Theorem fzo0addel

Description: Translate membership in a 0-based half-open integer range. (Contributed by AV, 30-Apr-2020)

	Ref	Expression
Assertion	fzo0addel	|- ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> ( A + D ) e. ( D ..^ ( C + D ) ) )

Proof

Step	Hyp	Ref	Expression
1		fzoaddel	|- ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> ( A + D ) e. ( ( 0 + D ) ..^ ( C + D ) ) )
2		zcn	|- ( D e. ZZ -> D e. CC )
3		addid2	|- ( D e. CC -> ( 0 + D ) = D )
4	3	eqcomd	|- ( D e. CC -> D = ( 0 + D ) )
5	2 4	syl	|- ( D e. ZZ -> D = ( 0 + D ) )
6	5	adantl	|- ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> D = ( 0 + D ) )
7	6	oveq1d	|- ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> ( D ..^ ( C + D ) ) = ( ( 0 + D ) ..^ ( C + D ) ) )
8	1 7	eleqtrrd	|- ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> ( A + D ) e. ( D ..^ ( C + D ) ) )