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Metamath Proof Explorer

Theorem itg2lecl

Description: If an S.2 integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014)

Ref Expression
Assertion itg2lecl ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ ℝ ∧ ( ∫2 ‘ 𝐹 ) ≤ 𝐴 ) → ( ∫2 ‘ 𝐹 ) ∈ ℝ )

Proof

Step Hyp Ref Expression
1 itg2cl ⊢ ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) → ( ∫2 ‘ 𝐹 ) ∈ ℝ* )
2 1 3ad2ant1 ⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ ℝ ∧ ( ∫2 ‘ 𝐹 ) ≤ 𝐴 ) → ( ∫2 ‘ 𝐹 ) ∈ ℝ* )
3 simp2 ⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ ℝ ∧ ( ∫2 ‘ 𝐹 ) ≤ 𝐴 ) → 𝐴 ∈ ℝ )
4 itg2ge0 ⊢ ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) → 0 ≤ ( ∫2 ‘ 𝐹 ) )
5 4 3ad2ant1 ⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ ℝ ∧ ( ∫2 ‘ 𝐹 ) ≤ 𝐴 ) → 0 ≤ ( ∫2 ‘ 𝐹 ) )
6 simp3 ⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ ℝ ∧ ( ∫2 ‘ 𝐹 ) ≤ 𝐴 ) → ( ∫2 ‘ 𝐹 ) ≤ 𝐴 )
7 xrrege0 ⊢ ( ( ( ( ∫2 ‘ 𝐹 ) ∈ ℝ* ∧ 𝐴 ∈ ℝ ) ∧ ( 0 ≤ ( ∫2 ‘ 𝐹 ) ∧ ( ∫2 ‘ 𝐹 ) ≤ 𝐴 ) ) → ( ∫2 ‘ 𝐹 ) ∈ ℝ )
8 2 3 5 6 7 syl22anc ⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ ℝ ∧ ( ∫2 ‘ 𝐹 ) ≤ 𝐴 ) → ( ∫2 ‘ 𝐹 ) ∈ ℝ )