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Theorem ply1termlem 19585
Description: Lemma for ply1term 19586. (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 731 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  N  e.  NN0 )
2 nn0uz 10262 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2373 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  N  e.  (
ZZ>= `  0 ) )
4 fzss1 10830 . . . . . 6  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( N ... N )  C_  (
0 ... N ) )
53, 4syl 15 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( N ... N )  C_  (
0 ... N ) )
6 elfz1eq 10807 . . . . . . . . 9  |-  ( k  e.  ( N ... N )  ->  k  =  N )
76adantl 452 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  k  =  N )
8 iftrue 3571 . . . . . . . 8  |-  ( k  =  N  ->  if ( k  =  N ,  A ,  0 )  =  A )
97, 8syl 15 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  if ( k  =  N ,  A ,  0 )  =  A )
10 simpll 730 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  A  e.  CC )
1110adantr 451 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  A  e.  CC )
129, 11eqeltrd 2357 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  if ( k  =  N ,  A ,  0 )  e.  CC )
13 simplr 731 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  z  e.  CC )
141adantr 451 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  N  e.  NN0 )
157, 14eqeltrd 2357 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  k  e.  NN0 )
1613, 15expcld 11245 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  (
z ^ k )  e.  CC )
1712, 16mulcld 8855 . . . . 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 3299 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... N )  \ 
( N ... N
) )  ->  -.  k  e.  ( N ... N ) )
1918adantl 452 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  -.  k  e.  ( N ... N
) )
201adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  N  e.  NN0 )
2120nn0zd 10115 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  N  e.  ZZ )
22 fzsn 10833 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  ( N ... N )  =  { N } )
2322eleq2d 2350 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
k  e.  ( N ... N )  <->  k  e.  { N } ) )
24 elsnc2g 3668 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
k  e.  { N } 
<->  k  =  N ) )
2523, 24bitrd 244 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
k  e.  ( N ... N )  <->  k  =  N ) )
2621, 25syl 15 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  ( k  e.  ( N ... N
)  <->  k  =  N ) )
2719, 26mtbid 291 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  -.  k  =  N )
28 iffalse 3572 . . . . . . . 8  |-  ( -.  k  =  N  ->  if ( k  =  N ,  A ,  0 )  =  0 )
2927, 28syl 15 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  if (
k  =  N ,  A ,  0 )  =  0 )
3029oveq1d 5873 . . . . . 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 447 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  z  e.  CC )
32 eldifi 3298 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... N )  \ 
( N ... N
) )  ->  k  e.  ( 0 ... N
) )
33 elfznn0 10822 . . . . . . . . 9  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
3432, 33syl 15 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... N )  \ 
( N ... N
) )  ->  k  e.  NN0 )
35 expcl 11121 . . . . . . . 8  |-  ( ( z  e.  CC  /\  k  e.  NN0 )  -> 
( z ^ k
)  e.  CC )
3631, 34, 35syl2an 463 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  ( z ^ k )  e.  CC )
3736mul02d 9010 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  ( 0  x.  ( z ^
k ) )  =  0 )
3830, 37eqtrd 2315 . . . . 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 11035 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( 0 ... N )  e.  Fin )
405, 17, 38, 39fsumss 12198 . . . 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 10115 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  N  e.  ZZ )
4231, 1expcld 11245 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( z ^ N )  e.  CC )
4310, 42mulcld 8855 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( A  x.  ( z ^ N
) )  e.  CC )
44 oveq2 5866 . . . . . . 7  |-  ( k  =  N  ->  (
z ^ k )  =  ( z ^ N ) )
458, 44oveq12d 5876 . . . . . 6  |-  ( k  =  N  ->  ( if ( k  =  N ,  A ,  0 )  x.  ( z ^ k ) )  =  ( A  x.  ( z ^ N
) ) )
4645fsum1 12214 . . . . 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 642 . . . 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 2317 . . 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 4106 . 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 2334 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   ifcif 3565   {csn 3640    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737    x. cmul 8742   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   ^cexp 11104   sum_csu 12158
This theorem is referenced by:  ply1term  19586  coe1termlem  19639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159
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