MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvply1 Structured version   Unicode version

Theorem dvply1 23223
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 6317 . 2  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
z ^ k ) ) ) ) )
3 eqid 2422 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
43cnfldtop 21790 . . . . 5  |-  ( TopOpen ` fld )  e.  Top
53cnfldtopon 21789 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
65toponunii 19933 . . . . . 6  |-  CC  =  U. ( TopOpen ` fld )
76restid 15319 . . . . 5  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
84, 7ax-mp 5 . . . 4  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
98eqcomi 2435 . . 3  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
10 cnelprrecn 9632 . . . 4  |-  CC  e.  { RR ,  CC }
1110a1i 11 . . 3  |-  ( ph  ->  CC  e.  { RR ,  CC } )
126topopn 19922 . . . 4  |-  ( (
TopOpen ` fld )  e.  Top  ->  CC  e.  ( TopOpen ` fld ) )
134, 12mp1i 13 . . 3  |-  ( ph  ->  CC  e.  ( TopOpen ` fld )
)
14 fzfid 12185 . . 3  |-  ( ph  ->  ( 0 ... N
)  e.  Fin )
15 dvply1.a . . . . . . 7  |-  ( ph  ->  A : NN0 --> CC )
16 elfznn0 11887 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
17 ffvelrn 6031 . . . . . . 7  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
1815, 16, 17syl2an 479 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
1918adantr 466 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  ( A `  k )  e.  CC )
20 simpr 462 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  z  e.  CC )
2116ad2antlr 731 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  k  e.  NN0 )
2220, 21expcld 12415 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  (
z ^ k )  e.  CC )
2319, 22mulcld 9663 . . . 4  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  (
( A `  k
)  x.  ( z ^ k ) )  e.  CC )
24233impa 1200 . . 3  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  ( ( A `  k )  x.  ( z ^
k ) )  e.  CC )
25183adant3 1025 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  ( A `
 k )  e.  CC )
26 0cnd 9636 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  k  =  0 )  -> 
0  e.  CC )
27 simpl2 1009 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
k  e.  ( 0 ... N ) )
2827, 16syl 17 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
k  e.  NN0 )
2928nn0cnd 10927 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
k  e.  CC )
30 simpl3 1010 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
z  e.  CC )
31 simpr 462 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  ->  -.  k  =  0
)
32 elnn0 10871 . . . . . . . . . 10  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
3328, 32sylib 199 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( k  e.  NN  \/  k  =  0
) )
34 orel2 384 . . . . . . . . 9  |-  ( -.  k  =  0  -> 
( ( k  e.  NN  \/  k  =  0 )  ->  k  e.  NN ) )
3531, 33, 34sylc 62 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
k  e.  NN )
36 nnm1nn0 10911 . . . . . . . 8  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
3735, 36syl 17 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( k  -  1 )  e.  NN0 )
3830, 37expcld 12415 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( z ^ (
k  -  1 ) )  e.  CC )
3929, 38mulcld 9663 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( k  x.  (
z ^ ( k  -  1 ) ) )  e.  CC )
4026, 39ifclda 3941 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) )  e.  CC )
4125, 40mulcld 9663 . . 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 9637 . . . . . 6  |-  0  e.  _V
44 ovex 6329 . . . . . 6  |-  ( k  x.  ( z ^
( k  -  1 ) ) )  e. 
