Theorem ply1termlem 15826

Description: Lemma for ply1term 15827. (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 533	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> N e. NN0 )
3		nn0uz 9940	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
4	2, 3	eleqtrdi 2331	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> N e. ( ZZ>= ` 0 ) )
5		fzss1 10452	. . . . . 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 10422	. . . . . . . . 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 3645	. . . . . . . 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 531	. . . . . . . 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 2315	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> if ( k = N , A , 0 ) e. CC )
14		simplr 533	. . . . . . 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 2315	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> k e. NN0 )
17	14, 16	expcld 11094	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> ( z ^ k ) e. CC )
18	13, 17	mulcld 8340	. . . . 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 3352	. . . . . . . . . 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 9749	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> N e. ZZ )
23		fzsn 10455	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( N ... N ) = { N } )
24	23	eleq2d 2308	. . . . . . . . . . 11 |- ( N e. ZZ -> ( k e. ( N ... N ) <-> k e. { N } ) )
25		elsn2g 3741	. . . . . . . . . . 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 683	. . . . . . . 8 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> -. k = N )
29	28	iffalsed 3650	. . . . . . 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 6094	. . . . . 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 3351	. . . . . . . . 9 |- ( k e. ( ( 0 ... N ) \ ( N ... N ) ) -> k e. ( 0 ... N ) )
33		elfznn0 10504	. . . . . . . . 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 10977	. . . . . . . 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 8713	. . . . . 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 2271	. . . . 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 10411	. . . . . . . 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 9749	. . . . . . . 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 10427	. . . . . . 7 |- ( ( w e. ZZ /\ N e. ZZ /\ N e. ZZ ) -> DECID w e. ( N ... N ) )
44	40, 42, 42, 43	syl3anc 1278	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ w e. ( 0 ... N ) ) -> DECID w e. ( N ... N ) )
45	44	ralrimiva 2623	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> A. w e. ( 0 ... N )DECID w e. ( N ... N ) )
46		0zd 9639	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> 0 e. ZZ )
47	46, 41	fzfigd 10851	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( 0 ... N ) e. Fin )
48	6, 18, 38, 45, 47	fisumss 12142	. . . 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 11094	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( z ^ N ) e. CC )
50	11, 49	mulcld 8340	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( A x. ( z ^ N ) ) e. CC )
51		oveq2 6087	. . . . . . 7 |- ( k = N -> ( z ^ k ) = ( z ^ N ) )
52	9, 51	oveq12d 6097	. . . . . 6 |- ( k = N -> ( if ( k = N , A , 0 ) x. ( z ^ k ) ) = ( A x. ( z ^ N ) ) )
53	52	fsum1 12162	. . . . 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 415	. . . 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 2273	. . 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 4219	. 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 2290	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 846   = wceq 1402   e. wcel 2209   \ cdif 3217   C_ wss 3220   ifcif 3638   {csn 3708   |-> cmpt 4190   ` cfv 5375  (class class class)co 6079   CCcc 8171   0cc0 8173   x. cmul 8178   NN0cn0 9546   ZZcz 9627   ZZ>=cuz 9904   ...cfz 10394   ^cexp 10958   sum_csu 12102

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

This theorem is referenced by:  ply1term  15827