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 ) |