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Metamath Proof Explorer

Theorem fzofzp1

Description: If a point is in a half-open range, the next point is in the closed range. (Contributed by Stefan O'Rear, 23-Aug-2015)

Ref Expression
Assertion fzofzp1 ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) → ( 𝐶 + 1 ) ∈ ( 𝐴 ... 𝐵 ) )

Proof

Step Hyp Ref Expression
1 elfzoel1 ⊢ ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) → 𝐴 ∈ ℤ )
2 uzid ⊢ ( 𝐴 ∈ ℤ → 𝐴 ∈ ( ℤ≥ ‘ 𝐴 ) )
3 peano2uz ⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 𝐴 + 1 ) ∈ ( ℤ≥ ‘ 𝐴 ) )
4 fzoss1 ⊢ ( ( 𝐴 + 1 ) ∈ ( ℤ≥ ‘ 𝐴 ) → ( ( 𝐴 + 1 ) ..^ ( 𝐵 + 1 ) ) ⊆ ( 𝐴 ..^ ( 𝐵 + 1 ) ) )
5 1 2 3 4 4syl ⊢ ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) → ( ( 𝐴 + 1 ) ..^ ( 𝐵 + 1 ) ) ⊆ ( 𝐴 ..^ ( 𝐵 + 1 ) ) )
6 1z ⊢ 1 ∈ ℤ
7 fzoaddel ⊢ ( ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) ∧ 1 ∈ ℤ ) → ( 𝐶 + 1 ) ∈ ( ( 𝐴 + 1 ) ..^ ( 𝐵 + 1 ) ) )
8 6 7 mpan2 ⊢ ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) → ( 𝐶 + 1 ) ∈ ( ( 𝐴 + 1 ) ..^ ( 𝐵 + 1 ) ) )
9 5 8 sseldd ⊢ ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) → ( 𝐶 + 1 ) ∈ ( 𝐴 ..^ ( 𝐵 + 1 ) ) )
10 elfzoel2 ⊢ ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) → 𝐵 ∈ ℤ )
11 fzval3 ⊢ ( 𝐵 ∈ ℤ → ( 𝐴 ... 𝐵 ) = ( 𝐴 ..^ ( 𝐵 + 1 ) ) )
12 10 11 syl ⊢ ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) → ( 𝐴 ... 𝐵 ) = ( 𝐴 ..^ ( 𝐵 + 1 ) ) )
13 9 12 eleqtrrd ⊢ ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) → ( 𝐶 + 1 ) ∈ ( 𝐴 ... 𝐵 ) )