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Metamath Proof Explorer

Theorem ioomidp

Description: The midpoint is an element of the open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Assertion ioomidp ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 + 𝐵 ) / 2 ) ∈ ( 𝐴 (,) 𝐵 ) )

Proof

Step Hyp Ref Expression
1 rexr ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ* )
2 1 3ad2ant1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → 𝐴 ∈ ℝ* )
3 rexr ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℝ* )
4 3 3ad2ant2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → 𝐵 ∈ ℝ* )
5 readdcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + 𝐵 ) ∈ ℝ )
6 5 rehalfcld ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 + 𝐵 ) / 2 ) ∈ ℝ )
7 6 3adant3 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 + 𝐵 ) / 2 ) ∈ ℝ )
8 avglt1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ 𝐴 < ( ( 𝐴 + 𝐵 ) / 2 ) ) )
9 8 biimp3a ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → 𝐴 < ( ( 𝐴 + 𝐵 ) / 2 ) )
10 avglt2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ( ( 𝐴 + 𝐵 ) / 2 ) < 𝐵 ) )
11 10 biimp3a ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 + 𝐵 ) / 2 ) < 𝐵 )
12 2 4 7 9 11 eliood ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 + 𝐵 ) / 2 ) ∈ ( 𝐴 (,) 𝐵 ) )