Theorem etransclem30

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

	Ref	Expression
Hypotheses	etransclem30.s	|- ( ph -> S e. { RR , CC } )
	etransclem30.a	|- ( ph -> X e. ( ( TopOpen ` CCfld ) |`t S ) )
	etransclem30.p	|- ( ph -> P e. NN )
	etransclem30.m	|- ( ph -> M e. NN0 )
	etransclem30.f	|- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
	etransclem30.n	|- ( ph -> N e. NN0 )
	etransclem30.h	|- H = ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
	etransclem30.c	|- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )
Assertion	etransclem30	|- ( 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		etransclem30.s	|- ( ph -> S e. { RR , CC } )
2		etransclem30.a	|- ( ph -> X e. ( ( TopOpen ` CCfld ) |`t S ) )
3		etransclem30.p	|- ( ph -> P e. NN )
4		etransclem30.m	|- ( ph -> M e. NN0 )
5		etransclem30.f	|- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
6		etransclem30.n	|- ( ph -> N e. NN0 )
7		etransclem30.h	|- H = ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
8		etransclem30.c	|- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )
9		eqid	|- ( x e. X |-> prod_ j e. ( 0 ... M ) ( ( H ` j ) ` x ) ) = ( x e. X |-> prod_ j e. ( 0 ... M ) ( ( H ` j ) ` x ) )
10	1 2 3 4 5 6 7 8 9	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 ) ) ) )