Step |
Hyp |
Ref |
Expression |
1 |
|
ovolval3.a |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
2 |
|
ovolval3.m |
⊢ 𝑀 = { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) } |
3 |
|
eqid |
⊢ { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( abs ∘ − ) ∘ 𝑓 ) ) ) } = { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( abs ∘ − ) ∘ 𝑓 ) ) ) } |
4 |
1 3
|
ovolval2 |
⊢ ( 𝜑 → ( vol* ‘ 𝐴 ) = inf ( { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( abs ∘ − ) ∘ 𝑓 ) ) ) } , ℝ* , < ) ) |
5 |
|
reex |
⊢ ℝ ∈ V |
6 |
5 5
|
xpex |
⊢ ( ℝ × ℝ ) ∈ V |
7 |
|
inss2 |
⊢ ( ≤ ∩ ( ℝ × ℝ ) ) ⊆ ( ℝ × ℝ ) |
8 |
|
mapss |
⊢ ( ( ( ℝ × ℝ ) ∈ V ∧ ( ≤ ∩ ( ℝ × ℝ ) ) ⊆ ( ℝ × ℝ ) ) → ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ⊆ ( ( ℝ × ℝ ) ↑m ℕ ) ) |
9 |
6 7 8
|
mp2an |
⊢ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ⊆ ( ( ℝ × ℝ ) ↑m ℕ ) |
10 |
9
|
sseli |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ) |
11 |
|
elmapi |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → 𝑓 : ℕ ⟶ ( ℝ × ℝ ) ) |
12 |
10 11
|
syl |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → 𝑓 : ℕ ⟶ ( ℝ × ℝ ) ) |
13 |
12
|
ffvelrnda |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 𝑓 ‘ 𝑛 ) ∈ ( ℝ × ℝ ) ) |
14 |
|
1st2nd2 |
⊢ ( ( 𝑓 ‘ 𝑛 ) ∈ ( ℝ × ℝ ) → ( 𝑓 ‘ 𝑛 ) = 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ) |
15 |
13 14
|
syl |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 𝑓 ‘ 𝑛 ) = 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ) |
16 |
15
|
fveq2d |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( (,) ‘ ( 𝑓 ‘ 𝑛 ) ) = ( (,) ‘ 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ) ) |
17 |
|
df-ov |
⊢ ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) = ( (,) ‘ 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ) |
18 |
17
|
eqcomi |
⊢ ( (,) ‘ 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ) = ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
19 |
18
|
a1i |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( (,) ‘ 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ) = ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
20 |
16 19
|
eqtrd |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( (,) ‘ ( 𝑓 ‘ 𝑛 ) ) = ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
21 |
20
|
fveq2d |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( vol ‘ ( (,) ‘ ( 𝑓 ‘ 𝑛 ) ) ) = ( vol ‘ ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) ) |
22 |
|
xp1st |
⊢ ( ( 𝑓 ‘ 𝑛 ) ∈ ( ℝ × ℝ ) → ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ) |
23 |
13 22
|
syl |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ) |
24 |
|
xp2nd |
⊢ ( ( 𝑓 ‘ 𝑛 ) ∈ ( ℝ × ℝ ) → ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ) |
25 |
13 24
|
syl |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ) |
26 |
|
elmapi |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → 𝑓 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → 𝑓 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
28 |
|
simpr |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℕ ) |
29 |
|
ovolfcl |
⊢ ( ( 𝑓 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ∧ 𝑛 ∈ ℕ ) → ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
30 |
27 28 29
|
syl2anc |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
31 |
30
|
simp3d |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
32 |
|
volioo |
⊢ ( ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) → ( vol ‘ ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) = ( ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
33 |
23 25 31 32
|
syl3anc |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( vol ‘ ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) = ( ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
34 |
21 33
|
