Theorem fz0add1fz1

Description: Translate membership in a 0-based half-open integer range into membership in a 1-based finite sequence of integers. (Contributed by Alexander van der Vekens, 23-Nov-2017)

	Ref	Expression
Assertion	fz0add1fz1	|- ( ( N e. NN0 /\ X e. ( 0 ..^ N ) ) -> ( X + 1 ) e. ( 1 ... N ) )

Proof

Step	Hyp	Ref	Expression
1		1z	|- 1 e. ZZ
2		fzoaddel	|- ( ( X e. ( 0 ..^ N ) /\ 1 e. ZZ ) -> ( X + 1 ) e. ( ( 0 + 1 ) ..^ ( N + 1 ) ) )
3	1 2	mpan2	|- ( X e. ( 0 ..^ N ) -> ( X + 1 ) e. ( ( 0 + 1 ) ..^ ( N + 1 ) ) )
4	3	adantl	|- ( ( N e. NN0 /\ X e. ( 0 ..^ N ) ) -> ( X + 1 ) e. ( ( 0 + 1 ) ..^ ( N + 1 ) ) )
5		0p1e1	|- ( 0 + 1 ) = 1
6	5	oveq1i	|- ( ( 0 + 1 ) ..^ ( N + 1 ) ) = ( 1 ..^ ( N + 1 ) )
7		nn0z	|- ( N e. NN0 -> N e. ZZ )
8		fzval3	|- ( N e. ZZ -> ( 1 ... N ) = ( 1 ..^ ( N + 1 ) ) )
9	8	eqcomd	|- ( N e. ZZ -> ( 1 ..^ ( N + 1 ) ) = ( 1 ... N ) )
10	7 9	syl	|- ( N e. NN0 -> ( 1 ..^ ( N + 1 ) ) = ( 1 ... N ) )
11	6 10	eqtrid	|- ( N e. NN0 -> ( ( 0 + 1 ) ..^ ( N + 1 ) ) = ( 1 ... N ) )
12	11	eleq2d	|- ( N e. NN0 -> ( ( X + 1 ) e. ( ( 0 + 1 ) ..^ ( N + 1 ) ) <-> ( X + 1 ) e. ( 1 ... N ) ) )
13	12	adantr	|- ( ( N e. NN0 /\ X e. ( 0 ..^ N ) ) -> ( ( X + 1 ) e. ( ( 0 + 1 ) ..^ ( N + 1 ) ) <-> ( X + 1 ) e. ( 1 ... N ) ) )
14	4 13	mpbid	|- ( ( N e. NN0 /\ X e. ( 0 ..^ N ) ) -> ( X + 1 ) e. ( 1 ... N ) )