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Theorem ply1termlem 20122
Description: Lemma for ply1term 20123. (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
StepHypRef Expression
1 simplr 732 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  N  e.  NN0 )
2 nn0uz 10520 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2526 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  N  e.  (
ZZ>= `  0 ) )
4 fzss1 11091 . . . . . 6  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( N ... N )  C_  (
0 ... N ) )
53, 4syl 16 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( N ... N )  C_  (
0 ... N ) )
6 elfz1eq 11068 . . . . . . . . 9  |-  ( k  e.  ( N ... N )  ->  k  =  N )
76adantl 453 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  k  =  N )
8 iftrue 3745 . . . . . . . 8  |-  ( k  =  N  ->  if ( k  =  N ,  A ,  0 )  =  A )
97, 8syl 16 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  if ( k  =  N ,  A ,  0 )  =  A )
10 simpll 731 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  A  e.  CC )
1110adantr 452 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  A  e.  CC )
129, 11eqeltrd 2510 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  if ( k  =  N ,  A ,  0 )  e.  CC )
13 simplr 732 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  z  e.  CC )
141adantr 452 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  N  e.  NN0 )
157, 14eqeltrd 2510 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  k  e.  NN0 )
1613, 15expcld 11523 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  (
z ^ k )  e.  CC )
1712, 16mulcld 9108 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  ( if ( k  =  N ,  A ,  0 )  x.  ( z ^ k ) )  e.  CC )
18 eldifn 3470 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... N )  \ 
( N ... N
) )  ->  -.  k  e.  ( N ... N ) )
1918adantl 453 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  -.  k  e.  ( N ... N
) )
201adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  N  e.  NN0 )
2120nn0zd 10373 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  N  e.  ZZ )
22 fzsn 11094 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  ( N ... N )  =  { N } )
2322eleq2d 2503 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
k  e.  ( N ... N )  <->  k  e.  { N } ) )
24 elsnc2g 3842 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
k  e.  { N } 
<->  k  =  N ) )
2523, 24bitrd 245 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
k  e.  ( N ... N )  <->  k  =  N ) )
2621, 25syl 16 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  ( k  e.  ( N ... N
)  <->  k  =  N ) )
2719, 26mtbid 292 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  -.  k  =  N )
28 iffalse 3746 . . . . . . . 8  |-  ( -.  k  =  N  ->  if ( k  =  N ,  A ,  0 )  =  0 )
2927, 28syl 16 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  if (
k  =  N ,  A ,  0 )  =  0 )
3029oveq1d 6096 . . . . . 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 448 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  z  e.  CC )
32 eldifi 3469 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... N )  \ 
( N ... N
) )  ->  k  e.  ( 0 ... N
) )
33 elfznn0 11083 . . . . . . . . 9  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
3432, 33syl 16 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... N )  \ 
( N ... N
) )  ->  k  e.  NN0 )
35 expcl 11399 . . . . . . . 8  |-  ( ( z  e.  CC  /\  k  e.  NN0 )  -> 
( z ^ k
)  e.  CC )
3631, 34, 35syl2an 464 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  ( z ^ k )  e.  CC )
3736mul02d 9264 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  ( 0  x.  ( z ^
k ) )  =  0 )
3830, 37eqtrd 2468 . . . . 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 fzfid 11312 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( 0 ... N )  e.  Fin )
405, 17, 38, 39fsumss 12519 . . . 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 ) ) )
411nn0zd 10373 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  N  e.  ZZ )
4231, 1expcld 11523 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( z ^ N )  e.  CC )
4310, 42mulcld 9108 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( A  x.  ( z ^ N
) )  e.  CC )
44 oveq2 6089 . . . . . . 7  |-  ( k  =  N  ->  (
z ^ k )  =  ( z ^ N ) )
458, 44oveq12d 6099 . . . . . 6  |-  ( k  =  N  ->  ( if ( k  =  N ,  A ,  0 )  x.  ( z ^ k ) )  =  ( A  x.  ( z ^ N
) ) )
4645fsum1 12535 . . . . 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 ) ) )
4741, 43, 46syl2anc 643 . . . 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 ) ) )
4840, 47eqtr3d 2470 . . 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 ) ) )
4948mpteq2dva 4295 . 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 ) ) ) )
50 ply1term.1 . 2  |-  F  =  ( z  e.  CC  |->  ( A  x.  (
z ^ N ) ) )
5149, 50syl6reqr 2487 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( if ( k  =  N ,  A ,  0 )  x.  ( z ^ k
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    \ cdif 3317    C_ wss 3320   ifcif 3739   {csn 3814    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990    x. cmul 8995   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043   ^cexp 11382   sum_csu 12479
This theorem is referenced by:  ply1term  20123  coe1termlem  20176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480
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