Theorem ply1termlem 15437

Description: Lemma for ply1term 15438. (Contributed by Mario Carneiro, 26-Jul-2014.)

Hypothesis

Ref	Expression
ply1term.1	|- F = ( z e. CC |-> ( A x. ( z ^ N ) ) )

Assertion

Ref	Expression
ply1termlem	|- ( ( A e. CC /\ N e. NN0 ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( if ( k = N , A , 0 ) x. ( z ^ k ) ) ) )

Distinct variable groups:   z, k, A   k, N, z

Allowed substitution hints:   F( z, k)

Proof of Theorem ply1termlem

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ply1term.1	. 2 |- F = ( z e. CC |-> ( A x. ( z ^ N ) ) )
2		simplr 528	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> N e. NN0 )
3		nn0uz 9774	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
4	2, 3	eleqtrdi 2322	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> N e. ( ZZ>= ` 0 ) )
5		fzss1 10276	. . . . . 6 |- ( N e. ( ZZ>= ` 0 ) -> ( N ... N ) C_ ( 0 ... N ) )
6	4, 5	syl 14	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( N ... N ) C_ ( 0 ... N ) )
7		elfz1eq 10248	. . . . . . . . 9 |- ( k e. ( N ... N ) -> k = N )
8	7	adantl 277	. . . . . . . 8 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> k = N )
9		iftrue 3607	. . . . . . . 8 |- ( k = N -> if ( k = N , A , 0 ) = A )
10	8, 9	syl 14	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> if ( k = N , A , 0 ) = A )
11		simpll 527	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> A e. CC )
12	11	adantr 276	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> A e. CC )
13	10, 12	eqeltrd 2306	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> if ( k = N , A , 0 ) e. CC )
14		simplr 528	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> z e. CC )
15	2	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> N e. NN0 )
16	8, 15	eqeltrd 2306	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> k e. NN0 )
17	14, 16	expcld 10912	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> ( z ^ k ) e. CC )
18	13, 17	mulcld 8183	. . . . 5 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> ( if ( k = N , A , 0 ) x. ( z ^ k ) ) e. CC )
19		eldifn 3327	. . . . . . . . . 10 |- ( k e. ( ( 0 ... N ) \ ( N ... N ) ) -> -. k e. ( N ... N ) )
20	19	adantl 277	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> -. k e. ( N ... N ) )
21	2	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> N e. NN0 )
22	21	nn0zd 9583	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> N e. ZZ )
23		fzsn 10279	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( N ... N ) = { N } )
24	23	eleq2d 2299	. . . . . . . . . . 11 |- ( N e. ZZ -> ( k e. ( N ... N ) <-> k e. { N } ) )
25		elsn2g 3699	. . . . . . . . . . 11 |- ( N e. ZZ -> ( k e. { N } <-> k = N ) )
26	24, 25	bitrd 188	. . . . . . . . . 10 |- ( N e. ZZ -> ( k e. ( N ... N ) <-> k = N ) )
27	22, 26	syl 14	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> ( k e. ( N ... N ) <-> k = N ) )
28	20, 27	mtbid 676	. . . . . . . 8 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> -. k = N )
29	28	iffalsed 3612	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> if ( k = N , A , 0 ) = 0 )
30	29	oveq1d 6025	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> ( if ( k = N , A , 0 ) x. ( z ^ k ) ) = ( 0 x. ( z ^ k ) ) )
31		simpr 110	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> z e. CC )
32		eldifi 3326	. . . . . . . . 9 |- ( k e. ( ( 0 ... N ) \ ( N ... N ) ) -> k e. ( 0 ... N ) )
33		elfznn0 10327	. . . . . . . . 9 |- ( k e. ( 0 ... N ) -> k e. NN0 )
34	32, 33	syl 14	. . . . . . . 8 |- ( k e. ( ( 0 ... N ) \ ( N ... N ) ) -> k e. NN0 )
