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Theorem dvply1 21709
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 6106 . 2  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
z ^ k ) ) ) ) )
3 eqid 2441 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
43cnfldtop 20322 . . . . 5  |-  ( TopOpen ` fld )  e.  Top
53cnfldtopon 20321 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
65toponunii 18496 . . . . . 6  |-  CC  =  U. ( TopOpen ` fld )
76restid 14368 . . . . 5  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
84, 7ax-mp 5 . . . 4  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
98eqcomi 2445 . . 3  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
10 cnelprrecn 9371 . . . 4  |-  CC  e.  { RR ,  CC }
1110a1i 11 . . 3  |-  ( ph  ->  CC  e.  { RR ,  CC } )
126topopn 18478 . . . 4  |-  ( (
TopOpen ` fld )  e.  Top  ->  CC  e.  ( TopOpen ` fld ) )
134, 12mp1i 12 . . 3  |-  ( ph  ->  CC  e.  ( TopOpen ` fld )
)
14 fzfid 11791 . . 3  |-  ( ph  ->  ( 0 ... N
)  e.  Fin )
15 dvply1.a . . . . . . 7  |-  ( ph  ->  A : NN0 --> CC )
16 elfznn0 11477 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
17 ffvelrn 5838 . . . . . . 7  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
1815, 16, 17syl2an 474 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
1918adantr 462 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  ( A `  k )  e.  CC )
20 simpr 458 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  z  e.  CC )
2116ad2antlr 721 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  k  e.  NN0 )
2220, 21expcld 12004 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  (
z ^ k )  e.  CC )
2319, 22mulcld 9402 . . . 4  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  (
( A `  k
)  x.  ( z ^ k ) )  e.  CC )
24233impa 1177 . . 3  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  ( ( A `  k )  x.  ( z ^
k ) )  e.  CC )
25183adant3 1003 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  ( A `
 k )  e.  CC )
26 0cnd 9375 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  k  =  0 )  -> 
0  e.  CC )
27 simpl2 987 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
k  e.  ( 0 ... N ) )
2827, 16syl 16 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
k  e.  NN0 )
2928nn0cnd 10634 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
k  e.  CC )
30 simpl3 988 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
z  e.  CC )
31 simpr 458 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  ->  -.  k  =  0
)
32 elnn0 10577 . . . . . . . . . 10  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
3328, 32sylib 196 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( k  e.  NN  \/  k  =  0
) )
34 orel2 383 . . . . . . . . 9  |-  ( -.  k  =  0  -> 
( ( k  e.  NN  \/  k  =  0 )  ->  k  e.  NN ) )
3531, 33, 34sylc 60 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
k  e.  NN )
36 nnm1nn0 10617 . . . . . . . 8  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
3735, 36syl 16 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( k  -  1 )  e.  NN0 )
3830, 37expcld 12004 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( z ^ (
k  -  1 ) )  e.  CC )
3929, 38mulcld 9402 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( k  x.  (
z ^ ( k  -  1 ) ) )  e.  CC )
4026, 39ifclda 3818 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) )  e.  CC )
4125, 40mulcld 9402 . . 3  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  ( ( A `  k )  x.  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ (
k  -  1 ) ) ) ) )  e.  CC )
4210a1i 11 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  CC  e.  { RR ,  CC } )
43 c0ex 9376 . . . . . 6  |-  0  e.  _V
44 ovex 6115 . . . . . 6  |-  ( k  x.  ( z ^
( k  -  1 ) ) )  e. 
