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Theorem dvply1 19666
Description: Derivative of a polynomial, explicit sum version. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
Hypotheses
Ref Expression
dvply1.f  |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( z ^ k
) ) ) )
dvply1.g  |-  ( ph  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( B `
 k )  x.  ( z ^ k
) ) ) )
dvply1.a  |-  ( ph  ->  A : NN0 --> CC )
dvply1.b  |-  B  =  ( k  e.  NN0  |->  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) ) )
dvply1.n  |-  ( ph  ->  N  e.  NN0 )
Assertion
Ref Expression
dvply1  |-  ( ph  ->  ( CC  _D  F
)  =  G )
Distinct variable groups:    ph, z, k   
z, A, k    z, B    k, N, z
Allowed substitution hints:    B( k)    F( z, k)    G( z, k)

Proof of Theorem dvply1
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 dvply1.f . . 3  |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( z ^ k
) ) ) )
21oveq2d 5876 . 2  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
z ^ k ) ) ) ) )
3 eqid 2285 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
43cnfldtop 18295 . . . . 5  |-  ( TopOpen ` fld )  e.  Top
53cnfldtopon 18294 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
65toponunii 16672 . . . . . 6  |-  CC  =  U. ( TopOpen ` fld )
76restid 13340 . . . . 5  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
84, 7ax-mp 8 . . . 4  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
98eqcomi 2289 . . 3  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
10 cnex 8820 . . . . 5  |-  CC  e.  _V
1110prid2 3737 . . . 4  |-  CC  e.  { RR ,  CC }
1211a1i 10 . . 3  |-  ( ph  ->  CC  e.  { RR ,  CC } )
136topopn 16654 . . . 4  |-  ( (
TopOpen ` fld )  e.  Top  ->  CC  e.  ( TopOpen ` fld ) )
144, 13mp1i 11 . . 3  |-  ( ph  ->  CC  e.  ( TopOpen ` fld )
)
15 fzfid 11037 . . 3  |-  ( ph  ->  ( 0 ... N
)  e.  Fin )
16 dvply1.a . . . . . . 7  |-  ( ph  ->  A : NN0 --> CC )
17 elfznn0 10824 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
18 ffvelrn 5665 . . . . . . 7  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
1916, 17, 18syl2an 463 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
2019adantr 451 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  ( A `  k )  e.  CC )
21 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  z  e.  CC )
2217ad2antlr 707 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  k  e.  NN0 )
2321, 22expcld 11247 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  (
z ^ k )  e.  CC )
2420, 23mulcld 8857 . . . 4  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  (
( A `  k
)  x.  ( z ^ k ) )  e.  CC )
25243impa 1146 . . 3  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  ( ( A `  k )  x.  ( z ^
k ) )  e.  CC )
26193adant3 975 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  ( A `
 k )  e.  CC )
27 0cn 8833 . . . . . 6  |-  0  e.  CC
2827a1i 10 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  k  =  0 )  -> 
0  e.  CC )
29 simpl2 959 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
k  e.  ( 0 ... N ) )
3029, 17syl 15 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
k  e.  NN0 )
3130nn0cnd 10022 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
k  e.  CC )
32 simpl3 960 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
z  e.  CC )
33 simpr 447 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  ->  -.  k  =  0
)
34 elnn0 9969 . . . . . . . . . 10  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
3530, 34sylib 188 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( k  e.  NN  \/  k  =  0
) )
36 orel2 372 . . . . . . . . 9  |-  ( -.  k  =  0  -> 
( ( k  e.  NN  \/  k  =  0 )  ->  k  e.  NN ) )
3733, 35, 36sylc 56 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
k  e.  NN )
38 nnm1nn0 10007 . . . . . . . 8  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
3937, 38syl 15 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( k  -  1 )  e.  NN0 )
4032, 39expcld 11247 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( z ^ (
k  -  1 ) )  e.  CC )
4131, 40mulcld 8857 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( k  x.  (
z ^ ( k  -  1 ) ) )  e.  CC )
4228, 41ifclda 3594 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) )  e.  CC )
4326, 42mulcld 8857 . . 3  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  ( ( A `  k )  x.  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ (
k  -  1 ) ) ) ) )  e.  CC )
4411a1i 10 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  CC  e.  { RR ,  CC } )
45 c0ex 8834 . . . . . 6  |-  0  e.  _V
46 ovex 5885 . . . . . 6  |-  ( k  x.  ( z ^
( k  -  1 ) ) )  e. 
