Theorem prodf1

Description: The value of the partial products in a one-valued infinite product. (Contributed by Scott Fenton, 5-Dec-2017)

	Ref	Expression
Hypothesis	prodf1.1	|- Z = ( ZZ>= ` M )
Assertion	prodf1	|- ( N e. Z -> ( seq M ( x. , ( Z X. { 1 } ) ) ` N ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		prodf1.1	|- Z = ( ZZ>= ` M )
2		1t1e1	|- ( 1 x. 1 ) = 1
3	2	a1i	|- ( N e. Z -> ( 1 x. 1 ) = 1 )
4	1	eleq2i	|- ( N e. Z <-> N e. ( ZZ>= ` M ) )
5	4	biimpi	|- ( N e. Z -> N e. ( ZZ>= ` M ) )
6		ax-1cn	|- 1 e. CC
7		elfzuz	|- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
8	7 1	eleqtrrdi	|- ( k e. ( M ... N ) -> k e. Z )
9	8	adantl	|- ( ( N e. Z /\ k e. ( M ... N ) ) -> k e. Z )
10		fvconst2g	|- ( ( 1 e. CC /\ k e. Z ) -> ( ( Z X. { 1 } ) ` k ) = 1 )
11	6 9 10	sylancr	|- ( ( N e. Z /\ k e. ( M ... N ) ) -> ( ( Z X. { 1 } ) ` k ) = 1 )
12	3 5 11	seqid3	|- ( N e. Z -> ( seq M ( x. , ( Z X. { 1 } ) ) ` N ) = 1 )