_V
4543, 44ifex 3855 . . . . 5  |-  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) )  e.  _V
4645a1i 11 . . . 4  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  e.  _V )
4716adantl 463 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
48 dvexp2 21387 . . . . 5  |-  ( k  e.  NN0  ->  ( CC 
_D  ( z  e.  CC  |->  ( z ^
k ) ) )  =  ( z  e.  CC  |->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ (
k  -  1 ) ) ) ) ) )
4947, 48syl 16 . . . 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 ) ) ) ) ) )
5042, 22, 46, 49, 18dvmptcmul 21397 . . 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 ) ) ) ) ) ) )
519, 3, 11, 13, 14, 24, 41, 50dvmptfsum 21406 . 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 ) ) ) ) ) ) )
52 elfznn 11474 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN )
5352nnne0d 10362 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... N )  ->  k  =/=  0 )
5453neneqd 2622 . . . . . . . . 9  |-  ( k  e.  ( 1 ... N )  ->  -.  k  =  0 )
5554adantl 463 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  -.  k  =  0 )
56 iffalse 3796 . . . . . . . 8  |-  ( -.  k  =  0  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  =  ( k  x.  ( z ^
( k  -  1 ) ) ) )
5755, 56syl 16 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  =  ( k  x.  ( z ^
( k  -  1 ) ) ) )
5857oveq2d 6106 . . . . . 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 ) ) ) ) )
5958sumeq2dv 13176 . . . . 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 ) ) ) ) )
60 1nn0 10591 . . . . . . . 8  |-  1  e.  NN0
61 nn0uz 10891 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
6260, 61eleqtri 2513 . . . . . . 7  |-  1  e.  ( ZZ>= `  0 )
63 fzss1 11493 . . . . . . 7  |-  ( 1  e.  ( ZZ>= `  0
)  ->  ( 1 ... N )  C_  ( 0 ... N
) )
6462, 63mp1i 12 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( 1 ... N )  C_  ( 0 ... N
) )
6515adantr 462 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  A : NN0
--> CC )
6652nnnn0d 10632 . . . . . . . 8  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN0 )
6765, 66, 17syl2an 474 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  ( A `  k )  e.  CC )
6853adantl 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  k  =/=  0 )
6968neneqd 2622 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  -.  k  =  0 )
7069, 56syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  =  ( k  x.  ( z ^
( k  -  1 ) ) ) )
7166adantl 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  k  e.  NN0 )
7271nn0cnd 10634 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  k  e.  CC )
73 simplr 749 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  z  e.  CC )
7452, 36syl 16 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... N )  ->  (
k  -  1 )  e.  NN0 )
7574adantl 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
k  -  1 )  e.  NN0 )
7673, 75expcld 12004 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
z ^ ( k  -  1 ) )  e.  CC )
7772, 76mulcld 9402 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
k  x.  ( z ^ ( k  - 
1 ) ) )  e.  CC )
7870, 77eqeltrd 2515 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  e.  CC )
7967, 78mulcld 9402 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
( A `  k
)  x.  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) ) )  e.  CC )
80 eldifn 3476 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... N )  \ 
( 1 ... N
) )  ->  -.  k  e.  ( 1 ... N ) )
81 0p1e1 10429 . . . . . . . . . . . . . 14  |-  ( 0  +  1 )  =  1
8281oveq1i 6100 . . . . . . . . . . . . 13  |-  ( ( 0  +  1 ) ... N )  =  ( 1 ... N
)
8382eleq2i 2505 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0  +  1 ) ... N )  <->  k  e.  ( 1 ... N
) )
8480, 83sylnibr 305 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... N )  \ 
( 1 ... N
) )  ->  -.  k  e.  ( (
0  +  1 ) ... N ) )
8584adantl 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  -.  k  e.  ( ( 0  +  1 ) ... N
) )
86 eldifi 3475 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... N )  \ 
( 1 ... N
) )  ->  k  e.  ( 0 ... N
) )
8786adantl 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  k  e.  ( 0 ... N
) )
88 dvply1.n . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  NN0 )
8988, 61syl6eleq 2531 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
9089ad2antrr 720 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  N  e.  ( ZZ>= `  0 )
)
91 elfzp12 11535 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( k  e.  ( 0 ... N
)  <->  ( k  =  0  \/  k  e.  ( ( 0  +  1 ) ... N
) ) ) )
9290, 91syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  ( k  e.  ( 0 ... N
)  <->  ( k  =  0  \/  k  e.  ( ( 0  +  1 ) ... N
) ) ) )
9387, 92mpbid 210 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  ( k  =  0  \/  k  e.  ( ( 0  +  1 ) ... N
) ) )
94 orel2 383 . . . . . . . . . 10  |-  ( -.  k  e.  ( ( 0  +  1 ) ... N )  -> 
( ( k  =  0  \/  k  e.  ( ( 0  +  1 ) ... N
) )  ->  k  =  0 ) )
9585, 93, 94sylc 60 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  k  = 
0 )
96 iftrue 3794 . . . . . . . . 9  |-  ( k  =  0  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  =  0 )
9795, 96syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  if (
k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) )  =  0 )
9897oveq2d 6106 . . . . . . 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 ) )
9965, 16, 17syl2an 474 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
10099mul01d 9564 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  0 )  =  0 )
10186, 100sylan2 471 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  ( ( A `  k )  x.  0 )  =  0 )
10298, 101eqtrd 2473 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  ( ( A `  k )  x.  if ( k  =  0 ,  0 ,  ( k  x.  (
z ^ ( k  -  1 ) ) ) ) )  =  0 )
103 fzfid 11791 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... N )  e. 
