This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem iccvolcl

Description: A closed real interval has finite volume. (Contributed by Mario Carneiro, 25-Aug-2014)

Ref Expression
Assertion iccvolcl ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) ∈ ℝ )

Proof

Step Hyp Ref Expression
1 iccmbl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 [,] 𝐵 ) ∈ dom vol )
2 mblvol ⊢ ( ( 𝐴 [,] 𝐵 ) ∈ dom vol → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) = ( vol* ‘ ( 𝐴 [,] 𝐵 ) ) )
3 1 2 syl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) = ( vol* ‘ ( 𝐴 [,] 𝐵 ) ) )
4 rexr ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ* )
5 rexr ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℝ* )
6 icc0 ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 [,] 𝐵 ) = ∅ ↔ 𝐵 < 𝐴 ) )
7 4 5 6 syl2an ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 [,] 𝐵 ) = ∅ ↔ 𝐵 < 𝐴 ) )
8 7 biimpar ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐵 < 𝐴 ) → ( 𝐴 [,] 𝐵 ) = ∅ )
9 fveq2 ⊢ ( ( 𝐴 [,] 𝐵 ) = ∅ → ( vol* ‘ ( 𝐴 [,] 𝐵 ) ) = ( vol* ‘ ∅ ) )
10 ovol0 ⊢ ( vol* ‘ ∅ ) = 0
11 9 10 eqtrdi ⊢ ( ( 𝐴 [,] 𝐵 ) = ∅ → ( vol* ‘ ( 𝐴 [,] 𝐵 ) ) = 0 )
12 0re ⊢ 0 ∈ ℝ
13 11 12 eqeltrdi ⊢ ( ( 𝐴 [,] 𝐵 ) = ∅ → ( vol* ‘ ( 𝐴 [,] 𝐵 ) ) ∈ ℝ )
14 8 13 syl ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐵 < 𝐴 ) → ( vol* ‘ ( 𝐴 [,] 𝐵 ) ) ∈ ℝ )
15 ovolicc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( vol* ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐵 − 𝐴 ) )
16 15 3expa ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≤ 𝐵 ) → ( vol* ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐵 − 𝐴 ) )
17 resubcl ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 𝐵 − 𝐴 ) ∈ ℝ )
18 17 ancoms ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 − 𝐴 ) ∈ ℝ )
19 18 adantr ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐵 − 𝐴 ) ∈ ℝ )
20 16 19 eqeltrd ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 ≤ 𝐵 ) → ( vol* ‘ ( 𝐴 [,] 𝐵 ) ) ∈ ℝ )
21 simpr ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 𝐵 ∈ ℝ )
22 simpl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 𝐴 ∈ ℝ )
23 14 20 21 22 ltlecasei ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( vol* ‘ ( 𝐴 [,] 𝐵 ) ) ∈ ℝ )
24 3 23 eqeltrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) ∈ ℝ )