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Metamath Proof Explorer

Theorem xrre

Description: A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006)

Ref Expression
Assertion xrre ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( -∞ < 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → 𝐴 ∈ ℝ )

Proof

Step Hyp Ref Expression
1 simprl ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( -∞ < 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → -∞ < 𝐴 )
2 ltpnf ⊢ ( 𝐵 ∈ ℝ → 𝐵 < +∞ )
3 2 adantl ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → 𝐵 < +∞ )
4 rexr ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℝ* )
5 pnfxr ⊢ +∞ ∈ ℝ*
6 xrlelttr ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞ ) → 𝐴 < +∞ ) )
7 5 6 mp3an3 ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞ ) → 𝐴 < +∞ ) )
8 4 7 sylan2 ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞ ) → 𝐴 < +∞ ) )
9 3 8 mpan2d ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 → 𝐴 < +∞ ) )
10 9 imp ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≤ 𝐵 ) → 𝐴 < +∞ )
11 10 adantrl ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( -∞ < 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → 𝐴 < +∞ )
12 xrrebnd ⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 ∈ ℝ ↔ ( -∞ < 𝐴 ∧ 𝐴 < +∞ ) ) )
13 12 ad2antrr ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( -∞ < 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ∈ ℝ ↔ ( -∞ < 𝐴 ∧ 𝐴 < +∞ ) ) )
14 1 11 13 mpbir2and ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( -∞ < 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → 𝐴 ∈ ℝ )