35		expcl 10796	. . . . . . . 8 |- ( ( z e. CC /\ k e. NN0 ) -> ( z ^ k ) e. CC )
36	31, 34, 35	syl2an 289	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> ( z ^ k ) e. CC )
37	36	mul02d 8554	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> ( 0 x. ( z ^ k ) ) = 0 )
38	30, 37	eqtrd 2262	. . . . 5 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> ( if ( k = N , A , 0 ) x. ( z ^ k ) ) = 0 )
39		elfzelz 10238	. . . . . . . 8 |- ( w e. ( 0 ... N ) -> w e. ZZ )
40	39	adantl 277	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ w e. ( 0 ... N ) ) -> w e. ZZ )
41	2	nn0zd 9583	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> N e. ZZ )
42	41	adantr 276	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ w e. ( 0 ... N ) ) -> N e. ZZ )
43		fzdcel 10253	. . . . . . 7 |- ( ( w e. ZZ /\ N e. ZZ /\ N e. ZZ ) -> DECID w e. ( N ... N ) )
44	40, 42, 42, 43	syl3anc 1271	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ w e. ( 0 ... N ) ) -> DECID w e. ( N ... N ) )
45	44	ralrimiva 2603	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> A. w e. ( 0 ... N )DECID w e. ( N ... N ) )
46		0zd 9474	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> 0 e. ZZ )
47	46, 41	fzfigd 10670	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( 0 ... N ) e. Fin )
48	6, 18, 38, 45, 47	fisumss 11924	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> sum_ k e. ( N ... N ) ( if ( k = N , A , 0 ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... N ) ( if ( k = N , A , 0 ) x. ( z ^ k ) ) )
49	31, 2	expcld 10912	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( z ^ N ) e. CC )
50	11, 49	mulcld 8183	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( A x. ( z ^ N ) ) e. CC )
51		oveq2 6018	. . . . . . 7 |- ( k = N -> ( z ^ k ) = ( z ^ N ) )
52	9, 51	oveq12d 6028	. . . . . 6 |- ( k = N -> ( if ( k = N , A , 0 ) x. ( z ^ k ) ) = ( A x. ( z ^ N ) ) )
53	52	fsum1 11944	. . . . 5 |- ( ( N e. ZZ /\ ( A x. ( z ^ N ) ) e. CC ) -> sum_ k e. ( N ... N ) ( if ( k = N , A , 0 ) x. ( z ^ k ) ) = ( A x. ( z ^ N ) ) )
54	41, 50, 53	syl2anc 411	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> sum_ k e. ( N ... N ) ( if ( k = N , A , 0 ) x. ( z ^ k ) ) = ( A x. ( z ^ N ) ) )
55	48, 54	eqtr3d 2264	. . 3 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> sum_ k e. ( 0 ... N ) ( if ( k = N , A , 0 ) x. ( z ^ k ) ) = ( A x. ( z ^ N ) ) )
56	55	mpteq2dva 4174	. 2 |- ( ( A e. CC /\ N e. NN0 ) -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( if ( k = N , A , 0 ) x. ( z ^ k ) ) ) = ( z e. CC |-> ( A x. ( z ^ N ) ) ) )
57	1, 56	eqtr4id 2281	1 |- ( ( A e. CC /\ N e. NN0 ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( if ( k = N , A , 0 ) x. ( z ^ k ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 839   = wceq 1395   e. wcel 2200   \ cdif 3194   C_ wss 3197   ifcif 3602   {csn 3666   |-> cmpt 4145   ` cfv 5321  (class class class)co 6010   CCcc 8013   0cc0 8015   x. cmul 8020   NN0cn0 9385   ZZcz 9462   ZZ>=cuz 9738   ...cfz 10221   ^cexp 10777   sum_csu 11885

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-isom 5330  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-frec 6548  df-1o 6573  df-oadd 6577  df-er 6693  df-en 6901  df-dom 6902  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-n0 9386  df-z 9463  df-uz 9739  df-q 9832  df-rp 9867  df-fz 10222  df-fzo 10356  df-seqfrec 10687  df-exp 10778  df-ihash 11015  df-cj 11374  df-re 11375  df-im 11376  df-rsqrt 11530  df-abs 11531  df-clim 11811  df-sumdc 11886

This theorem is referenced by:  ply1term  15438