Step |
Hyp |
Ref |
Expression |
1 |
|
ovolval4lem2.a |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
2 |
|
ovolval4lem2.m |
⊢ 𝑀 = { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) } |
3 |
|
ovolval4lem2.g |
⊢ 𝐺 = ( 𝑛 ∈ ℕ ↦ 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) 〉 ) |
4 |
|
iftrue |
⊢ ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) → if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) = ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
5 |
4
|
opeq2d |
⊢ ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) → 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) 〉 = 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ) |
6 |
5
|
adantl |
⊢ ( ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) ∧ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) → 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) 〉 = 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ) |
7 |
|
df-br |
⊢ ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ↔ 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ∈ ≤ ) |
8 |
7
|
biimpi |
⊢ ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) → 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ∈ ≤ ) |
9 |
8
|
adantl |
⊢ ( ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) ∧ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) → 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ∈ ≤ ) |
10 |
6 9
|
eqeltrd |
⊢ ( ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) ∧ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) → 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) 〉 ∈ ≤ ) |
11 |
|
iffalse |
⊢ ( ¬ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) → if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) = ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
12 |
11
|
opeq2d |
⊢ ( ¬ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) → 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) 〉 = 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ) |
13 |
12
|
adantl |
⊢ ( ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) → 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) 〉 = 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ) |
14 |
|
elmapi |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → 𝑓 : ℕ ⟶ ( ℝ × ℝ ) ) |
15 |
14
|
ffvelrnda |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 𝑓 ‘ 𝑛 ) ∈ ( ℝ × ℝ ) ) |
16 |
|
xp1st |
⊢ ( ( 𝑓 ‘ 𝑛 ) ∈ ( ℝ × ℝ ) → ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ) |
17 |
15 16
|
syl |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ) |
18 |
17
|
leidd |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
19 |
|
df-br |
⊢ ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ↔ 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ∈ ≤ ) |
20 |
18 19
|
sylib |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ∈ ≤ ) |
21 |
20
|
adantr |
⊢ ( ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) → 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) 〉 ∈ ≤ ) |
22 |
13 21
|
eqeltrd |
⊢ ( ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) → 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) 〉 ∈ ≤ ) |
23 |
10 22
|
pm2.61dan |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) 〉 ∈ ≤ ) |
24 |
|
xp2nd |
⊢ ( ( 𝑓 ‘ 𝑛 ) ∈ ( ℝ × ℝ ) → ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ) |
25 |
15 24
|
syl |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ) |
26 |
25 17
|
ifcld |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) ∈ ℝ ) |
27 |
|
opelxpi |
⊢ ( ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ∈ ℝ ∧ if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) ∈ ℝ ) → 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) 〉 ∈ ( ℝ × ℝ ) ) |
28 |
17 26 27
|
syl2anc |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) 〉 ∈ ( ℝ × ℝ ) ) |
29 |
23 28
|
elind |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑛 ∈ ℕ ) → 〈 ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) 〉 ∈ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
30 |
29 3
|
fmptd |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → 𝐺 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
31 |
|
reex |
⊢ ℝ ∈ V |
32 |
31 31
|
xpex |
⊢ ( ℝ × ℝ ) ∈ V |
33 |
32
|
inex2 |
⊢ ( ≤ ∩ ( ℝ × ℝ ) ) ∈ V |
34 |
33
|
