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Theorem dvply1 22414
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 6298 . 2  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
z ^ k ) ) ) ) )
3 eqid 2467 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
43cnfldtop 21026 . . . . 5  |-  ( TopOpen ` fld )  e.  Top
53cnfldtopon 21025 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
65toponunii 19200 . . . . . 6  |-  CC  =  U. ( TopOpen ` fld )
76restid 14685 . . . . 5  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
84, 7ax-mp 5 . . . 4  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
98eqcomi 2480 . . 3  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
10 cnelprrecn 9581 . . . 4  |-  CC  e.  { RR ,  CC }
1110a1i 11 . . 3  |-  ( ph  ->  CC  e.  { RR ,  CC } )
126topopn 19182 . . . 4  |-  ( (
TopOpen ` fld )  e.  Top  ->  CC  e.  ( TopOpen ` fld ) )
134, 12mp1i 12 . . 3  |-  ( ph  ->  CC  e.  ( TopOpen ` fld )
)
14 fzfid 12047 . . 3  |-  ( ph  ->  ( 0 ... N
)  e.  Fin )
15 dvply1.a . . . . . . 7  |-  ( ph  ->  A : NN0 --> CC )
16 elfznn0 11766 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
17 ffvelrn 6017 . . . . . . 7  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
1815, 16, 17syl2an 477 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
1918adantr 465 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  ( A `  k )  e.  CC )
20 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  z  e.  CC )
2116ad2antlr 726 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  k  e.  NN0 )
2220, 21expcld 12274 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  (
z ^ k )  e.  CC )
2319, 22mulcld 9612 . . . 4  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
) )  /\  z  e.  CC )  ->  (
( A `  k
)  x.  ( z ^ k ) )  e.  CC )
24233impa 1191 . . 3  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  ( ( A `  k )  x.  ( z ^
k ) )  e.  CC )
25183adant3 1016 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  ( A `
 k )  e.  CC )
26 0cnd 9585 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  k  =  0 )  -> 
0  e.  CC )
27 simpl2 1000 . . . . . . . 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 10850 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
k  e.  CC )
30 simpl3 1001 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
z  e.  CC )
31 simpr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  ->  -.  k  =  0
)
32 elnn0 10793 . . . . . . . . . 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 10833 . . . . . . . 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 12274 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( z ^ (
k  -  1 ) )  e.  CC )
3929, 38mulcld 9612 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  /\  -.  k  =  0 )  -> 
( k  x.  (
z ^ ( k  -  1 ) ) )  e.  CC )
4026, 39ifclda 3971 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
)  /\  z  e.  CC )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^
( k  -  1 ) ) ) )  e.  CC )
4125, 40mulcld 9612 . . 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 9586 . . . . . 6  |-  0  e.  _V
44 ovex 6307 . . . . . 6  |-  ( k  x.  ( z ^
( k  -  1 ) ) )  e. 