_V
4745, 46ifex 3625 . . . . 5  |-  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) )  e.  _V
4847a1i 10 . . . 4  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  e.  _V )
4917adantl 452 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
50 dvexp2 19305 . . . . 5  |-  ( k  e.  NN0  ->  ( CC 
_D  ( z  e.  CC  |->  ( z ^
k ) ) )  =  ( z  e.  CC  |->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ (
k  -  1 ) ) ) ) ) )
5149, 50syl 15 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( CC  _D  ( z  e.  CC  |->  ( z ^
k ) ) )  =  ( z  e.  CC  |->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ (
k  -  1 ) ) ) ) ) )
5244, 23, 48, 51, 19dvmptcmul 19315 . . 3  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( CC  _D  ( z  e.  CC  |->  ( ( A `
 k )  x.  ( z ^ k
) ) ) )  =  ( z  e.  CC  |->  ( ( A `
 k )  x.  if ( k  =  0 ,  0 ,  ( k  x.  (
z ^ ( k  -  1 ) ) ) ) ) ) )
539, 3, 12, 14, 15, 25, 43, 52dvmptfsum 19324 . 2  |-  ( ph  ->  ( CC  _D  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) ) ) ) )
54 elfznn 10821 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN )
5554nnne0d 9792 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... N )  ->  k  =/=  0 )
5655neneqd 2464 . . . . . . . . 9  |-  ( k  e.  ( 1 ... N )  ->  -.  k  =  0 )
5756adantl 452 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  -.  k  =  0 )
58 iffalse 3574 . . . . . . . 8  |-  ( -.  k  =  0  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  =  ( k  x.  ( z ^
( k  -  1 ) ) ) )
5957, 58syl 15 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  =  ( k  x.  ( z ^
( k  -  1 ) ) ) )
6059oveq2d 5876 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
( A `  k
)  x.  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) ) )  =  ( ( A `  k )  x.  ( k  x.  ( z ^ (
k  -  1 ) ) ) ) )
6160sumeq2dv 12178 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 1 ... N
) ( ( A `
 k )  x.  if ( k  =  0 ,  0 ,  ( k  x.  (
z ^ ( k  -  1 ) ) ) ) )  = 
sum_ k  e.  ( 1 ... N ) ( ( A `  k )  x.  (
k  x.  ( z ^ ( k  - 
1 ) ) ) ) )
62 1nn0 9983 . . . . . . . 8  |-  1  e.  NN0
63 nn0uz 10264 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
6462, 63eleqtri 2357 . . . . . . 7  |-  1  e.  ( ZZ>= `  0 )
65 fzss1 10832 . . . . . . 7  |-  ( 1  e.  ( ZZ>= `  0
)  ->  ( 1 ... N )  C_  ( 0 ... N
) )
6664, 65mp1i 11 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( 1 ... N )  C_  ( 0 ... N
) )
6716adantr 451 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  A : NN0
--> CC )
6854nnnn0d 10020 . . . . . . . 8  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN0 )
6967, 68, 18syl2an 463 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  ( A `  k )  e.  CC )
7055adantl 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  k  =/=  0 )
7170neneqd 2464 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  -.  k  =  0 )
7271, 58syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  =  ( k  x.  ( z ^
( k  -  1 ) ) ) )
7368adantl 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  k  e.  NN0 )
7473nn0cnd 10022 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  k  e.  CC )
75 simplr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  z  e.  CC )
7654, 38syl 15 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... N )  ->  (
k  -  1 )  e.  NN0 )
7776adantl 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
k  -  1 )  e.  NN0 )
7875, 77expcld 11247 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
z ^ ( k  -  1 ) )  e.  CC )
7974, 78mulcld 8857 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
k  x.  ( z ^ ( k  - 
1 ) ) )  e.  CC )
8072, 79eqeltrd 2359 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  e.  CC )
8169, 80mulcld 8857 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
( A `  k
)  x.  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) ) )  e.  CC )
82 eldifn 3301 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... N )  \ 
( 1 ... N
) )  ->  -.  k  e.  ( 1 ... N ) )