_V
4543, 44ifex 3977 . . . . 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 467 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
48 dvexp2 22894 . . . . 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 17 . . . 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 22904 . . 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 22913 . 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 11828 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN )
5352nnne0d 10654 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... N )  ->  k  =/=  0 )
5453neneqd 2625 . . . . . . . . 9  |-  ( k  e.  ( 1 ... N )  ->  -.  k  =  0 )
5554adantl 467 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  -.  k  =  0 )
5655iffalsed 3920 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  =  ( k  x.  ( z ^
( k  -  1 ) ) ) )
5756oveq2d 6317 . . . . . 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 ) ) ) ) )
5857sumeq2dv 13756 . . . . 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 ) ) ) ) )
59 1eluzge0 11202 . . . . . . 7  |-  1  e.  ( ZZ>= `  0 )
60 fzss1 11837 . . . . . . 7  |-  ( 1  e.  ( ZZ>= `  0
)  ->  ( 1 ... N )  C_  ( 0 ... N
) )
6159, 60mp1i 13 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( 1 ... N )  C_  ( 0 ... N
) )
6215adantr 466 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  A : NN0
--> CC )
6352nnnn0d 10925 . . . . . . . 8  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN0 )
6462, 63, 17syl2an 479 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  ( A `  k )  e.  CC )
6553adantl 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  k  =/=  0 )
6665neneqd 2625 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  -.  k  =  0 )
6766iffalsed 3920 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  =  ( k  x.  ( z ^
( k  -  1 ) ) ) )
6863adantl 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  k  e.  NN0 )
6968nn0cnd 10927 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  k  e.  CC )
70 simplr 760 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  z  e.  CC )
7152, 36syl 17 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... N )  ->  (
k  -  1 )  e.  NN0 )
7271adantl 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
k  -  1 )  e.  NN0 )
7370, 72expcld 12415 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
z ^ ( k  -  1 ) )  e.  CC )
7469, 73mulcld 9663 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
k  x.  ( z ^ ( k  - 
1 ) ) )  e.  CC )
7567, 74eqeltrd 2510 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  e.  CC )
7664, 75mulcld 9663 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
( A `  k
)  x.  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) ) )  e.  CC )
77 eldifn 3588 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... N )  \ 
( 1 ... N
) )  ->  -.  k  e.  ( 1 ... N ) )
78 0p1e1 10721 . . . . . . . . . . . . . 14  |-  ( 0  +  1 )  =  1
7978oveq1i 6311 . . . . . . . . . . . . 13  |-  ( ( 0  +  1 ) ... N )  =  ( 1 ... N
)
8079eleq2i 2500 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0  +  1 ) ... N )  <->  k  e.  ( 1 ... N
) )
8177, 80sylnibr 306 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... N )  \ 
( 1 ... N
) )  ->  -.  k  e.  ( (
0  +  1 ) ... N ) )
8281adantl 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  -.  k  e.  ( ( 0  +  1 ) ... N
) )
83 eldifi 3587 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... N )  \ 
( 1 ... N
) )  ->  k  e.  ( 0 ... N
) )
8483adantl 467 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  k  e.  ( 0 ... N
) )
85 dvply1.n . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  NN0 )
86 nn0uz 11193 . . . . . . . . . . . . . 14  |-  NN0  =  ( ZZ>= `  0 )
8785, 86syl6eleq 2520 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
8887ad2antrr 730 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  N  e.  ( ZZ>= `  0 )
)
89 elfzp12 11873 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( k  e.  ( 0 ... N
)  <->  ( k  =  0  \/  k  e.  ( ( 0  +  1 ) ... N
) ) ) )
9088, 89syl 17 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  ( k  e.  ( 0 ... N
)  <->  ( k  =  0  \/  k  e.  ( ( 0  +  1 ) ... N
) ) ) )
9184, 90mpbid 213 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  ( k  =  0  \/  k  e.  ( ( 0  +  1 ) ... N
) ) )
92 orel2 384 . . . . . . . . . 10  |-  ( -.  k  e.  ( ( 0  +  1 ) ... N )  -> 
( ( k  =  0  \/  k  e.  ( ( 0  +  1 ) ... N
) )  ->  k  =  0 ) )
9382, 91, 92sylc 62 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  k  = 
0 )
9493iftrued 3917 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  if (
k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) )  =  0 )
9594oveq2d 6317 . . . . . . 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 ) )
9662, 16, 17syl2an 479 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
9796mul01d 9832 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  0 )  =  0 )
9883, 97sylan2 476 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  ( ( A `  k )  x.  0 )  =  0 )
9995, 98eqtrd 2463 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  ( ( A `  k )  x.  if ( k  =  0 ,  0 ,  ( k  x.  (
z ^ ( k  -  1 ) ) ) ) )  =  0 )
100 fzfid 12185 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... N )  e. 
