Step |
Hyp |
Ref |
Expression |
1 |
|
voliccico.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
voliccico.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
iftrue |
⊢ ( 𝐴 < 𝐵 → if ( 𝐴 < 𝐵 , ( 𝐵 − 𝐴 ) , 0 ) = ( 𝐵 − 𝐴 ) ) |
4 |
3
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐴 < 𝐵 ) → if ( 𝐴 < 𝐵 , ( 𝐵 − 𝐴 ) , 0 ) = ( 𝐵 − 𝐴 ) ) |
5 |
2
|
recnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
6 |
5
|
subidd |
⊢ ( 𝜑 → ( 𝐵 − 𝐵 ) = 0 ) |
7 |
6
|
eqcomd |
⊢ ( 𝜑 → 0 = ( 𝐵 − 𝐵 ) ) |
8 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ ¬ 𝐴 < 𝐵 ) → 0 = ( 𝐵 − 𝐵 ) ) |
9 |
|
iffalse |
⊢ ( ¬ 𝐴 < 𝐵 → if ( 𝐴 < 𝐵 , ( 𝐵 − 𝐴 ) , 0 ) = 0 ) |
10 |
9
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ ¬ 𝐴 < 𝐵 ) → if ( 𝐴 < 𝐵 , ( 𝐵 − 𝐴 ) , 0 ) = 0 ) |
11 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ ¬ 𝐴 < 𝐵 ) → 𝜑 ) |
12 |
11 1
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ ¬ 𝐴 < 𝐵 ) → 𝐴 ∈ ℝ ) |
13 |
11 2
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ ¬ 𝐴 < 𝐵 ) → 𝐵 ∈ ℝ ) |
14 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ≤ 𝐵 ) |
15 |
14
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ ¬ 𝐴 < 𝐵 ) → 𝐴 ≤ 𝐵 ) |
16 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ ¬ 𝐴 < 𝐵 ) → ¬ 𝐴 < 𝐵 ) |
17 |
12 13 15 16
|
lenlteq |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ ¬ 𝐴 < 𝐵 ) → 𝐴 = 𝐵 ) |
18 |
|
oveq2 |
⊢ ( 𝐴 = 𝐵 → ( 𝐵 − 𝐴 ) = ( 𝐵 − 𝐵 ) ) |
19 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( 𝐵 − 𝐴 ) = ( 𝐵 − 𝐵 ) ) |
20 |
11 17 19
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ ¬ 𝐴 < 𝐵 ) → ( 𝐵 − 𝐴 ) = ( 𝐵 − 𝐵 ) ) |
21 |
8 10 20
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ ¬ 𝐴 < 𝐵 ) → if ( 𝐴 < 𝐵 , ( 𝐵 − 𝐴 ) , 0 ) = ( 𝐵 − 𝐴 ) ) |
22 |
4 21
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → if ( 𝐴 < 𝐵 , ( 𝐵 − 𝐴 ) , 0 ) = ( 𝐵 − 𝐴 ) ) |
23 |
22
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → ( 𝐵 − 𝐴 ) = if ( 𝐴 < 𝐵 , ( 𝐵 − 𝐴 ) , 0 ) ) |
24 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ℝ ) |
25 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ℝ ) |
26 |
|
volicc |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐵 − 𝐴 ) ) |
27 |
24 25 14 26
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐵 − 𝐴 ) ) |
28 |
|
volico |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( vol ‘ ( 𝐴 [,) 𝐵 ) ) = if ( 𝐴 < 𝐵 , ( 𝐵 − 𝐴 ) , 0 ) ) |
29 |
1 2 28
|
syl2anc |
⊢ ( 𝜑 → ( vol ‘ ( 𝐴 [,) 𝐵 ) ) = if ( 𝐴 < 𝐵 , ( 𝐵 − 𝐴 ) , 0 ) ) |
30 |
29
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → ( vol ‘ ( 𝐴 [,) 𝐵 ) ) = if ( 𝐴 < 𝐵 , ( 𝐵 − 𝐴 ) , 0 ) ) |
31 |
23 27 30
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) = ( vol ‘ ( 𝐴 [,) 𝐵 ) ) ) |
32 |
|
simpl |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ≤ 𝐵 ) → 𝜑 ) |
33 |
|
simpr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ≤ 𝐵 ) → ¬ 𝐴 ≤ 𝐵 ) |
34 |
32 2
|
syl |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ℝ ) |
35 |
32 1
|
syl |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ℝ ) |
36 |
34 35
|
ltnled |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ≤ 𝐵 ) → ( 𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵 ) ) |
37 |
33 36
|
mpbird |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ≤ 𝐵 ) → 𝐵 < 𝐴 ) |
38 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → 𝐵 < 𝐴 ) |
39 |
1
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
40 |
2
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
41 |
|
icc0 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 [,] 𝐵 ) = ∅ ↔ 𝐵 < 𝐴 ) ) |
42 |
39 40 41
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐴 [,] 𝐵 ) = ∅ ↔ 𝐵 < 𝐴 ) ) |
43 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ( ( 𝐴 [,] 𝐵 ) = ∅ ↔ 𝐵 < 𝐴 ) ) |
44 |
38 43
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ( 𝐴 [,] 𝐵 ) = ∅ ) |
45 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → 𝐵 ∈ ℝ ) |
46 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → 𝐴 ∈ ℝ ) |
47 |
45 46 38
|
ltled |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → 𝐵 ≤ 𝐴 ) |
48 |
46
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → 𝐴 ∈ ℝ* ) |
49 |
45
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → 𝐵 ∈ ℝ* ) |
50 |
|
ico0 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 [,) 𝐵 ) = ∅ ↔ 𝐵 ≤ 𝐴 ) ) |
51 |
48 49 50
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ( ( 𝐴 [,) 𝐵 ) = ∅ ↔ 𝐵 ≤ 𝐴 ) ) |
52 |
47 51
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ( 𝐴 [,) 𝐵 ) = ∅ ) |
53 |
44 52
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ( 𝐴 [,] 𝐵 ) = ( 𝐴 [,) 𝐵 ) ) |
54 |
53
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) = ( vol ‘ ( 𝐴 [,) 𝐵 ) ) ) |
55 |
32 37 54
|
syl2anc |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ≤ 𝐵 ) → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) = ( vol ‘ ( 𝐴 [,) 𝐵 ) ) ) |
56 |
31 55
|
pm2.61dan |
⊢ ( 𝜑 → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) = ( vol ‘ ( 𝐴 [,) 𝐵 ) ) ) |