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Metamath Proof Explorer

Theorem fzo0addelr

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

Ref Expression
Assertion fzo0addelr ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝐴 + 𝐷 ) ∈ ( 𝐷 ..^ ( 𝐷 + 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 fzo0addel ⊢ ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝐴 + 𝐷 ) ∈ ( 𝐷 ..^ ( 𝐶 + 𝐷 ) ) )
2 zcn ⊢ ( 𝐷 ∈ ℤ → 𝐷 ∈ ℂ )
3 elfzoel2 ⊢ ( 𝐴 ∈ ( 0 ..^ 𝐶 ) → 𝐶 ∈ ℤ )
4 3 zcnd ⊢ ( 𝐴 ∈ ( 0 ..^ 𝐶 ) → 𝐶 ∈ ℂ )
5 addcom ⊢ ( ( 𝐷 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐷 + 𝐶 ) = ( 𝐶 + 𝐷 ) )
6 2 4 5 syl2anr ⊢ ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝐷 + 𝐶 ) = ( 𝐶 + 𝐷 ) )
7 6 oveq2d ⊢ ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝐷 ..^ ( 𝐷 + 𝐶 ) ) = ( 𝐷 ..^ ( 𝐶 + 𝐷 ) ) )
8 1 7 eleqtrrd ⊢ ( ( 𝐴 ∈ ( 0 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝐴 + 𝐷 ) ∈ ( 𝐷 ..^ ( 𝐷 + 𝐶 ) ) )