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Metamath Proof Explorer

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 addridd ⊢ ( ( 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 ) )