eqtrd |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( vol ‘ ( (,) ‘ ( 𝑓 ‘ 𝑛 ) ) ) = ( ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
35 |
|
ioof |
⊢ (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ |
36 |
|
ffun |
⊢ ( (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ → Fun (,) ) |
37 |
35 36
|
ax-mp |
⊢ Fun (,) |
38 |
37
|
a1i |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → Fun (,) ) |
39 |
|
rexpssxrxp |
⊢ ( ℝ × ℝ ) ⊆ ( ℝ* × ℝ* ) |
40 |
39 13
|
sseldi |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 𝑓 ‘ 𝑛 ) ∈ ( ℝ* × ℝ* ) ) |
41 |
35
|
fdmi |
⊢ dom (,) = ( ℝ* × ℝ* ) |
42 |
41
|
eqcomi |
⊢ ( ℝ* × ℝ* ) = dom (,) |
43 |
42
|
a1i |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( ℝ* × ℝ* ) = dom (,) ) |
44 |
40 43
|
eleqtrd |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 𝑓 ‘ 𝑛 ) ∈ dom (,) ) |
45 |
|
fvco |
⊢ ( ( Fun (,) ∧ ( 𝑓 ‘ 𝑛 ) ∈ dom (,) ) → ( ( vol ∘ (,) ) ‘ ( 𝑓 ‘ 𝑛 ) ) = ( vol ‘ ( (,) ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
46 |
38 44 45
|
syl2anc |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( ( vol ∘ (,) ) ‘ ( 𝑓 ‘ 𝑛 ) ) = ( vol ‘ ( (,) ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
47 |
15
|
fveq2d |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑛 ) ) = ( ( abs ∘ − ) ‘ 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ) ) |
48 |
|
df-ov |
⊢ ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ( abs ∘ − ) ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) = ( ( abs ∘ − ) ‘ 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ) |
49 |
48
|
eqcomi |
⊢ ( ( abs ∘ − ) ‘ 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ) = ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ( abs ∘ − ) ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
50 |
49
|
a1i |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( ( abs ∘ − ) ‘ 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ) = ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ( abs ∘ − ) ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
51 |
23
|
recnd |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℂ ) |
52 |
25
|
recnd |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℂ ) |
53 |
|
eqid |
⊢ ( abs ∘ − ) = ( abs ∘ − ) |
54 |
53
|
cnmetdval |
⊢ ( ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℂ ∧ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℂ ) → ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ( abs ∘ − ) ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) = ( abs ‘ ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) − ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) ) |
55 |
51 52 54
|
syl2anc |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ( abs ∘ − ) ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) = ( abs ‘ ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) − ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) ) |
56 |
|
abssub |
⊢ ( ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℂ ∧ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℂ ) → ( abs ‘ ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) − ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) = ( abs ‘ ( ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) ) |
57 |
51 52 56
|
syl2anc |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( abs ‘ ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) − ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) = ( abs ‘ ( ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) ) |
58 |
23 25 31
|
abssubge0d |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( abs ‘ ( ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) = ( ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
59 |
55 57 58
|
