Step |
Hyp |
Ref |
Expression |
1 |
|
ditgcl.x |
⊢ ( 𝜑 → 𝑋 ∈ ℝ ) |
2 |
|
ditgcl.y |
⊢ ( 𝜑 → 𝑌 ∈ ℝ ) |
3 |
|
ditgcl.a |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝑋 [,] 𝑌 ) ) |
4 |
|
ditgcl.b |
⊢ ( 𝜑 → 𝐵 ∈ ( 𝑋 [,] 𝑌 ) ) |
5 |
|
ditgcl.c |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑋 (,) 𝑌 ) ) → 𝐶 ∈ 𝑉 ) |
6 |
|
ditgcl.i |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝑋 (,) 𝑌 ) ↦ 𝐶 ) ∈ 𝐿1 ) |
7 |
|
elicc2 |
⊢ ( ( 𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ ) → ( 𝐴 ∈ ( 𝑋 [,] 𝑌 ) ↔ ( 𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌 ) ) ) |
8 |
1 2 7
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 ∈ ( 𝑋 [,] 𝑌 ) ↔ ( 𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌 ) ) ) |
9 |
3 8
|
mpbid |
⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌 ) ) |
10 |
9
|
simp1d |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
11 |
|
elicc2 |
⊢ ( ( 𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ ) → ( 𝐵 ∈ ( 𝑋 [,] 𝑌 ) ↔ ( 𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌 ) ) ) |
12 |
1 2 11
|
syl2anc |
⊢ ( 𝜑 → ( 𝐵 ∈ ( 𝑋 [,] 𝑌 ) ↔ ( 𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌 ) ) ) |
13 |
4 12
|
mpbid |
⊢ ( 𝜑 → ( 𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌 ) ) |
14 |
13
|
simp1d |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
15 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ≤ 𝐵 ) |
16 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ℝ ) |
17 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ℝ ) |
18 |
15 16 17
|
ditgneg |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → ⨜ [ 𝐵 → 𝐴 ] 𝐶 d 𝑥 = - ∫ ( 𝐴 (,) 𝐵 ) 𝐶 d 𝑥 ) |
19 |
15
|
ditgpos |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → ⨜ [ 𝐴 → 𝐵 ] 𝐶 d 𝑥 = ∫ ( 𝐴 (,) 𝐵 ) 𝐶 d 𝑥 ) |
20 |
19
|
negeqd |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → - ⨜ [ 𝐴 → 𝐵 ] 𝐶 d 𝑥 = - ∫ ( 𝐴 (,) 𝐵 ) 𝐶 d 𝑥 ) |
21 |
18 20
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → ⨜ [ 𝐵 → 𝐴 ] 𝐶 d 𝑥 = - ⨜ [ 𝐴 → 𝐵 ] 𝐶 d 𝑥 ) |
22 |
1
|
rexrd |
⊢ ( 𝜑 → 𝑋 ∈ ℝ* ) |
23 |
13
|
simp2d |
⊢ ( 𝜑 → 𝑋 ≤ 𝐵 ) |
24 |
|
iooss1 |
⊢ ( ( 𝑋 ∈ ℝ* ∧ 𝑋 ≤ 𝐵 ) → ( 𝐵 (,) 𝐴 ) ⊆ ( 𝑋 (,) 𝐴 ) ) |
25 |
22 23 24
|
syl2anc |
⊢ ( 𝜑 → ( 𝐵 (,) 𝐴 ) ⊆ ( 𝑋 (,) 𝐴 ) ) |
26 |
2
|
rexrd |
⊢ ( 𝜑 → 𝑌 ∈ ℝ* ) |
27 |
9
|
simp3d |
⊢ ( 𝜑 → 𝐴 ≤ 𝑌 ) |
28 |
|
iooss2 |
⊢ ( ( 𝑌 ∈ ℝ* ∧ 𝐴 ≤ 𝑌 ) → ( 𝑋 (,) 𝐴 ) ⊆ ( 𝑋 (,) 𝑌 ) ) |
29 |
26 27 28
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 (,) 𝐴 ) ⊆ ( 𝑋 (,) 𝑌 ) ) |
30 |
25 29
|
sstrd |
⊢ ( 𝜑 → ( 𝐵 (,) 𝐴 ) ⊆ ( 𝑋 (,) 𝑌 ) ) |
31 |
30
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐵 (,) 𝐴 ) ) → 𝑥 ∈ ( 𝑋 (,) 𝑌 ) ) |
32 |
|
iblmbf |
⊢ ( ( 𝑥 ∈ ( 𝑋 (,) 𝑌 ) ↦ 𝐶 ) ∈ 𝐿1 → ( 𝑥 ∈ ( 𝑋 (,) 𝑌 ) ↦ 𝐶 ) ∈ MblFn ) |
33 |
6 32
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝑋 (,) 𝑌 ) ↦ 𝐶 ) ∈ MblFn ) |
34 |
33 5
|
mbfmptcl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑋 (,) 𝑌 ) ) → 𝐶 ∈ ℂ ) |
35 |
31 34
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐵 (,) 𝐴 ) ) → 𝐶 ∈ ℂ ) |
36 |
|
ioombl |
⊢ ( 𝐵 (,) 𝐴 ) ∈ dom vol |
37 |
36
|
a1i |
⊢ ( 𝜑 → ( 𝐵 (,) 𝐴 ) ∈ dom vol ) |
38 |
30 37 5 6
|
iblss |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐵 (,) 𝐴 ) ↦ 𝐶 ) ∈ 𝐿1 ) |
39 |
35 38
|
itgcl |
⊢ ( 𝜑 → ∫ ( 𝐵 (,) 𝐴 ) 𝐶 d 𝑥 ∈ ℂ ) |
40 |
39
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → ∫ ( 𝐵 (,) 𝐴 ) 𝐶 d 𝑥 ∈ ℂ ) |
41 |
40
|
negnegd |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → - - ∫ ( 𝐵 (,) 𝐴 ) 𝐶 d 𝑥 = ∫ ( 𝐵 (,) 𝐴 ) 𝐶 d 𝑥 ) |
42 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → 𝐵 ≤ 𝐴 ) |
43 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → 𝐵 ∈ ℝ ) |
44 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → 𝐴 ∈ ℝ ) |
45 |
42 43 44
|
ditgneg |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → ⨜ [ 𝐴 → 𝐵 ] 𝐶 d 𝑥 = - ∫ ( 𝐵 (,) 𝐴 ) 𝐶 d 𝑥 ) |
46 |
45
|
negeqd |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → - ⨜ [ 𝐴 → 𝐵 ] 𝐶 d 𝑥 = - - ∫ ( 𝐵 (,) 𝐴 ) 𝐶 d 𝑥 ) |
47 |
42
|
ditgpos |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → ⨜ [ 𝐵 → 𝐴 ] 𝐶 d 𝑥 = ∫ ( 𝐵 (,) 𝐴 ) 𝐶 d 𝑥 ) |
48 |
41 46 47
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → ⨜ [ 𝐵 → 𝐴 ] 𝐶 d 𝑥 = - ⨜ [ 𝐴 → 𝐵 ] 𝐶 d 𝑥 ) |
49 |
10 14 21 48
|
lecasei |
⊢ ( 𝜑 → ⨜ [ 𝐵 → 𝐴 ] 𝐶 d 𝑥 = - ⨜ [ 𝐴 → 𝐵 ] 𝐶 d 𝑥 ) |