Fin )
10161, 76, 99, 100fsumss 13778 . . . . 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 ) ) ) ) ) )
102 elfznn0 11887 . . . . . . . . . . . . . . 15  |-  ( j  e.  ( 0 ... ( N  -  1 ) )  ->  j  e.  NN0 )
103102adantl 467 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  j  e.  NN0 )
104103nn0cnd 10927 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  j  e.  CC )
105 ax-1cn 9597 . . . . . . . . . . . . 13  |-  1  e.  CC
106 pncan 9881 . . . . . . . . . . . . 13  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( ( j  +  1 )  -  1 )  =  j )
107104, 105, 106sylancl 666 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( j  +  1 )  -  1 )  =  j )
108107oveq2d 6317 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
z ^ ( ( j  +  1 )  -  1 ) )  =  ( z ^
j ) )
109108oveq2d 6317 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( j  +  1 )  x.  ( z ^ ( ( j  +  1 )  - 
1 ) ) )  =  ( ( j  +  1 )  x.  ( z ^ j
) ) )
110109oveq2d 6317 . . . . . . . . 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 ) ) ) )
11115ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  A : NN0 --> CC )
112 peano2nn0 10910 . . . . . . . . . . . . 13  |-  ( j  e.  NN0  ->  ( j  +  1 )  e. 
NN0 )
113102, 112syl 17 . . . . . . . . . . . 12  |-  ( j  e.  ( 0 ... ( N  -  1 ) )  ->  (
j  +  1 )  e.  NN0 )
114113adantl 467 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
j  +  1 )  e.  NN0 )
115111, 114ffvelrnd 6034 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( A `  ( j  +  1 ) )  e.  CC )
116114nn0cnd 10927 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
j  +  1 )  e.  CC )
117 simplr 760 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  z  e.  CC )
118117, 103expcld 12415 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
z ^ j )  e.  CC )
119115, 116, 118mulassd 9666 . . . . . . . . 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 ) ) ) )
120115, 116mulcomd 9664 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( A `  (
j  +  1 ) )  x.  ( j  +  1 ) )  =  ( ( j  +  1 )  x.  ( A `  (
j  +  1 ) ) ) )
121120oveq1d 6316 . . . . . . . . 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
) ) )
122110, 119, 1213eqtr2d 2469 . . . . . . . 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
) ) )
123122sumeq2dv 13756 . . . . . . 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 ) ) )
124 1m1e0 10678 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
125124oveq1i 6311 . . . . . . . 8  |-  ( ( 1  -  1 ) ... ( N  - 
1 ) )  =  ( 0 ... ( N  -  1 ) )
126125sumeq1i 13751 . . . . . . 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 ) ) ) )
127 oveq1 6308 . . . . . . . . . 10  |-  ( k  =  j  ->  (
k  +  1 )  =  ( j  +  1 ) )
128127fveq2d 5881 . . . . . . . . . 10  |-  ( k  =  j  ->  ( A `  ( k  +  1 ) )  =  ( A `  ( j  +  1 ) ) )
129127, 128oveq12d 6319 . . . . . . . . 9  |-  ( k  =  j  ->  (
( k  +  1 )  x.  ( A `
 ( k  +  1 ) ) )  =  ( ( j  +  1 )  x.  ( A `  (
j  +  1 ) ) ) )
130 oveq2 6309 . . . . . . . . 9  |-  ( k  =  j  ->  (
z ^ k )  =  ( z ^
j ) )
131129, 130oveq12d 6319 . . . . . . . 8  |-  ( k  =  j  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( z ^ k ) )  =  ( ( ( j  +  1 )  x.  ( A `  ( j  +  1 ) ) )  x.  ( z ^ j
) ) )
132131cbvsumv 13749 . . . . . . 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 ) )
133123, 126, 1323eqtr4g 2488 . . . . . 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 ) ) )
134 1zzd 10968 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  1  e.  ZZ )
13585adantr 466 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  N  e. 