83 0p1e1 9841 . . . . . . . . . . . . . 14  |-  ( 0  +  1 )  =  1
8483oveq1i 5870 . . . . . . . . . . . . 13  |-  ( ( 0  +  1 ) ... N )  =  ( 1 ... N
)
8584eleq2i 2349 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0  +  1 ) ... N )  <->  k  e.  ( 1 ... N
) )
8682, 85sylnibr 296 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... N )  \ 
( 1 ... N
) )  ->  -.  k  e.  ( (
0  +  1 ) ... N ) )
8786adantl 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  -.  k  e.  ( ( 0  +  1 ) ... N
) )
88 eldifi 3300 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... N )  \ 
( 1 ... N
) )  ->  k  e.  ( 0 ... N
) )
8988adantl 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  k  e.  ( 0 ... N
) )
90 dvply1.n . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  NN0 )
9190, 63syl6eleq 2375 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
9291ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  N  e.  ( ZZ>= `  0 )
)
93 elfzp12 10863 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( k  e.  ( 0 ... N
)  <->  ( k  =  0  \/  k  e.  ( ( 0  +  1 ) ... N
) ) ) )
9492, 93syl 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  ( k  e.  ( 0 ... N
)  <->  ( k  =  0  \/  k  e.  ( ( 0  +  1 ) ... N
) ) ) )
9589, 94mpbid 201 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  ( k  =  0  \/  k  e.  ( ( 0  +  1 ) ... N
) ) )
96 orel2 372 . . . . . . . . . 10  |-  ( -.  k  e.  ( ( 0  +  1 ) ... N )  -> 
( ( k  =  0  \/  k  e.  ( ( 0  +  1 ) ... N
) )  ->  k  =  0 ) )
9787, 95, 96sylc 56 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  k  = 
0 )
98 iftrue 3573 . . . . . . . . 9  |-  ( k  =  0  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  =  0 )
9997, 98syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  if (
k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) )  =  0 )
10099oveq2d 5876 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  ( ( A `  k )  x.  if ( k  =  0 ,  0 ,  ( k  x.  (
z ^ ( k  -  1 ) ) ) ) )  =  ( ( A `  k )  x.  0 ) )
10167, 17, 18syl2an 463 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
102101mul01d 9013 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  0 )  =  0 )
10388, 102sylan2 460 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  ( ( A `  k )  x.  0 )  =  0 )
104100, 103eqtrd 2317 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  ( ( A `  k )  x.  if ( k  =  0 ,  0 ,  ( k  x.  (
z ^ ( k  -  1 ) ) ) ) )  =  0 )
105 fzfid 11037 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... N )  e. 
Fin )
10666, 81, 104, 105fsumss 12200 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 1 ... N
) ( ( A `
 k )  x.  if ( k  =  0 ,  0 ,  ( k  x.  (
z ^ ( k  -  1 ) ) ) ) )  = 
sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) ) ) )
107 elfznn0 10824 . . . . . . . . . . . . . . 15  |-  ( j  e.  ( 0 ... ( N  -  1 ) )  ->  j  e.  NN0 )
108107adantl 452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  j  e.  NN0 )
109108nn0cnd 10022 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  j  e.  CC )
110 ax-1cn 8797 . . . . . . . . . . . . 13  |-  1  e.  CC
111 pncan 9059 . . . . . . . . . . . . 13  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( ( j  +  1 )  -  1 )  =  j )
112109, 110, 111sylancl 643 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( j  +  1 )  -  1 )  =  j )
113112oveq2d 5876 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
z ^ ( ( j  +  1 )  -  1 ) )  =  ( z ^
j ) )
114113oveq2d 5876 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( j  +  1 )  x.  ( z ^ ( ( j  +  1 )  - 
1 ) ) )  =  ( ( j  +  1 )  x.  ( z ^ j
) ) )
115114oveq2d 5876 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( A `  (
j  +  1 ) )  x.  ( ( j  +  1 )  x.  ( z ^
( ( j  +  1 )  -  1 ) ) ) )  =  ( ( A `
 ( j  +  1 ) )  x.  ( ( j  +  1 )  x.  (
z ^ j ) ) ) )
11616ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  A : NN0 --> CC )
117 peano2nn0 10006 . . . . . . . . . . . . 13  |-  ( j  e.  NN0  ->  ( j  +  1 )  e. 
