Metamath Proof Explorer

Theorem etransclem29

Description: The N -th derivative of F . (Contributed by Glauco Siliprandi, 5-Apr-2020)

Ref Expression
Hypotheses etranslemdvnf2lemlem.s 
|- ( ph -> S e. { RR , CC } )
etransclem29.a 
|- ( ph -> X e. ( ( TopOpen ` CCfld ) |`t S ) )
etransclem29.p 
|- ( ph -> P e. NN )
etransclem29.m 
|- ( ph -> M e. NN0 )
etransclem29.f 
|- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
etransclem29.n 
|- ( ph -> N e. NN0 )
etransclem29.h 
|- H = ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
etransclem29.c 
|- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )
etransclem29.e 
|- E = ( x e. X |-> prod_ j e. ( 0 ... M ) ( ( H ` j ) ` x ) )
Assertion etransclem29 
|- ( ph -> ( ( S Dn F ) ` N ) = ( x e. X |-> sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` x ) ) ) )

Proof

Step Hyp Ref Expression
1 etranslemdvnf2lemlem.s 
 |-  ( ph -> S e. { RR , CC } )
2 etransclem29.a 
 |-  ( ph -> X e. ( ( TopOpen ` CCfld ) |`t S ) )
3 etransclem29.p 
 |-  ( ph -> P e. NN )
4 etransclem29.m 
 |-  ( ph -> M e. NN0 )
5 etransclem29.f 
 |-  F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
6 etransclem29.n 
 |-  ( ph -> N e. NN0 )
7 etransclem29.h 
 |-  H = ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
8 etransclem29.c 
 |-  C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )
9 etransclem29.e 
 |-  E = ( x e. X |-> prod_ j e. ( 0 ... M ) ( ( H ` j ) ` x ) )
10 1 2 dvdmsscn 
 |-  ( ph -> X C_ CC )
11 10 3 4 5 7 9 etransclem4 
 |-  ( ph -> F = E )
12 11 oveq2d 
 |-  ( ph -> ( S Dn F ) = ( S Dn E ) )
13 12 fveq1d 
 |-  ( ph -> ( ( S Dn F ) ` N ) = ( ( S Dn E ) ` N ) )
14 fzfid 
 |-  ( ph -> ( 0 ... M ) e. Fin )
15 10 adantr 
 |-  ( ( ph /\ j e. ( 0 ... M ) ) -> X C_ CC )
16 3 adantr 
 |-  ( ( ph /\ j e. ( 0 ... M ) ) -> P e. NN )
17 simpr 
 |-  ( ( ph /\ j e. ( 0 ... M ) ) -> j e. ( 0 ... M ) )
18 15 16 7 17 etransclem1 
 |-  ( ( ph /\ j e. ( 0 ... M ) ) -> ( H ` j ) : X --> CC )
19 1 3ad2ant1 
 |-  ( ( ph /\ j e. ( 0 ... M ) /\ i e. ( 0 ... N ) ) -> S e. { RR , CC } )
20 2 3ad2ant1 
 |-  ( ( ph /\ j e. ( 0 ... M ) /\ i e. ( 0 ... N ) ) -> X e. ( ( TopOpen ` CCfld ) |`t S ) )
21 3 3ad2ant1 
 |-  ( ( ph /\ j e. ( 0 ... M ) /\ i e. ( 0 ... N ) ) -> P e. NN )
22 etransclem5 
 |-  ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) = ( k e. ( 0 ... M ) |-> ( y e. X |-> ( ( y - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) )
23 7 22 eqtri 
 |-  H = ( k e. ( 0 ... M ) |-> ( y e. X |-> ( ( y - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) )
24 simp2 
 |-  ( ( ph /\ j e. ( 0 ... M ) /\ i e. ( 0 ... N ) ) -> j e. ( 0 ... M ) )
25 elfznn0 
 |-  ( i e. ( 0 ... N ) -> i e. NN0 )
26 25 3ad2ant3 
 |-  ( ( ph /\ j e. ( 0 ... M ) /\ i e. ( 0 ... N ) ) -> i e. NN0 )
27 19 20 21 23 24 26 etransclem20 
 |-  ( ( ph /\ j e. ( 0 ... M ) /\ i e. ( 0 ... N ) ) -> ( ( S Dn ( H ` j ) ) ` i ) : X --> CC )
28 1 2 14 18 6 27 9 8 dvnprod 
 |-  ( ph -> ( ( S Dn E ) ` N ) = ( x e. X |-> sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` x ) ) ) )
29 13 28 eqtrd 
 |-  ( ph -> ( ( S Dn F ) ` N ) = ( x e. X |-> sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` x ) ) ) )