 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 11715	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) = ( ( * ` ( A ` k ) ) x. ( * ` ( ( * ` z ) ^ k ) ) ) )
53	21	adantlr 481	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> A : NN0 --> CC )
54		fvco3 5773	. . . . . . . 8 |- ( ( A : NN0 --> CC /\ k e. NN0 ) -> ( ( * o. A ) ` k ) = ( * ` ( A ` k ) ) )
55	53, 48, 54	syl2anc 415	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( * o. A ) ` k ) = ( * ` ( A ` k ) ) )
56	47, 48	cjexpd 11707	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( ( * ` z ) ^ k ) ) = ( ( * ` ( * ` z ) ) ^ k ) )
57		cjcj 11631	. . . . . . . . . 10 |- ( z e. CC -> ( * ` ( * ` z ) ) = z )
58	57	ad2antlr 493	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( * ` z ) ) = z )
59	58	oveq1d 6094	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( * ` ( * ` z ) ) ^ k ) = ( z ^ k ) )
60	56, 59	eqtr2d 2272	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( z ^ k ) = ( * ` ( ( * ` z ) ^ k ) ) )
61	55, 60	oveq12d 6097	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) = ( ( * ` ( A ` k ) ) x. ( * ` ( ( * ` z ) ^ k ) ) ) )
62	52, 61	eqtr4d 2274	. . . . 5 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( * ` ( ( A ` k ) x. ( ( * ` z ) ^ k ) ) ) = ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) )
63	62	sumeq2dv 12117	. . . 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 2271	. . 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 4219	. 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 2271	1 |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   u. cun 3218   C_ wss 3220   {csn 3708   |-> cmpt 4190   o. ccom 4776   -->wf 5371   ` cfv 5375  (class class class)co 6079   Fincfn 7016   CCcc 8171   0cc0 8173   x. cmul 8178   NN0cn0 9546   ...cfz 10394   ^cexp 10958   *ccj 11587   sum_csu 12102  Polycply 15812

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-oadd 6685  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-seqfrec 10868  df-exp 10959  df-ihash 11198  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-clim 12028  df-sumdc 12103  df-ply 15814

This theorem is referenced by:  plycj  15845