_V
4543, 44ifex 4008 . . . . 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 466 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
48 dvexp2 22092 . . . . 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 22102 . . 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 22111 . 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 11710 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN )
5352nnne0d 10576 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... N )  ->  k  =/=  0 )
5453neneqd 2669 . . . . . . . . 9  |-  ( k  e.  ( 1 ... N )  ->  -.  k  =  0 )
5554adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  -.  k  =  0 )
56 iffalse 3948 . . . . . . . 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 6298 . . . . . 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 13484 . . . . 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 10807 . . . . . . . 8  |-  1  e.  NN0
61 nn0uz 11112 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
6260, 61eleqtri 2553 . . . . . . 7  |-  1  e.  ( ZZ>= `  0 )
63 fzss1 11718 . . . . . . 7  |-  ( 1  e.  ( ZZ>= `  0
)  ->  ( 1 ... N )  C_  ( 0 ... N
) )
6462, 63mp1i 12 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( 1 ... N )  C_  ( 0 ... N
) )
6515adantr 465 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  A : NN0
--> CC )
6652nnnn0d 10848 . . . . . . . 8  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN0 )
6765, 66, 17syl2an 477 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  ( A `  k )  e.  CC )
6853adantl 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  k  =/=  0 )
6968neneqd 2669 . . . . . . . . 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 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  k  e.  NN0 )
7271nn0cnd 10850 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  k  e.  CC )
73 simplr 754 . . . . . . . . . 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 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
k  -  1 )  e.  NN0 )
7673, 75expcld 12274 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
z ^ ( k  -  1 ) )  e.  CC )
7772, 76mulcld 9612 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
k  x.  ( z ^ ( k  - 
1 ) ) )  e.  CC )
7870, 77eqeltrd 2555 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  if ( k  =  0 ,  0 ,  ( k  x.  ( z ^ ( k  - 
1 ) ) ) )  e.  CC )
7967, 78mulcld 9612 . . . . . 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 3627 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... N )  \ 
( 1 ... N
) )  ->  -.  k  e.  ( 1 ... N ) )
81 0p1e1 10643 . . . . . . . . . . . . . 14  |-  ( 0  +  1 )  =  1
8281oveq1i 6292 . . . . . . . . . . . . 13  |-  ( ( 0  +  1 ) ... N )  =  ( 1 ... N
)
8382eleq2i 2545 . . . . . . . . . . . 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 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  -.  k  e.  ( ( 0  +  1 ) ... N
) )
86 eldifi 3626 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... N )  \ 
( 1 ... N
) )  ->  k  e.  ( 0 ... N
) )
8786adantl 466 . . . . . . . . . . 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 2565 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
9089ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  N  e.  ( ZZ>= `  0 )
)
91 elfzp12 11753 . . . . . . . . . . . 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 3945 . . . . . . . . 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 6298 . . . . . . 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 477 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
10099mul01d 9774 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  0 )  =  0 )
10186, 100sylan2 474 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... N )  \  (
1 ... N ) ) )  ->  ( ( A `  k )  x.  0 )  =  0 )
10298, 101eqtrd 2508 . . . . . 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 12047 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... N )  e. 
Fin )
10464, 79, 102, 103fsumss 13506 . . . . 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 11766 . . . . . . . . . . . . . . 15  |-  ( j  e.  ( 0 ... ( N  -  1 ) )  ->  j  e.  NN0 )
106105adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  j  e.  NN0 )
107106nn0cnd 10850 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  j  e.  CC )
108 ax-1cn 9546 . . . . . . . . . . . . 13  |-  1  e.  CC
109 pncan 9822 . . . . . . . . . . . . 13  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( ( j  +  1 )  -  1 )  =  j )
110107, 108, 109sylancl 662 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( j  +  1 )  -  1 )  =  j )
111110oveq2d 6298 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
z ^ ( ( j  +  1 )  -  1 ) )  =  ( z ^
j ) )
112111oveq2d 6298 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( j  +  1 )  x.  ( z ^ ( ( j  +  1 )  - 
1 ) ) )  =  ( ( j  +  1 )  x.  ( z ^ j
) ) )
113112oveq2d 6298 . . . . . . . . 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 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  A : NN0 --> CC )
115 peano2nn0 10832 . . . . . . . . . . . . 13  |-  ( j  e.  NN0  ->  ( j  +  1 )  e. 
