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Theorem dvply1 19680
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 5890 . 2  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
z ^ k ) ) ) ) )
3 eqid 2296 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
43cnfldtop 18309 . . . . 5  |-  ( TopOpen ` fld )  e.  Top
53cnfldtopon 18308 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
65toponunii 16686 . . . . . 6  |-  CC  =  U. ( TopOpen ` fld )
76restid 13354 . . . . 5  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
84, 7ax-mp 8 . . . 4  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
98eqcomi 2300 . . 3  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
10 cnex 8834 . . . . 5  |-  CC  e.  _V
1110prid2 3748 . . . 4  |-  CC  e.  { RR ,  CC }
1211a1i 10 . . 3  |-  ( ph  ->  CC  e.  { RR ,  CC } )
136topopn 16668 . . . 4  |-  ( (
TopOpen ` fld )  e.  Top  ->  CC  e.  ( TopOpen ` fld ) )
144, 13mp1i 11 . . 3  |-  ( ph  ->  CC  e.  ( TopOpen ` fld )
)
15 fzfid 11051 . . 3  |-  ( ph  ->  ( 0 ... N
)  e.  Fin )
16 dvply1.a . . . . . . 7  |-  ( ph  ->  A : NN0 --> CC )
17 elfznn0 10838 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
18 ffvelrn 5679 . . . . . . 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 11261 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  (
z ^ k )  e.  CC )
2420, 23mulcld 8871 . . . 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 8847 . . . . . 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 10036 . . . . . 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 9983 . . . . . . . . . 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 10021 . . . . . . . 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 11261 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( z ^ (
k  -  1 ) )  e.  CC )
4131, 40mulcld 8871 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( k  x.  (
z ^ ( k  -  1 ) ) )  e.  CC )
4228, 41ifclda 3605 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) )  e.  CC )
4326, 42mulcld 8871 . . 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 8848 . . . . . 6  |-  0  e.  _V
46 ovex 5899 . . . . . 6  |-  ( k  x.  ( z ^
( k  -  1 ) ) )  e. 
_V
4745, 46ifex 3636 . . . . 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 19319 . . . . 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 19329 . . 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 19338 . 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 10835 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN )
5554nnne0d 9806 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... N )  ->  k  =/=  0 )
5655neneqd 2475 . . . . . . . . 9  |-  ( k  e.  ( 1 ... N )  ->  -.  k  =  0 )
5756adantl 452 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  -.  k  =  0 )
58 iffalse 3585 . . . . . . . 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 5890 . . . . . 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 12192 . . . . 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 9997 . . . . . . . 8  |-  1  e.  NN0
63 nn0uz 10278 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
6462, 63eleqtri 2368 . . . . . . 7  |-  1  e.  ( ZZ>= `  0 )
65 fzss1 10846 . . . . . . 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 10034 . . . . . . . 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 2475 . . . . . . . . 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 10036 . . . . . . . . 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 11261 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
z ^ ( k  -  1 ) )  e.  CC )
7974, 78mulcld 8871 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
k  x.  ( z ^ ( k  - 
1 ) ) )  e.  CC )
8072, 79eqeltrd 2370 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  e.  CC )
8169, 80mulcld 8871 . . . . . 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 3312 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... N )  \ 
( 1 ... N
) )  ->  -.  k  e.  ( 1 ... N ) )
83 0p1e1 9855 . . . . . . . . . . . . . 14  |-  ( 0  +  1 )  =  1
8483oveq1i 5884 . . . . . . . . . . . . 13  |-  ( ( 0  +  1 ) ... N )  =  ( 1 ... N
)
8584eleq2i 2360 . . . . . . . . . . . 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 3311 . . . . . . . . . . . 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 2386 . . . . . . . . . . . . 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 10877 . . . . . . . . . . . 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 3584 . . . . . . . . 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 5890 . . . . . . 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 9027 . . . . . . . 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 2328 . . . . . 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 11051 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... N )  e. 
