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Theorem ply1termlem 23022
Description: Lemma for ply1term 23023. (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 760 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  N  e.  NN0 )
2 nn0uz 11182 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2518 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  N  e.  (
ZZ>= `  0 ) )
4 fzss1 11824 . . . . . 6  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( N ... N )  C_  (
0 ... N ) )
53, 4syl 17 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( N ... N )  C_  (
0 ... N ) )
6 elfz1eq 11797 . . . . . . . . 9  |-  ( k  e.  ( N ... N )  ->  k  =  N )
76adantl 467 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  k  =  N )
8 iftrue 3912 . . . . . . . 8  |-  ( k  =  N  ->  if ( k  =  N ,  A ,  0 )  =  A )
97, 8syl 17 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  if ( k  =  N ,  A ,  0 )  =  A )
10 simpll 758 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  A  e.  CC )
1110adantr 466 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  A  e.  CC )
129, 11eqeltrd 2508 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  if ( k  =  N ,  A ,  0 )  e.  CC )
13 simplr 760 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  z  e.  CC )
141adantr 466 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  N  e.  NN0 )
157, 14eqeltrd 2508 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  k  e.  NN0 )
1613, 15expcld 12402 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  (
z ^ k )  e.  CC )
1712, 16mulcld 9652 . . . . 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 3585 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... N )  \ 
( N ... N
) )  ->  -.  k  e.  ( N ... N ) )
1918adantl 467 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  -.  k  e.  ( N ... N
) )
201adantr 466 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  N  e.  NN0 )
2120nn0zd 11027 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  N  e.  ZZ )
22 fzsn 11827 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  ( N ... N )  =  { N } )
2322eleq2d 2490 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
k  e.  ( N ... N )  <->  k  e.  { N } ) )
24 elsnc2g 4023 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
k  e.  { N } 
<->  k  =  N ) )
2523, 24bitrd 256 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
k  e.  ( N ... N )  <->  k  =  N ) )
2621, 25syl 17 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  ( k  e.  ( N ... N
)  <->  k  =  N ) )
2719, 26mtbid 301 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  -.  k  =  N )
2827iffalsed 3917 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  if (
k  =  N ,  A ,  0 )  =  0 )
2928oveq1d 6311 . . . . . 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
) ) )
30 simpr 462 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  z  e.  CC )
31 eldifi 3584 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... N )  \ 
( N ... N
) )  ->  k  e.  ( 0 ... N
) )
32 elfznn0 11874 . . . . . . . . 9  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
3331, 32syl 17 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... N )  \ 
( N ... N
) )  ->  k  e.  NN0 )
34 expcl 12276 . . . . . . . 8  |-  ( ( z  e.  CC  /\  k  e.  NN0 )  -> 
( z ^ k
)  e.  CC )
3530, 33, 34syl2an 479 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  ( z ^ k )  e.  CC )
3635mul02d 9820 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  ( 0  x.  ( z ^
k ) )  =  0 )
3729, 36eqtrd 2461 . . . . 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 )
38 fzfid 12172 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( 0 ... N )  e.  Fin )
395, 17, 37, 38fsumss 13758 . . . 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 ) ) )
401nn0zd 11027 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  N  e.  ZZ )
4130, 1expcld 12402 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( z ^ N )  e.  CC )
4210, 41mulcld 9652 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( A  x.  ( z ^ N
) )  e.  CC )
43 oveq2 6304 . . . . . . 7  |-  ( k  =  N  ->  (
z ^ k )  =  ( z ^ N ) )
448, 43oveq12d 6314 . . . . . 6  |-  ( k  =  N  ->  ( if ( k  =  N ,  A ,  0 )  x.  ( z ^ k ) )  =  ( A  x.  ( z ^ N
) ) )
4544fsum1 13775 . . . . 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 ) ) )
4640, 42, 45syl2anc 665 . . . 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 ) ) )
4739, 46eqtr3d 2463 . . 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 ) ) )
4847mpteq2dva 4503 . 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 ) ) ) )
49 ply1term.1 . 2  |-  F  =  ( z  e.  CC  |->  ( A  x.  (
z ^ N ) ) )
5048, 49syl6reqr 2480 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 187    /\ wa 370    = wceq 1437    e. wcel 1867    \ cdif 3430    C_ wss 3433   ifcif 3906   {csn 3993    |-> cmpt 4475   ` cfv 5592  (class class class)co 6296   CCcc 9526   0cc0 9528    x. cmul 9533   NN0cn0 10858   ZZcz 10926   ZZ>=cuz 11148   ...cfz 11771   ^cexp 12258   sum_csu 13719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-oi 8016  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-fz 11772  df-fzo 11903  df-seq 12200  df-exp 12259  df-hash 12502  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-clim 13519  df-sum 13720
This theorem is referenced by:  ply1term  23023  coe1termlem  23077
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