Theorem etransclem42

Description: The N -th derivative of F applied to J is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020)

	Ref	Expression
Hypotheses	etransclem42.s	|- ( ph -> S e. { RR , CC } )
	etransclem42.x	|- ( ph -> X e. ( ( TopOpen ` CCfld ) |`t S ) )
	etransclem42.p	|- ( ph -> P e. NN )
	etransclem42.m	|- ( ph -> M e. NN0 )
	etransclem42.f	|- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
	etransclem42.n	|- ( ph -> N e. NN0 )
	etransclem42.jx	|- ( ph -> J e. X )
	etransclem42.jz	|- ( ph -> J e. ZZ )
Assertion	etransclem42	|- ( ph -> ( ( ( S Dn F ) ` N ) ` J ) e. ZZ )

Proof

Step	Hyp	Ref	Expression
1		etransclem42.s	|- ( ph -> S e. { RR , CC } )
2		etransclem42.x	|- ( ph -> X e. ( ( TopOpen ` CCfld ) |`t S ) )
3		etransclem42.p	|- ( ph -> P e. NN )
4		etransclem42.m	|- ( ph -> M e. NN0 )
5		etransclem42.f	|- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
6		etransclem42.n	|- ( ph -> N e. NN0 )
7		etransclem42.jx	|- ( ph -> J e. X )
8		etransclem42.jz	|- ( ph -> J e. ZZ )
9		etransclem5	|- ( k e. ( 0 ... M ) |-> ( y e. X |-> ( ( y - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) ) = ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
10		etransclem11	|- ( m e. NN0 |-> { d e. ( ( 0 ... m ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( d ` k ) = m } ) = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )
11	1 2 3 4 5 6 9 7 8 10	etransclem36	|- ( ph -> ( ( ( S Dn F ) ` N ) ` J ) e. ZZ )