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Theorem ply1termlem 19587
Description: Lemma for ply1term 19588. (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 10264 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2375 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  N  e.  (
ZZ>= `  0 ) )
4 fzss1 10832 . . . . . 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 10809 . . . . . . . . 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 3573 . . . . . . . 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 2359 . . . . . 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 2359 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  k  e.  NN0 )
1613, 15expcld 11247 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( N ... N
) )  ->  (
z ^ k )  e.  CC )
1712, 16mulcld 8857 . . . . 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 3301 . . . . . . . . . 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 10117 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  N  e.  ZZ )
22 fzsn 10835 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  ( N ... N )  =  { N } )
2322eleq2d 2352 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
k  e.  ( N ... N )  <->  k  e.  { N } ) )
24 elsnc2g 3670 . . . . . . . . . . 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 3574 . . . . . . . 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 5875 . . . . . 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 3300 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... N )  \ 
( N ... N
) )  ->  k  e.  ( 0 ... N
) )
33 elfznn0 10824 . . . . . . . . 9  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
3432, 33syl 15 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... N )  \ 
( N ... N
) )  ->  k  e.  NN0 )
35 expcl 11123 . . . . . . . 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 9012 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e. 
NN0 )  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  ( N ... N ) ) )  ->  ( 0  x.  ( z ^
k ) )  =  0 )
3830, 37eqtrd 2317 . . . . 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 11037 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( 0 ... N )  e.  Fin )
405, 17, 38, 39fsumss 12200 . . . 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 10117 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  N  e.  ZZ )
4231, 1expcld 11247 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( z ^ N )  e.  CC )
4310, 42mulcld 8857 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  z  e.  CC )  ->  ( A  x.  ( z ^ N
) )  e.  CC )
44 oveq2 5868 . . . . . . 7  |-  ( k  =  N  ->  (
z ^ k )  =  ( z ^ N ) )
458, 44oveq12d 5878 . . . . . 6  |-  ( k  =  N  ->  ( if ( k  =  N ,  A ,  0 )  x.  ( z ^ k ) )  =  ( A  x.  ( z ^ N
) ) )
4645fsum1 12216 . . . . 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 2319 . . 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 4108 . 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 2336 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 1625    e. wcel 1686    \ cdif 3151    C_ wss 3154   ifcif 3567   {csn 3642    e. cmpt 4079   ` cfv 5257  (class class class)co 5860   CCcc 8737   0cc0 8739    x. cmul 8744   NN0cn0 9967   ZZcz 10026   ZZ>=cuz 10232   ...cfz 10784   ^cexp 11106   sum_csu 12160
This theorem is referenced by:  ply1term  19588  coe1termlem  19641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-oi 7227  df-card 7574  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-rp 10357  df-fz 10785  df-fzo 10873  df-seq 11049  df-exp 11107  df-hash 11340  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-clim 11964  df-sum 12161
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