NN0 )
116105, 115syl 16 . . . . . . . . . . . 12  |-  ( j  e.  ( 0 ... ( N  -  1 ) )  ->  (
j  +  1 )  e.  NN0 )
117116adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
j  +  1 )  e.  NN0 )
118114, 117ffvelrnd 6020 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( A `  ( j  +  1 ) )  e.  CC )
119117nn0cnd 10850 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
j  +  1 )  e.  CC )
120 simplr 754 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  z  e.  CC )
121120, 106expcld 12274 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
z ^ j )  e.  CC )
122118, 119, 121mulassd 9615 . . . . . . . . 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 9613 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  j  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( A `  (
j  +  1 ) )  x.  ( j  +  1 ) )  =  ( ( j  +  1 )  x.  ( A `  (
j  +  1 ) ) ) )
124123oveq1d 6297 . . . . . . . . 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 2514 . . . . . . . 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 13484 . . . . . . 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 10600 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
128127oveq1i 6292 . . . . . . . 8  |-  ( ( 1  -  1 ) ... ( N  - 
1 ) )  =  ( 0 ... ( N  -  1 ) )
129128sumeq1i 13479 . . . . . . 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 6289 . . . . . . . . . 10  |-  ( k  =  j  ->  (
k  +  1 )  =  ( j  +  1 ) )
131130fveq2d 5868 . . . . . . . . . 10  |-  ( k  =  j  ->  ( A `  ( k  +  1 ) )  =  ( A `  ( j  +  1 ) ) )
132130, 131oveq12d 6300 . . . . . . . . 9  |-  ( k  =  j  ->  (
( k  +  1 )  x.  ( A `
 ( k  +  1 ) ) )  =  ( ( j  +  1 )  x.  ( A `  (
j  +  1 ) ) ) )
133 oveq2 6290 . . . . . . . . 9  |-  ( k  =  j  ->  (
z ^ k )  =  ( z ^
j ) )
134132, 133oveq12d 6300 . . . . . . . 8  |-  ( k  =  j  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( z ^ k ) )  =  ( ( ( j  +  1 )  x.  ( A `  ( j  +  1 ) ) )  x.  ( z ^ j
) ) )
135134cbvsumv 13477 . . . . . . 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 2533 . . . . . 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 10891 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  1  e.  ZZ )
13888adantr 465 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  N  e. 
NN0 )
139138nn0zd 10960 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  N  e.  ZZ )
14067, 77mulcld 9612 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 1 ... N
) )  ->  (
( A `  k
)  x.  ( k  x.  ( z ^
( k  -  1 ) ) ) )  e.  CC )
141 fveq2 5864 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  ( A `  k )  =  ( A `  ( j  +  1 ) ) )
142 id 22 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  k  =  ( j  +  1 ) )
143 oveq1 6289 . . . . . . . . . 10  |-  ( k  =  ( j  +  1 )  ->  (
k  -  1 )  =  ( ( j  +  1 )  - 
1 ) )
144143oveq2d 6298 . . . . . . . . 9  |-  ( k  =  ( j  +  1 )  ->  (
z ^ ( k  -  1 ) )  =  ( z ^
( ( j  +  1 )  -  1 ) ) )
145142, 144oveq12d 6300 . . . . . . . 8  |-  ( k  =  ( j  +  1 )  ->  (
k  x.  ( z ^ ( k  - 
1 ) ) )  =  ( ( j  +  1 )  x.  ( z ^ (
( j  +  1 )  -  1 ) ) ) )
146141, 145oveq12d 6300 . . . . . . 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 13555 . . . . . 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 11766 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
149148adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  k  e.  NN0 )
150 ovex 6307 . . . . . . . . 9  |-  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  e. 
_V
151 dvply1.b . . . . . . . . . 10  |-  B  =  ( k  e.  NN0  |->  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) ) )
152151fvmpt2 5955 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  e.  _V )  ->  ( B `  k
)  =  ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) ) )
153149, 150, 152sylancl 662 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( B `  k )  =  ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) ) )
154153oveq1d 6297 . . . . . . 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 13484 . . . . . 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 2518 . . . . 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 2516 . . . 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 4533 . . 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 2511 . 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 2512 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    \ cdif 3473    C_ wss 3476   ifcif 3939   {cpr 4029    |-> cmpt 4505   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    - cmin 9801   NNcn 10532   NN0cn0 10791   ZZ>=cuz 11078   ...cfz 11668   ^cexp 12130   sum_csu 13467   ↾t crest 14672   TopOpenctopn 14673  ℂfldccnfld 18191   Topctop 19161    _D cdv 22002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-icc 11532  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cncf 21117  df-limc 22005  df-dv 22006
This theorem is referenced by:  dvply2g  22415
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