Step |
Hyp |
Ref |
Expression |
1 |
|
itgioo.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
itgioo.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
itgioo.3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐶 ∈ ℂ ) |
4 |
|
ioossicc |
⊢ ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) |
5 |
4
|
a1i |
⊢ ( 𝜑 → ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
6 |
|
iccssre |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
7 |
1 2 6
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
8 |
1
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
9 |
2
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
10 |
|
icc0 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 [,] 𝐵 ) = ∅ ↔ 𝐵 < 𝐴 ) ) |
11 |
8 9 10
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐴 [,] 𝐵 ) = ∅ ↔ 𝐵 < 𝐴 ) ) |
12 |
11
|
biimpar |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ( 𝐴 [,] 𝐵 ) = ∅ ) |
13 |
12
|
difeq1d |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ( ( 𝐴 [,] 𝐵 ) ∖ ( 𝐴 (,) 𝐵 ) ) = ( ∅ ∖ ( 𝐴 (,) 𝐵 ) ) ) |
14 |
|
0dif |
⊢ ( ∅ ∖ ( 𝐴 (,) 𝐵 ) ) = ∅ |
15 |
|
0ss |
⊢ ∅ ⊆ { 𝐴 , 𝐵 } |
16 |
14 15
|
eqsstri |
⊢ ( ∅ ∖ ( 𝐴 (,) 𝐵 ) ) ⊆ { 𝐴 , 𝐵 } |
17 |
13 16
|
eqsstrdi |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ( ( 𝐴 [,] 𝐵 ) ∖ ( 𝐴 (,) 𝐵 ) ) ⊆ { 𝐴 , 𝐵 } ) |
18 |
|
uncom |
⊢ ( { 𝐴 , 𝐵 } ∪ ( 𝐴 (,) 𝐵 ) ) = ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) |
19 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ℝ* ) |
20 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ℝ* ) |
21 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ≤ 𝐵 ) |
22 |
|
prunioo |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) = ( 𝐴 [,] 𝐵 ) ) |
23 |
19 20 21 22
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) = ( 𝐴 [,] 𝐵 ) ) |
24 |
18 23
|
eqtr2id |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 [,] 𝐵 ) = ( { 𝐴 , 𝐵 } ∪ ( 𝐴 (,) 𝐵 ) ) ) |
25 |
24
|
difeq1d |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 [,] 𝐵 ) ∖ ( 𝐴 (,) 𝐵 ) ) = ( ( { 𝐴 , 𝐵 } ∪ ( 𝐴 (,) 𝐵 ) ) ∖ ( 𝐴 (,) 𝐵 ) ) ) |
26 |
|
difun2 |
⊢ ( ( { 𝐴 , 𝐵 } ∪ ( 𝐴 (,) 𝐵 ) ) ∖ ( 𝐴 (,) 𝐵 ) ) = ( { 𝐴 , 𝐵 } ∖ ( 𝐴 (,) 𝐵 ) ) |
27 |
25 26
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 [,] 𝐵 ) ∖ ( 𝐴 (,) 𝐵 ) ) = ( { 𝐴 , 𝐵 } ∖ ( 𝐴 (,) 𝐵 ) ) ) |
28 |
|
difss |
⊢ ( { 𝐴 , 𝐵 } ∖ ( 𝐴 (,) 𝐵 ) ) ⊆ { 𝐴 , 𝐵 } |
29 |
27 28
|
eqsstrdi |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 [,] 𝐵 ) ∖ ( 𝐴 (,) 𝐵 ) ) ⊆ { 𝐴 , 𝐵 } ) |
30 |
17 29 2 1
|
ltlecasei |
⊢ ( 𝜑 → ( ( 𝐴 [,] 𝐵 ) ∖ ( 𝐴 (,) 𝐵 ) ) ⊆ { 𝐴 , 𝐵 } ) |
31 |
1 2
|
prssd |
⊢ ( 𝜑 → { 𝐴 , 𝐵 } ⊆ ℝ ) |
32 |
|
prfi |
⊢ { 𝐴 , 𝐵 } ∈ Fin |
33 |
|
ovolfi |
⊢ ( ( { 𝐴 , 𝐵 } ∈ Fin ∧ { 𝐴 , 𝐵 } ⊆ ℝ ) → ( vol* ‘ { 𝐴 , 𝐵 } ) = 0 ) |
34 |
32 31 33
|
sylancr |
⊢ ( 𝜑 → ( vol* ‘ { 𝐴 , 𝐵 } ) = 0 ) |
35 |
|
ovolssnul |
⊢ ( ( ( ( 𝐴 [,] 𝐵 ) ∖ ( 𝐴 (,) 𝐵 ) ) ⊆ { 𝐴 , 𝐵 } ∧ { 𝐴 , 𝐵 } ⊆ ℝ ∧ ( vol* ‘ { 𝐴 , 𝐵 } ) = 0 ) → ( vol* ‘ ( ( 𝐴 [,] 𝐵 ) ∖ ( 𝐴 (,) 𝐵 ) ) ) = 0 ) |
36 |
30 31 34 35
|
syl3anc |
⊢ ( 𝜑 → ( vol* ‘ ( ( 𝐴 [,] 𝐵 ) ∖ ( 𝐴 (,) 𝐵 ) ) ) = 0 ) |
37 |
5 7 36 3
|
itgss3 |
⊢ ( 𝜑 → ( ( ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ 𝐶 ) ∈ 𝐿1 ↔ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝐶 ) ∈ 𝐿1 ) ∧ ∫ ( 𝐴 (,) 𝐵 ) 𝐶 d 𝑥 = ∫ ( 𝐴 [,] 𝐵 ) 𝐶 d 𝑥 ) ) |
38 |
37
|
simprd |
⊢ ( 𝜑 → ∫ ( 𝐴 (,) 𝐵 ) 𝐶 d 𝑥 = ∫ ( 𝐴 [,] 𝐵 ) 𝐶 d 𝑥 ) |