NN0 )
136135nn0zd 11038 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  N  e.  ZZ )
13764, 74mulcld 9663 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
( A `  k
)  x.  ( k  x.  ( z ^
( k  -  1 ) ) ) )  e.  CC )
138 fveq2 5877 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  ( A `  k )  =  ( A `  ( j  +  1 ) ) )
139 id 23 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  k  =  ( j  +  1 ) )
140 oveq1 6308 . . . . . . . . . 10  |-  ( k  =  ( j  +  1 )  ->  (
k  -  1 )  =  ( ( j  +  1 )  - 
1 ) )
141140oveq2d 6317 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  (
z ^ ( k  -  1 ) )  =  ( z ^
( ( j  +  1 )  -  1 ) ) )
142139, 141oveq12d 6319 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
k  x.  ( z ^ ( k  - 
1 ) ) )  =  ( ( j  +  1 )  x.  ( z ^ (
( j  +  1 )  -  1 ) ) ) )
143138, 142oveq12d 6319 . . . . . . 7  |-  ( k  =  ( j  +  1 )  ->  (
( A `  k
)  x.  ( k  x.  ( z ^
( k  -  1 ) ) ) )  =  ( ( A `
 ( j  +  1 ) )  x.  ( ( j  +  1 )  x.  (
z ^ ( ( j  +  1 )  -  1 ) ) ) ) )
144134, 134, 136, 137, 143fsumshftm 13829 . . . . . 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 ) ) ) ) )
145 elfznn0 11887 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
146145adantl 467 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  k  e.  NN0 )
147 ovex 6329 . . . . . . . . 9  |-  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  e. 
_V
148 dvply1.b . . . . . . . . . 10  |-  B  =  ( k  e.  NN0  |->  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) ) )
149148fvmpt2 5969 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  e.  _V )  ->  ( B `  k
)  =  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) ) )
150146, 147, 149sylancl 666 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( B `  k )  =  ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) ) )
151150oveq1d 6316 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( B `  k
)  x.  ( z ^ k ) )  =  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( z ^ k
) ) )
152151sumeq2dv 13756 . . . . . 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 ) ) )
153133, 144, 1523eqtr4d 2473 . . . . 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 ) ) )
15458, 101, 1533eqtr3d 2471 . . . 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 ) ) )
155154mpteq2dva 4507 . . 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 ) ) ) )
156 dvply1.g . . 3  |-  ( ph  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( B `
 k )  x.  ( z ^ k
) ) ) )
157155, 156eqtr4d 2466 . 2  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) ) ) )  =  G )
1582, 51, 1573eqtrd 2467 1  |-  ( ph  ->  ( CC  _D  F
)  =  G )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   _Vcvv 3081    \ cdif 3433    C_ wss 3436   ifcif 3909   {cpr 3998    |-> cmpt 4479   -->wf 5593   ` cfv 5597  (class class class)co 6301   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    - cmin 9860   NNcn 10609   NN0cn0 10869   ZZ>=cuz 11159   ...cfz 11784   ^cexp 12271   sum_csu 13739   ↾t crest 15306   TopOpenctopn 15307  ℂfldccnfld 18957   Topctop 19903    _D cdv 22804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-pm 7479  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12213  df-exp 12272  df-hash 12515  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-clim 13539  df-sum 13740  df-struct 15110  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-mulr 15191  df-starv 15192  df-sca 15193  df-vsca 15194  df-ip 15195  df-tset 15196  df-ple 15197  df-ds 15199  df-unif 15200  df-hom 15201  df-cco 15202  df-rest 15308  df-topn 15309  df-0g 15327  df-gsum 15328  df-topgen 15329  df-pt 15330  df-prds 15333  df-xrs 15387  df-qtop 15393  df-imas 15394  df-xps 15397  df-mre 15479  df-mrc 15480  df-acs 15482  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-submnd 16570  df-mulg 16663  df-cntz 16958  df-cmn 17419  df-psmet 18949  df-xmet 18950  df-met 18951  df-bl 18952  df-mopn 18953  df-fbas 18954  df-fg 18955  df-cnfld 18958  df-top 19907  df-bases 19908  df-topon 19909  df-topsp 19910  df-cld 20020  df-ntr 20021  df-cls 20022  df-nei 20100  df-lp 20138  df-perf 20139  df-cn 20229  df-cnp 20230  df-haus 20317  df-tx 20563  df-hmeo 20756  df-fil 20847  df-fm 20939  df-flim 20940  df-flf 20941  df-xms 21321  df-ms 21322  df-tms 21323  df-cncf 21896  df-limc 22807  df-dv 22808
This theorem is referenced by:  dvply2g  23224
  Copyright terms: Public domain W3C validator