Fin )
10464, 79, 102, 103fsumss 13198 . . . . 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 ) ) ) ) ) )
105 elfznn0 11477 . . . . . . . . . . . . . . 15  |-  ( j  e.  ( 0 ... ( N  -  1 ) )  ->  j  e.  NN0 )
106105adantl 463 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  j  e.  NN0 )
107106nn0cnd 10634 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  j  e.  CC )
108 ax-1cn 9336 . . . . . . . . . . . . 13  |-  1  e.  CC
109 pncan 9612 . . . . . . . . . . . . 13  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( ( j  +  1 )  -  1 )  =  j )
110107, 108, 109sylancl 657 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( j  +  1 )  -  1 )  =  j )
111110oveq2d 6106 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
z ^ ( ( j  +  1 )  -  1 ) )  =  ( z ^
j ) )
112111oveq2d 6106 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( j  +  1 )  x.  ( z ^ ( ( j  +  1 )  - 
1 ) ) )  =  ( ( j  +  1 )  x.  ( z ^ j
) ) )
113112oveq2d 6106 . . . . . . . . 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 ) ) ) )
11415ad2antrr 720 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  A : NN0 --> CC )
115 peano2nn0 10616 . . . . . . . . . . . . 13  |-  ( j  e.  NN0  ->  ( j  +  1 )  e. 
NN0 )
116105, 115syl 16 . . . . . . . . . . . 12  |-  ( j  e.  ( 0 ... ( N  -  1 ) )  ->  (
j  +  1 )  e.  NN0 )
117116adantl 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
j  +  1 )  e.  NN0 )
118114, 117ffvelrnd 5841 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( A `  ( j  +  1 ) )  e.  CC )
119117nn0cnd 10634 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
j  +  1 )  e.  CC )
120 simplr 749 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  z  e.  CC )
121120, 106expcld 12004 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
z ^ j )  e.  CC )
122118, 119, 121mulassd 9405 . . . . . . . . 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 ) ) ) )
123118, 119mulcomd 9403 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( A `  (
j  +  1 ) )  x.  ( j  +  1 ) )  =  ( ( j  +  1 )  x.  ( A `  (
j  +  1 ) ) ) )
124123oveq1d 6105 . . . . . . . . 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
) ) )
125113, 122, 1243eqtr2d 2479 . . . . . . . 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
) ) )
126125sumeq2dv 13176 . . . . . . 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 ) ) )
127 1m1e0 10386 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
128127oveq1i 6100 . . . . . . . 8  |-  ( ( 1  -  1 ) ... ( N  - 
1 ) )  =  ( 0 ... ( N  -  1 ) )
129128sumeq1i 13171 . . . . . . 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 ) ) ) )
130 oveq1 6097 . . . . . . . . . 10  |-  ( k  =  j  ->  (
k  +  1 )  =  ( j  +  1 ) )
131130fveq2d 5692 . . . . . . . . . 10  |-  ( k  =  j  ->  ( A `  ( k  +  1 ) )  =  ( A `  ( j  +  1 ) ) )
132130, 131oveq12d 6108 . . . . . . . . 9  |-  ( k  =  j  ->  (
( k  +  1 )  x.  ( A `
 ( k  +  1 ) ) )  =  ( ( j  +  1 )  x.  ( A `  (
j  +  1 ) ) ) )
133 oveq2 6098 . . . . . . . . 9  |-  ( k  =  j  ->  (
z ^ k )  =  ( z ^
j ) )
134132, 133oveq12d 6108 . . . . . . . 8  |-  ( k  =  j  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( z ^ k ) )  =  ( ( ( j  +  1 )  x.  ( A `  ( j  +  1 ) ) )  x.  ( z ^ j
) ) )
135134cbvsumv 13169 . . . . . . 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 ) )
136126, 129, 1353eqtr4g 2498 . . . . . 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 ) ) )
137 1zzd 10673 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  1  e.  ZZ )
13888adantr 462 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  N  e. 
