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Theorem ply1termlem 22427
Description: Lemma for ply1term 22428. (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 754 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  N  e.  NN0 )
2 nn0uz 11117 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2565 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  N  e.  (
ZZ>= `  0 ) )
4 fzss1 11723 . . . . . 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 11698 . . . . . . . . 9  |-  ( k  e.  ( N ... N )  ->  k  =  N )
76adantl 466 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  k  =  N )
8 iftrue 3945 . . . . . . . 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 753 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  A  e.  CC )
1110adantr 465 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  A  e.  CC )
129, 11eqeltrd 2555 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  if ( k  =  N ,  A ,  0 )  e.  CC )
13 simplr 754 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  z  e.  CC )
141adantr 465 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  N  e.  NN0 )
157, 14eqeltrd 2555 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  k  e.  NN0 )
1613, 15expcld 12279 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  (
z ^ k )  e.  CC )
1712, 16mulcld 9617 . . . . 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 3627 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... N )  \ 
( N ... N
) )  ->  -.  k  e.  ( N ... N ) )
1918adantl 466 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  -.  k  e.  ( N ... N
) )
201adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  N  e.  NN0 )
2120nn0zd 10965 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  N  e.  ZZ )
22 fzsn 11726 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  ( N ... N )  =  { N } )
2322eleq2d 2537 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
k  e.  ( N ... N )  <->  k  e.  { N } ) )
24 elsnc2g 4057 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
k  e.  { N } 
<->  k  =  N ) )
2523, 24bitrd 253 . . . . . . . . . 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 300 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  -.  k  =  N )
28 iffalse 3948 . . . . . . . 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 6300 . . . . . 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 461 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  z  e.  CC )
32 eldifi 3626 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... N )  \ 
( N ... N
) )  ->  k  e.  ( 0 ... N
) )
33 elfznn0 11771 . . . . . . . . 9  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
3432, 33syl 16 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... N )  \ 
( N ... N
) )  ->  k  e.  NN0 )
35 expcl 12153 . . . . . . . 8  |-  ( ( z  e.  CC  /\  k  e.  NN0 )  -> 
( z ^ k
)  e.  CC )
3631, 34, 35syl2an 477 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  ( z ^ k )  e.  CC )
3736mul02d 9778 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  ( 0  x.  ( z ^
k ) )  =  0 )
3830, 37eqtrd 2508 . . . . 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 12052 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( 0 ... N )  e.  Fin )
405, 17, 38, 39fsumss 13513 . . . 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 10965 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  N  e.  ZZ )
4231, 1expcld 12279 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( z ^ N )  e.  CC )
4310, 42mulcld 9617 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( A  x.  ( z ^ N
) )  e.  CC )
44 oveq2 6293 . . . . . . 7  |-  ( k  =  N  ->  (
z ^ k )  =  ( z ^ N ) )
458, 44oveq12d 6303 . . . . . 6  |-  ( k  =  N  ->  ( if ( k  =  N ,  A ,  0 )  x.  ( z ^ k ) )  =  ( A  x.  ( z ^ N
) ) )
4645fsum1 13530 . . . . 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 661 . . . 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 2510 . . 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 4533 . 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 2527 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3473    C_ wss 3476   ifcif 3939   {csn 4027    |-> cmpt 4505   ` cfv 5588  (class class class)co 6285   CCcc 9491   0cc0 9493    x. cmul 9498   NN0cn0 10796   ZZcz 10865   ZZ>=cuz 11083   ...cfz 11673   ^cexp 12135   sum_csu 13474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-oi 7936  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-fz 11674  df-fzo 11794  df-seq 12077  df-exp 12136  df-hash 12375  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-clim 13277  df-sum 13475
This theorem is referenced by:  ply1term  22428  coe1termlem  22481
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