Theorem etransclem40

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

	Ref	Expression
Hypotheses	etransclem40.s	|- ( ph -> S e. { RR , CC } )
	etransclem40.a	|- ( ph -> X e. ( ( TopOpen ` CCfld ) |`t S ) )
	etransclem40.p	|- ( ph -> P e. NN )
	etransclem40.m	|- ( ph -> M e. NN0 )
	etransclem40.f	|- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ k e. ( 1 ... M ) ( ( x - k ) ^ P ) ) )
	etransclem40.6	|- ( ph -> N e. NN0 )
Assertion	etransclem40	|- ( ph -> ( ( S Dn F ) ` N ) e. ( X -cn-> CC ) )

Proof

Step	Hyp	Ref	Expression
1		etransclem40.s	|- ( ph -> S e. { RR , CC } )
2		etransclem40.a	|- ( ph -> X e. ( ( TopOpen ` CCfld ) |`t S ) )
3		etransclem40.p	|- ( ph -> P e. NN )
4		etransclem40.m	|- ( ph -> M e. NN0 )
5		etransclem40.f	|- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ k e. ( 1 ... M ) ( ( x - k ) ^ P ) ) )
6		etransclem40.6	|- ( ph -> N e. NN0 )
7		etransclem5	|- ( j e. ( 0 ... M ) |-> ( y e. X |-> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) = ( k e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) )
8		etransclem11	|- ( m e. NN0 |-> { d e. ( ( 0 ... m ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( d ` j ) = m } ) = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( c ` k ) = n } )
9	1 2 3 4 5 6 7 8	etransclem34	|- ( ph -> ( ( S Dn F ) ` N ) e. ( X -cn-> CC ) )