Theorem ply1termlem 15495

Description: Lemma for ply1term 15496. (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 529	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> N e. NN0 )
3		nn0uz 9796	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
4	2, 3	eleqtrdi 2323	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> N e. ( ZZ>= ` 0 ) )
5		fzss1 10303	. . . . . 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 10275	. . . . . . . . 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 3611	. . . . . . . 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 2307	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> if ( k = N , A , 0 ) e. CC )
14		simplr 529	. . . . . . 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 2307	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> k e. NN0 )
17	14, 16	expcld 10941	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> ( z ^ k ) e. CC )
18	13, 17	mulcld 8205	. . . . 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 3329	. . . . . . . . . 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 9605	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> N e. ZZ )
23		fzsn 10306	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( N ... N ) = { N } )
24	23	eleq2d 2300	. . . . . . . . . . 11 |- ( N e. ZZ -> ( k e. ( N ... N ) <-> k e. { N } ) )
25		elsn2g 3703	. . . . . . . . . . 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 678	. . . . . . . 8 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> -. k = N )
29	28	iffalsed 3616	. . . . . . 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 6038	. . . . . 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 3328	. . . . . . . . 9 |- ( k e. ( ( 0 ... N ) \ ( N ... N ) ) -> k e. ( 0 ... N ) )
33		elfznn0 10354	. . . . . . . . 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 10825	. . . . . . . 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 8576	. . . . . 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 2263	. . . . 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 10265	. . . . . . . 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 9605	. . . . . . . 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 10280	. . . . . . 7 |- ( ( w e. ZZ /\ N e. ZZ /\ N e. ZZ ) -> DECID w e. ( N ... N ) )
44	40, 42, 42, 43	syl3anc 1273	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ w e. ( 0 ... N ) ) -> DECID w e. ( N ... N ) )
45	44	ralrimiva 2604	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> A. w e. ( 0 ... N )DECID w e. ( N ... N ) )
46		0zd 9496	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> 0 e. ZZ )
47	46, 41	fzfigd 10699	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( 0 ... N ) e. Fin )
48	6, 18, 38, 45, 47	fisumss 11976	. . . 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 10941	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( z ^ N ) e. CC )
50	11, 49	mulcld 8205	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( A x. ( z ^ N ) ) e. CC )
51		oveq2 6031	. . . . . . 7 |- ( k = N -> ( z ^ k ) = ( z ^ N ) )
52	9, 51	oveq12d 6041	. . . . . 6 |- ( k = N -> ( if ( k = N , A , 0 ) x. ( z ^ k ) ) = ( A x. ( z ^ N ) ) )
53	52	fsum1 11996	. . . . 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 2265	. . 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 4180	. 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 2282	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 841   = wceq 1397   e. wcel 2201   \ cdif 3196   C_ wss 3199   ifcif 3604   {csn 3670   |-> cmpt 4151   ` cfv 5328  (class class class)co 6023   CCcc 8035   0cc0 8037   x. cmul 8042   NN0cn0 9407   ZZcz 9484   ZZ>=cuz 9760   ...cfz 10248   ^cexp 10806   sum_csu 11936

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-mulrcl 8136  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-precex 8147  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153  ax-pre-mulgt0 8154  ax-pre-mulext 8155  ax-arch 8156  ax-caucvg 8157

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-ilim 4468  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-isom 5337  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-irdg 6541  df-frec 6562  df-1o 6587  df-oadd 6591  df-er 6707  df-en 6915  df-dom 6916  df-fin 6917  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-reap 8760  df-ap 8767  df-div 8858  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-n0 9408  df-z 9485  df-uz 9761  df-q 9859  df-rp 9894  df-fz 10249  df-fzo 10383  df-seqfrec 10716  df-exp 10807  df-ihash 11044  df-cj 11425  df-re 11426  df-im 11427  df-rsqrt 11581  df-abs 11582  df-clim 11862  df-sumdc 11937

This theorem is referenced by:  ply1term  15496