Step |
Hyp |
Ref |
Expression |
1 |
|
itggt0.1 |
⊢ ( 𝜑 → 0 < ( vol ‘ 𝐴 ) ) |
2 |
|
itggt0.2 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 ) |
3 |
|
itggt0.3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ+ ) |
4 |
|
iblmbf |
⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn ) |
5 |
2 4
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn ) |
6 |
5 3
|
mbfdm2 |
⊢ ( 𝜑 → 𝐴 ∈ dom vol ) |
7 |
3
|
rpred |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
8 |
3
|
rpge0d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ 𝐵 ) |
9 |
|
elrege0 |
⊢ ( 𝐵 ∈ ( 0 [,) +∞ ) ↔ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) |
10 |
7 8 9
|
sylanbrc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,) +∞ ) ) |
11 |
|
0e0icopnf |
⊢ 0 ∈ ( 0 [,) +∞ ) |
12 |
11
|
a1i |
⊢ ( ( 𝜑 ∧ ¬ 𝑥 ∈ 𝐴 ) → 0 ∈ ( 0 [,) +∞ ) ) |
13 |
10 12
|
ifclda |
⊢ ( 𝜑 → if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ∈ ( 0 [,) +∞ ) ) |
14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ∈ ( 0 [,) +∞ ) ) |
15 |
14
|
fmpttd |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
16 |
|
mblss |
⊢ ( 𝐴 ∈ dom vol → 𝐴 ⊆ ℝ ) |
17 |
6 16
|
syl |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
18 |
|
rembl |
⊢ ℝ ∈ dom vol |
19 |
18
|
a1i |
⊢ ( 𝜑 → ℝ ∈ dom vol ) |
20 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ∈ ( 0 [,) +∞ ) ) |
21 |
|
eldifn |
⊢ ( 𝑥 ∈ ( ℝ ∖ 𝐴 ) → ¬ 𝑥 ∈ 𝐴 ) |
22 |
21
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ℝ ∖ 𝐴 ) ) → ¬ 𝑥 ∈ 𝐴 ) |
23 |
22
|
iffalsed |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ℝ ∖ 𝐴 ) ) → if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) = 0 ) |
24 |
|
iftrue |
⊢ ( 𝑥 ∈ 𝐴 → if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) = 𝐵 ) |
25 |
24
|
mpteq2ia |
⊢ ( 𝑥 ∈ 𝐴 ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
26 |
25 5
|
eqeltrid |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ∈ MblFn ) |
27 |
17 19 20 23 26
|
mbfss |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ∈ MblFn ) |
28 |
3
|
rpgt0d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 < 𝐵 ) |
29 |
17
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ℝ ) |
30 |
24
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) = 𝐵 ) |
31 |
30 3
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ∈ ℝ+ ) |
32 |
|
eqid |
⊢ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) |
33 |
32
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ ℝ ∧ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ∈ ℝ+ ) → ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑥 ) = if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) |
34 |
29 31 33
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑥 ) = if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) |
35 |
34 30
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑥 ) = 𝐵 ) |
36 |
28 35
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 < ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑥 ) ) |
37 |
36
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 0 < ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑥 ) ) |
38 |
|
nfcv |
⊢ Ⅎ 𝑥 0 |
39 |
|
nfcv |
⊢ Ⅎ 𝑥 < |
40 |
|
nffvmpt1 |
⊢ Ⅎ 𝑥 ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑦 ) |
41 |
38 39 40
|
nfbr |
⊢ Ⅎ 𝑥 0 < ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑦 ) |
42 |
|
nfv |
⊢ Ⅎ 𝑦 0 < ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑥 ) |
43 |
|
fveq2 |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑦 ) = ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑥 ) ) |
44 |
43
|
breq2d |
⊢ ( 𝑦 = 𝑥 → ( 0 < ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑦 ) ↔ 0 < ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑥 ) ) ) |
45 |
41 42 44
|
cbvralw |
⊢ ( ∀ 𝑦 ∈ 𝐴 0 < ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑦 ) ↔ ∀ 𝑥 ∈ 𝐴 0 < ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑥 ) ) |
46 |
37 45
|
sylibr |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐴 0 < ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑦 ) ) |
47 |
46
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → 0 < ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ‘ 𝑦 ) ) |
48 |
6 1 15 27 47
|
itg2gt0 |
⊢ ( 𝜑 → 0 < ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ) ) |
49 |
7 2 8
|
itgposval |
⊢ ( 𝜑 → ∫ 𝐴 𝐵 d 𝑥 = ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , 𝐵 , 0 ) ) ) ) |
50 |
48 49
|
breqtrrd |
⊢ ( 𝜑 → 0 < ∫ 𝐴 𝐵 d 𝑥 ) |