NN0 )
118107, 117syl 15 . . . . . . . . . . . 12  |-  ( j  e.  ( 0 ... ( N  -  1 ) )  ->  (
j  +  1 )  e.  NN0 )
119118adantl 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
j  +  1 )  e.  NN0 )
120116, 119ffvelrnd 5668 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( A `  ( j  +  1 ) )  e.  CC )
121119nn0cnd 10022 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
j  +  1 )  e.  CC )
122 simplr 731 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  z  e.  CC )
123122, 108expcld 11247 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
z ^ j )  e.  CC )
124120, 121, 123mulassd 8860 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( A `  ( j  +  1 ) )  x.  (
j  +  1 ) )  x.  ( z ^ j ) )  =  ( ( A `
 ( j  +  1 ) )  x.  ( ( j  +  1 )  x.  (
z ^ j ) ) ) )
125120, 121mulcomd 8858 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( A `  (
j  +  1 ) )  x.  ( j  +  1 ) )  =  ( ( j  +  1 )  x.  ( A `  (
j  +  1 ) ) ) )
126125oveq1d 5875 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( A `  ( j  +  1 ) )  x.  (
j  +  1 ) )  x.  ( z ^ j ) )  =  ( ( ( j  +  1 )  x.  ( A `  ( j  +  1 ) ) )  x.  ( z ^ j
) ) )
127115, 124, 1263eqtr2d 2323 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( A `  (
j  +  1 ) )  x.  ( ( j  +  1 )  x.  ( z ^
( ( j  +  1 )  -  1 ) ) ) )  =  ( ( ( j  +  1 )  x.  ( A `  ( j  +  1 ) ) )  x.  ( z ^ j
) ) )
128127sumeq2dv 12178 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( ( A `
 ( j  +  1 ) )  x.  ( ( j  +  1 )  x.  (
z ^ ( ( j  +  1 )  -  1 ) ) ) )  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( ( ( j  +  1 )  x.  ( A `  ( j  +  1 ) ) )  x.  ( z ^ j ) ) )
129 1m1e0 9816 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
130129oveq1i 5870 . . . . . . . 8  |-  ( ( 1  -  1 ) ... ( N  - 
1 ) )  =  ( 0 ... ( N  -  1 ) )
131130sumeq1i 12173 . . . . . . 7  |-  sum_ j  e.  ( ( 1  -  1 ) ... ( N  -  1 ) ) ( ( A `
 ( j  +  1 ) )  x.  ( ( j  +  1 )  x.  (
z ^ ( ( j  +  1 )  -  1 ) ) ) )  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( ( A `  (
j  +  1 ) )  x.  ( ( j  +  1 )  x.  ( z ^
( ( j  +  1 )  -  1 ) ) ) )
132 oveq1 5867 . . . . . . . . . 10  |-  ( k  =  j  ->  (
k  +  1 )  =  ( j  +  1 ) )
133132fveq2d 5531 . . . . . . . . . 10  |-  ( k  =  j  ->  ( A `  ( k  +  1 ) )  =  ( A `  ( j  +  1 ) ) )
134132, 133oveq12d 5878 . . . . . . . . 9  |-  ( k  =  j  ->  (
( k  +  1 )  x.  ( A `
 ( k  +  1 ) ) )  =  ( ( j  +  1 )  x.  ( A `  (
j  +  1 ) ) ) )
135 oveq2 5868 . . . . . . . . 9  |-  ( k  =  j  ->  (
z ^ k )  =  ( z ^
j ) )
136134, 135oveq12d 5878 . . . . . . . 8  |-  ( k  =  j  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( z ^ k ) )  =  ( ( ( j  +  1 )  x.  ( A `  ( j  +  1 ) ) )  x.  ( z ^ j
) ) )
137136cbvsumv 12171 . . . . . . 7  |-  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( z ^ k
) )  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( ( ( j  +  1 )  x.  ( A `  ( j  +  1 ) ) )  x.  ( z ^ j ) )
138128, 131, 1373eqtr4g 2342 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ j  e.  ( ( 1  -  1 ) ... ( N  -  1 ) ) ( ( A `
 ( j  +  1 ) )  x.  ( ( j  +  1 )  x.  (
z ^ ( ( j  +  1 )  -  1 ) ) ) )  =  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( z ^ k ) ) )
139 1z 10055 . . . . . . . 8  |-  1  e.  ZZ
140139a1i 10 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  1  e.  ZZ )
14190adantr 451 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  N  e. 