NN0 )
139138nn0zd 10741 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  N  e.  ZZ )
14067, 77mulcld 9402 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
( A `  k
)  x.  ( k  x.  ( z ^
( k  -  1 ) ) ) )  e.  CC )
141 fveq2 5688 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  ( A `  k )  =  ( A `  ( j  +  1 ) ) )
142 id 22 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  k  =  ( j  +  1 ) )
143 oveq1 6097 . . . . . . . . . 10  |-  ( k  =  ( j  +  1 )  ->  (
k  -  1 )  =  ( ( j  +  1 )  - 
1 ) )
144143oveq2d 6106 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  (
z ^ ( k  -  1 ) )  =  ( z ^
( ( j  +  1 )  -  1 ) ) )
145142, 144oveq12d 6108 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
k  x.  ( z ^ ( k  - 
1 ) ) )  =  ( ( j  +  1 )  x.  ( z ^ (
( j  +  1 )  -  1 ) ) ) )
146141, 145oveq12d 6108 . . . . . . 7  |-  ( k  =  ( j  +  1 )  ->  (
( A `  k
)  x.  ( k  x.  ( z ^
( k  -  1 ) ) ) )  =  ( ( A `
 ( j  +  1 ) )  x.  ( ( j  +  1 )  x.  (
z ^ ( ( j  +  1 )  -  1 ) ) ) ) )
147137, 137, 139, 140, 146fsumshftm 13244 . . . . . 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 ) ) ) ) )
148 elfznn0 11477 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
149148adantl 463 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  k  e.  NN0 )
150 ovex 6115 . . . . . . . . 9  |-  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  e. 
_V
151 dvply1.b . . . . . . . . . 10  |-  B  =  ( k  e.  NN0  |->  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) ) )
152151fvmpt2 5778 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  e.  _V )  ->  ( B `  k
)  =  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) ) )
153149, 150, 152sylancl 657 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( B `  k )  =  ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) ) )
154153oveq1d 6105 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( B `  k
)  x.  ( z ^ k ) )  =  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( z ^ k
) ) )
155154sumeq2dv 13176 . . . . . 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 ) ) )
156136, 147, 1553eqtr4d 2483 . . . . 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 ) ) )
15759, 104, 1563eqtr3d 2481 . . . 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 ) ) )
158157mpteq2dva 4375 . . 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 ) ) ) )
159 dvply1.g . . 3  |-  ( ph  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( B `
 k )  x.  ( z ^ k
) ) ) )
160158, 159eqtr4d 2476 . 2  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) ) ) )  =  G )
1612, 51, 1603eqtrd 2477 1  |-  ( ph  ->  ( CC  _D  F
)  =  G )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970    \ cdif 3322    C_ wss 3325   ifcif 3788   {cpr 3876    e. cmpt 4347   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    - cmin 9591   NNcn 10318   NN0cn0 10575   ZZ>=cuz 10857   ...cfz 11433   ^cexp 11861   sum_csu 13159   ↾t crest 14355   TopOpenctopn 14356  ℂfldccnfld 17777   Topctop 18457    _D cdv 21297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301
This theorem is referenced by:  dvply2g  21710
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