Theorem fzosubel3

Description: Membership in a translated half-open integer range when the original range is zero-based. (Contributed by Stefan O'Rear, 15-Aug-2015)

	Ref	Expression
Assertion	fzosubel3	⊢ ( ( 𝐴 ∈ ( 𝐵 ..^ ( 𝐵 + 𝐷 ) ) ∧ 𝐷 ∈ ℤ ) → ( 𝐴 − 𝐵 ) ∈ ( 0 ..^ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝐴 ∈ ( 𝐵 ..^ ( 𝐵 + 𝐷 ) ) ∧ 𝐷 ∈ ℤ ) → 𝐴 ∈ ( 𝐵 ..^ ( 𝐵 + 𝐷 ) ) )
2		elfzoel1	⊢ ( 𝐴 ∈ ( 𝐵 ..^ ( 𝐵 + 𝐷 ) ) → 𝐵 ∈ ℤ )
3	2	adantr	⊢ ( ( 𝐴 ∈ ( 𝐵 ..^ ( 𝐵 + 𝐷 ) ) ∧ 𝐷 ∈ ℤ ) → 𝐵 ∈ ℤ )
4	3	zcnd	⊢ ( ( 𝐴 ∈ ( 𝐵 ..^ ( 𝐵 + 𝐷 ) ) ∧ 𝐷 ∈ ℤ ) → 𝐵 ∈ ℂ )
5	4	addid1d	⊢ ( ( 𝐴 ∈ ( 𝐵 ..^ ( 𝐵 + 𝐷 ) ) ∧ 𝐷 ∈ ℤ ) → ( 𝐵 + 0 ) = 𝐵 )
6	5	oveq1d	⊢ ( ( 𝐴 ∈ ( 𝐵 ..^ ( 𝐵 + 𝐷 ) ) ∧ 𝐷 ∈ ℤ ) → ( ( 𝐵 + 0 ) ..^ ( 𝐵 + 𝐷 ) ) = ( 𝐵 ..^ ( 𝐵 + 𝐷 ) ) )
7	1 6	eleqtrrd	⊢ ( ( 𝐴 ∈ ( 𝐵 ..^ ( 𝐵 + 𝐷 ) ) ∧ 𝐷 ∈ ℤ ) → 𝐴 ∈ ( ( 𝐵 + 0 ) ..^ ( 𝐵 + 𝐷 ) ) )
8		0zd	⊢ ( ( 𝐴 ∈ ( 𝐵 ..^ ( 𝐵 + 𝐷 ) ) ∧ 𝐷 ∈ ℤ ) → 0 ∈ ℤ )
9		simpr	⊢ ( ( 𝐴 ∈ ( 𝐵 ..^ ( 𝐵 + 𝐷 ) ) ∧ 𝐷 ∈ ℤ ) → 𝐷 ∈ ℤ )
10		fzosubel2	⊢ ( ( 𝐴 ∈ ( ( 𝐵 + 0 ) ..^ ( 𝐵 + 𝐷 ) ) ∧ ( 𝐵 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝐷 ∈ ℤ ) ) → ( 𝐴 − 𝐵 ) ∈ ( 0 ..^ 𝐷 ) )
11	7 3 8 9 10	syl13anc	⊢ ( ( 𝐴 ∈ ( 𝐵 ..^ ( 𝐵 + 𝐷 ) ) ∧ 𝐷 ∈ ℤ ) → ( 𝐴 − 𝐵 ) ∈ ( 0 ..^ 𝐷 ) )