3eqtrd |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ( abs ∘ − ) ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) = ( ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
60 |
47 50 59
|
3eqtrd |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑛 ) ) = ( ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
61 |
34 46 60
|
3eqtr4d |
⊢ ( ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( ( vol ∘ (,) ) ‘ ( 𝑓 ‘ 𝑛 ) ) = ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
62 |
61
|
mpteq2dva |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → ( 𝑛 ∈ ℕ ↦ ( ( vol ∘ (,) ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) = ( 𝑛 ∈ ℕ ↦ ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
63 |
|
volioof |
⊢ ( vol ∘ (,) ) : ( ℝ* × ℝ* ) ⟶ ( 0 [,] +∞ ) |
64 |
63
|
a1i |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → ( vol ∘ (,) ) : ( ℝ* × ℝ* ) ⟶ ( 0 [,] +∞ ) ) |
65 |
39
|
a1i |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → ( ℝ × ℝ ) ⊆ ( ℝ* × ℝ* ) ) |
66 |
12 65
|
fssd |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → 𝑓 : ℕ ⟶ ( ℝ* × ℝ* ) ) |
67 |
|
fcompt |
⊢ ( ( ( vol ∘ (,) ) : ( ℝ* × ℝ* ) ⟶ ( 0 [,] +∞ ) ∧ 𝑓 : ℕ ⟶ ( ℝ* × ℝ* ) ) → ( ( vol ∘ (,) ) ∘ 𝑓 ) = ( 𝑛 ∈ ℕ ↦ ( ( vol ∘ (,) ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
68 |
64 66 67
|
syl2anc |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → ( ( vol ∘ (,) ) ∘ 𝑓 ) = ( 𝑛 ∈ ℕ ↦ ( ( vol ∘ (,) ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
69 |
|
absf |
⊢ abs : ℂ ⟶ ℝ |
70 |
|
subf |
⊢ − : ( ℂ × ℂ ) ⟶ ℂ |
71 |
|
fco |
⊢ ( ( abs : ℂ ⟶ ℝ ∧ − : ( ℂ × ℂ ) ⟶ ℂ ) → ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ ) |
72 |
69 70 71
|
mp2an |
⊢ ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ |
73 |
72
|
a1i |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ ) |
74 |
|
rr2sscn2 |
⊢ ( ℝ × ℝ ) ⊆ ( ℂ × ℂ ) |
75 |
74
|
a1i |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → ( ℝ × ℝ ) ⊆ ( ℂ × ℂ ) ) |
76 |
12 75
|
fssd |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → 𝑓 : ℕ ⟶ ( ℂ × ℂ ) ) |
77 |
|
fcompt |
⊢ ( ( ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ ∧ 𝑓 : ℕ ⟶ ( ℂ × ℂ ) ) → ( ( abs ∘ − ) ∘ 𝑓 ) = ( 𝑛 ∈ ℕ ↦ ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
78 |
73 76 77
|
syl2anc |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → ( ( abs ∘ − ) ∘ 𝑓 ) = ( 𝑛 ∈ ℕ ↦ ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
79 |
62 68 78
|
3eqtr4d |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → ( ( vol ∘ (,) ) ∘ 𝑓 ) = ( ( abs ∘ − ) ∘ 𝑓 ) ) |
80 |
79
|
fveq2d |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) = ( Σ^ ‘ ( ( abs ∘ − ) ∘ 𝑓 ) ) ) |
81 |
80
|
eqeq2d |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → ( 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ↔ 𝑦 = ( Σ^ ‘ ( ( abs ∘ − ) ∘ 𝑓 ) ) ) ) |
82 |
81
|
anbi2d |
⊢ ( 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → ( ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ↔ ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( abs ∘ − ) ∘ 𝑓 ) ) ) ) ) |
83 |
82
|
rexbiia |
⊢ ( ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ↔ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( abs ∘ − ) ∘ 𝑓 ) ) ) ) |
84 |
83
|
rabbii |
⊢ { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) } = { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( abs ∘ − ) ∘ 𝑓 ) ) ) } |
85 |
2 84
|
eqtr2i |
⊢ { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( abs ∘ − ) ∘ 𝑓 ) ) ) } = 𝑀 |
86 |
85
|
infeq1i |
⊢ inf ( { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( abs ∘ − ) ∘ 𝑓 ) ) ) } , ℝ* , < ) = inf ( 𝑀 , ℝ* , < ) |
87 |
86
|
a1i |
⊢ ( 𝜑 → inf ( { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( abs ∘ − ) ∘ 𝑓 ) ) ) } , ℝ* , < ) = inf ( 𝑀 , ℝ* , < ) ) |
88 |
4 87
|
eqtrd |
⊢ ( 𝜑 → ( vol* ‘ 𝐴 ) = inf ( 𝑀 , ℝ* , < ) ) |