NN0 )
142141nn0zd 10117 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  N  e.  ZZ )
14369, 79mulcld 8857 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
( A `  k
)  x.  ( k  x.  ( z ^
( k  -  1 ) ) ) )  e.  CC )
144 fveq2 5527 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  ( A `  k )  =  ( A `  ( j  +  1 ) ) )
145 id 19 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  k  =  ( j  +  1 ) )
146 oveq1 5867 . . . . . . . . . 10  |-  ( k  =  ( j  +  1 )  ->  (
k  -  1 )  =  ( ( j  +  1 )  - 
1 ) )
147146oveq2d 5876 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  (
z ^ ( k  -  1 ) )  =  ( z ^
( ( j  +  1 )  -  1 ) ) )
148145, 147oveq12d 5878 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
k  x.  ( z ^ ( k  - 
1 ) ) )  =  ( ( j  +  1 )  x.  ( z ^ (
( j  +  1 )  -  1 ) ) ) )
149144, 148oveq12d 5878 . . . . . . 7  |-  ( k  =  ( j  +  1 )  ->  (
( A `  k
)  x.  ( k  x.  ( z ^
( k  -  1 ) ) ) )  =  ( ( A `
 ( j  +  1 ) )  x.  ( ( j  +  1 )  x.  (
z ^ ( ( j  +  1 )  -  1 ) ) ) ) )
150140, 140, 142, 143, 149fsumshftm 12245 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 1 ... N
) ( ( A `
 k )  x.  ( k  x.  (
z ^ ( k  -  1 ) ) ) )  =  sum_ j  e.  ( (
1  -  1 ) ... ( N  - 
1 ) ) ( ( A `  (
j  +  1 ) )  x.  ( ( j  +  1 )  x.  ( z ^
( ( j  +  1 )  -  1 ) ) ) ) )
151 elfznn0 10824 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
152151adantl 452 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  k  e.  NN0 )
153 ovex 5885 . . . . . . . . 9  |-  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  e. 
_V
154 dvply1.b . . . . . . . . . 10  |-  B  =  ( k  e.  NN0  |->  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) ) )
155154fvmpt2 5610 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  e.  _V )  ->  ( B `  k
)  =  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) ) )
156152, 153, 155sylancl 643 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( B `  k )  =  ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) ) )
157156oveq1d 5875 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( B `  k
)  x.  ( z ^ k ) )  =  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( z ^ k
) ) )
158157sumeq2dv 12178 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( B `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( z ^ k ) ) )
159138, 150, 1583eqtr4d 2327 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 1 ... N
) ( ( A `
 k )  x.  ( k  x.  (
z ^ ( k  -  1 ) ) ) )  =  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( B `  k
)  x.  ( z ^ k ) ) )
16061, 106, 1593eqtr3d 2325 . . . 4  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  if ( k  =  0 ,  0 ,  ( k  x.  (
z ^ ( k  -  1 ) ) ) ) )  = 
sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( B `  k )  x.  (
z ^ k ) ) )
161160mpteq2dva 4108 . . 3  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( B `  k
)  x.  ( z ^ k ) ) ) )
162 dvply1.g . . 3  |-  ( ph  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( B `
 k )  x.  ( z ^ k
) ) ) )
163161, 162eqtr4d 2320 . 2  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) ) ) )  =  G )
1642, 53, 1633eqtrd 2321 1  |-  ( ph  ->  ( CC  _D  F
)  =  G )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   _Vcvv 2790    \ cdif 3151    C_ wss 3154   ifcif 3567   {cpr 3643    e. cmpt 4079   -->wf 5253   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    + caddc 8742    x. cmul 8744    - cmin 9039   NNcn 9748   NN0cn0 9967   ZZcz 10026   ZZ>=cuz 10232   ...cfz 10784   ^cexp 11106   sum_csu 12160   ↾t crest 13327   TopOpenctopn 13328  ℂfldccnfld 16379   Topctop 16633    _D cdv 19215
This theorem is referenced by:  dvply2g  19667
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  ax-addf 8818  ax-mulf 8819
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-iin 3910  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-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  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-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-icc 10665  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  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-limc 19218  df-dv 19219
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