Theorem ply1termlem 15594

Description: Lemma for ply1term 15595. (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 9885	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
4	2, 3	eleqtrdi 2325	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> N e. ( ZZ>= ` 0 ) )
5		fzss1 10393	. . . . . 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 10365	. . . . . . . . 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 3626	. . . . . . . 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 2309	. . . . . 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 2309	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> k e. NN0 )
17	14, 16	expcld 11031	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( N ... N ) ) -> ( z ^ k ) e. CC )
18	13, 17	mulcld 8290	. . . . 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 3341	. . . . . . . . . 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 9694	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> N e. ZZ )
23		fzsn 10396	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( N ... N ) = { N } )
24	23	eleq2d 2302	. . . . . . . . . . 11 |- ( N e. ZZ -> ( k e. ( N ... N ) <-> k e. { N } ) )
25		elsn2g 3721	. . . . . . . . . . 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 679	. . . . . . . 8 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( N ... N ) ) ) -> -. k = N )
29	28	iffalsed 3631	. . . . . . 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 6064	. . . . . 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 3340	. . . . . . . . 9 |- ( k e. ( ( 0 ... N ) \ ( N ... N ) ) -> k e. ( 0 ... N ) )
33		elfznn0 10444	. . . . . . . . 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 10915	. . . . . . . 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 8661	. . . . . 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 2265	. . . . 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 10355	. . . . . . . 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 9694	. . . . . . . 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 10370	. . . . . . 7 |- ( ( w e. ZZ /\ N e. ZZ /\ N e. ZZ ) -> DECID w e. ( N ... N ) )
44	40, 42, 42, 43	syl3anc 1274	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) /\ w e. ( 0 ... N ) ) -> DECID w e. ( N ... N ) )
45	44	ralrimiva 2615	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> A. w e. ( 0 ... N )DECID w e. ( N ... N ) )
46		0zd 9585	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> 0 e. ZZ )
47	46, 41	fzfigd 10789	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( 0 ... N ) e. Fin )
48	6, 18, 38, 45, 47	fisumss 12071	. . . 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 11031	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( z ^ N ) e. CC )
50	11, 49	mulcld 8290	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ z e. CC ) -> ( A x. ( z ^ N ) ) e. CC )
51		oveq2 6057	. . . . . . 7 |- ( k = N -> ( z ^ k ) = ( z ^ N ) )
52	9, 51	oveq12d 6067	. . . . . 6 |- ( k = N -> ( if ( k = N , A , 0 ) x. ( z ^ k ) ) = ( A x. ( z ^ N ) ) )
53	52	fsum1 12091	. . . . 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 2267	. . 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 4199	. 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 2284	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 842   = wceq 1398   e. wcel 2203   \ cdif 3207   C_ wss 3210   ifcif 3619   {csn 3688   |-> cmpt 4170   ` cfv 5351  (class class class)co 6049   CCcc 8121   0cc0 8123   x. cmul 8128   NN0cn0 9492   ZZcz 9573   ZZ>=cuz 9849   ...cfz 10338   ^cexp 10896   sum_csu 12031

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241  ax-arch 8242  ax-caucvg 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-oadd 6650  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-n0 9493  df-z 9574  df-uz 9850  df-q 9948  df-rp 9983  df-fz 10339  df-fzo 10473  df-seqfrec 10806  df-exp 10897  df-ihash 11134  df-cj 11520  df-re 11521  df-im 11522  df-rsqrt 11676  df-abs 11677  df-clim 11957  df-sumdc 12032

This theorem is referenced by:  ply1term  15595