Fin )
10666, 81, 104, 105fsumss 12214 . . . . 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 10838 . . . . . . . . . . . . . . 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 10036 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  j  e.  CC )
110 ax-1cn 8811 . . . . . . . . . . . . 13  |-  1  e.  CC
111 pncan 9073 . . . . . . . . . . . . 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 5890 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
z ^ ( ( j  +  1 )  -  1 ) )  =  ( z ^
j ) )
114113oveq2d 5890 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( j  +  1 )  x.  ( z ^ ( ( j  +  1 )  - 
1 ) ) )  =  ( ( j  +  1 )  x.  ( z ^ j
) ) )
115114oveq2d 5890 . . . . . . . . 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 10020 . . . . . . . . . . . . 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 5682 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( A `  ( j  +  1 ) )  e.  CC )
121119nn0cnd 10036 . . . . . . . . . 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 11261 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
z ^ j )  e.  CC )
124120, 121, 123mulassd 8874 . . . . . . . . 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 8872 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( A `  (
j  +  1 ) )  x.  ( j  +  1 ) )  =  ( ( j  +  1 )  x.  ( A `  (
j  +  1 ) ) ) )
126125oveq1d 5889 . . . . . . . . 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 2334 . . . . . . . 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 12192 . . . . . . 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 9830 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
130129oveq1i 5884 . . . . . . . 8  |-  ( ( 1  -  1 ) ... ( N  - 
1 ) )  =  ( 0 ... ( N  -  1 ) )
131130sumeq1i 12187 . . . . . . 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 5881 . . . . . . . . . 10  |-  ( k  =  j  ->  (
k  +  1 )  =  ( j  +  1 ) )
133132fveq2d 5545 . . . . . . . . . 10  |-  ( k  =  j  ->  ( A `  ( k  +  1 ) )  =  ( A `  ( j  +  1 ) ) )
134132, 133oveq12d 5892 . . . . . . . . 9  |-  ( k  =  j  ->  (
( k  +  1 )  x.  ( A `
 ( k  +  1 ) ) )  =  ( ( j  +  1 )  x.  ( A `  (
j  +  1 ) ) ) )
135 oveq2 5882 . . . . . . . . 9  |-  ( k  =  j  ->  (
z ^ k )  =  ( z ^
j ) )
136134, 135oveq12d 5892 . . . . . . . 8  |-  ( k  =  j  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( z ^ k ) )  =  ( ( ( j  +  1 )  x.  ( A `  ( j  +  1 ) ) )  x.  ( z ^ j
) ) )
137136cbvsumv 12185 . . . . . . 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 2353 . . . . . 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 10069 . . . . . . . 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 10131 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  N  e.  ZZ )
14369, 79mulcld 8871 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
( A `  k
)  x.  ( k  x.  ( z ^
( k  -  1 ) ) ) )  e.  CC )
144 fveq2 5541 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  ( A `  k )  =  ( A `  ( j  +  1 ) ) )
145 id 19 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  k  =  ( j  +  1 ) )
146 oveq1 5881 . . . . . . . . . 10  |-  ( k  =  ( j  +  1 )  ->  (
k  -  1 )  =  ( ( j  +  1 )  - 
1 ) )
147146oveq2d 5890 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  (
z ^ ( k  -  1 ) )  =  ( z ^
( ( j  +  1 )  -  1 ) ) )
148145, 147oveq12d 5892 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
k  x.  ( z ^ ( k  - 
1 ) ) )  =  ( ( j  +  1 )  x.  ( z ^ (
( j  +  1 )  -  1 ) ) ) )
149144, 148oveq12d 5892 . . . . . . 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 12259 . . . . . 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 10838 . . . . . . . . . 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 5899 . . . . . . . . 9  |-  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  e. 
_V
154 dvply1.b . . . . . . . . . 10  |-  B  =  ( k  e.  NN0  |->  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) ) )
155154fvmpt2 5624 . . . . . . . . 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 5889 . . . . . . 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 12192 . . . . . 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 2338 . . . . 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 2336 . . . 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 4122 . . 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 2331 . 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 2332 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 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    \ cdif 3162    C_ wss 3165   ifcif 3578   {cpr 3654    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   ^cexp 11120   sum_csu 12174   ↾t crest 13341   TopOpenctopn 13342  ℂfldccnfld 16393   Topctop 16647    _D cdv 19229
This theorem is referenced by:  dvply2g  19681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233
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