Theorem ply1termlem 15845

Description: Lemma for ply1term 15846. (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 9959	. . . . . . 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 10471	. . . . . 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 10441	. . . . . . . . 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 11113	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> ( z ^ k ) e. CC )
18	13, 17	mulcld 8346	. . . . 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 9768	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> N e. ZZ )
23		fzsn 10474	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( N ... N ) = { N } )
24	23	eleq2d 2308	. . . . . . . . . . 11 |- ( N e. ZZ -> ( k e. ( N ... N ) <-> k e. { N } ) )
25		elsn2g 3742	. . . . . . . . . . 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 6100	. . . . . 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 10523	. . . . . . . . 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 10996	. . . . . . . 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 8719	. . . . . 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 10430	. . . . . . . 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 9768	. . . . . . . 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 10446	. . . . . . 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 9658	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> 0 e. ZZ )
47	46, 41	fzfigd 10870	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( 0 ... N ) e. Fin )
48	6, 18, 38, 45, 47	fisumss 12161	. . . 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 11113	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( z ^ N ) e. CC )
50	11, 49	mulcld 8346	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( A x. ( z ^ N ) ) e. CC )
51		oveq2 6093	. . . . . . 7 |- ( k = N -> ( z ^ k ) = ( z ^ N ) )
52	9, 51	oveq12d 6103	. . . . . 6 |- ( k = N -> ( if ( k = N , A , 0 ) x. ( z ^ k ) ) = ( A x. ( z ^ N ) ) )
53	52	fsum1 12181	. . . . 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 4221	. 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 ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   = wceq 1402   e. wcel 2209   \ cdif 3217   C_ wss 3220   ifcif 3638   {csn 3709   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   x. cmul 8184   NN0cn0 9565   ZZcz 9646   ZZ>=cuz 9923   ...cfz 10413   ^cexp 10977   sum_csu 12121

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-fz 10414  df-fzo 10552  df-seqfrec 10887  df-exp 10978  df-ihash 11217  df-cj 11609  df-re 11610  df-im 11611  df-rsqrt 11766  df-abs 11767  df-clim 12047  df-sumdc 12122

This theorem is used by:  ply1term  15846