a1i |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( ≤ ∩ ( ℝ × ℝ ) ) ∈ V ) |
35 |
|
nnex |
⊢ ℕ ∈ V |
36 |
35
|
a1i |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ℕ ∈ V ) |
37 |
34 36
|
elmapd |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( 𝐺 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ↔ 𝐺 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) ) |
38 |
30 37
|
mpbird |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → 𝐺 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ) |
39 |
38
|
adantr |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ) → 𝐺 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ) |
40 |
|
simpr |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ) → 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ) |
41 |
|
rexpssxrxp |
⊢ ( ℝ × ℝ ) ⊆ ( ℝ* × ℝ* ) |
42 |
41
|
a1i |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( ℝ × ℝ ) ⊆ ( ℝ* × ℝ* ) ) |
43 |
14 42
|
fssd |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → 𝑓 : ℕ ⟶ ( ℝ* × ℝ* ) ) |
44 |
|
2fveq3 |
⊢ ( 𝑘 = 𝑛 → ( 1st ‘ ( 𝑓 ‘ 𝑘 ) ) = ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
45 |
|
2fveq3 |
⊢ ( 𝑘 = 𝑛 → ( 2nd ‘ ( 𝑓 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
46 |
44 45
|
breq12d |
⊢ ( 𝑘 = 𝑛 → ( ( 1st ‘ ( 𝑓 ‘ 𝑘 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑘 ) ) ↔ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
47 |
46
|
cbvrabv |
⊢ { 𝑘 ∈ ℕ ∣ ( 1st ‘ ( 𝑓 ‘ 𝑘 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑘 ) ) } = { 𝑛 ∈ ℕ ∣ ( 1st ‘ ( 𝑓 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝑓 ‘ 𝑛 ) ) } |
48 |
43 3 47
|
ovolval4lem1 |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( ∪ ran ( (,) ∘ 𝑓 ) = ∪ ran ( (,) ∘ 𝐺 ) ∧ ( vol ∘ ( (,) ∘ 𝑓 ) ) = ( vol ∘ ( (,) ∘ 𝐺 ) ) ) ) |
49 |
48
|
simpld |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ∪ ran ( (,) ∘ 𝑓 ) = ∪ ran ( (,) ∘ 𝐺 ) ) |
50 |
49
|
adantr |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ) → ∪ ran ( (,) ∘ 𝑓 ) = ∪ ran ( (,) ∘ 𝐺 ) ) |
51 |
40 50
|
sseqtrd |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ) → 𝐴 ⊆ ∪ ran ( (,) ∘ 𝐺 ) ) |
52 |
51
|
adantrr |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ) → 𝐴 ⊆ ∪ ran ( (,) ∘ 𝐺 ) ) |
53 |
|
simpr |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) → 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) |
54 |
48
|
simprd |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( vol ∘ ( (,) ∘ 𝑓 ) ) = ( vol ∘ ( (,) ∘ 𝐺 ) ) ) |
55 |
|
coass |
⊢ ( ( vol ∘ (,) ) ∘ 𝑓 ) = ( vol ∘ ( (,) ∘ 𝑓 ) ) |
56 |
55
|
a1i |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( ( vol ∘ (,) ) ∘ 𝑓 ) = ( vol ∘ ( (,) ∘ 𝑓 ) ) ) |
57 |
|
coass |
⊢ ( ( vol ∘ (,) ) ∘ 𝐺 ) = ( vol ∘ ( (,) ∘ 𝐺 ) ) |
58 |
57
|
a1i |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( ( vol ∘ (,) ) ∘ 𝐺 ) = ( vol ∘ ( (,) ∘ 𝐺 ) ) ) |
59 |
54 56 58
|
3eqtr4d |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( ( vol ∘ (,) ) ∘ 𝑓 ) = ( ( vol ∘ (,) ) ∘ 𝐺 ) ) |
60 |
59
|
fveq2d |
⊢ ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) ) |
61 |
60
|
adantr |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) → ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) ) |
62 |
53 61
|
eqtrd |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) → 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) ) |
63 |
62
|
adantrl |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ) → 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) ) |
64 |
52 63
|
jca |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ) → ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝐺 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) ) ) |
65 |
|
coeq2 |
⊢ ( 𝑔 = 𝐺 → ( (,) ∘ 𝑔 ) = ( (,) ∘ 𝐺 ) ) |
66 |
65
|
rneqd |
⊢ ( 𝑔 = 𝐺 → ran ( (,) ∘ 𝑔 ) = ran ( (,) ∘ 𝐺 ) ) |
67 |
66
|
unieqd |
⊢ ( 𝑔 = 𝐺 → ∪ ran ( (,) ∘ 𝑔 ) = ∪ ran ( (,) ∘ 𝐺 ) ) |
68 |
67
|
sseq2d |
⊢ ( 𝑔 = 𝐺 → ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ↔ 𝐴 ⊆ ∪ ran ( (,) ∘ 𝐺 ) ) ) |
69 |
|
coeq2 |
⊢ ( 𝑔 = 𝐺 → ( ( vol ∘ (,) ) ∘ 𝑔 ) = ( ( vol ∘ (,) ) ∘ 𝐺 ) ) |
70 |
69
|
fveq2d |
⊢ ( 𝑔 = 𝐺 → ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) ) |
71 |
70
|
eqeq2d |
⊢ ( 𝑔 = 𝐺 → ( 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ↔ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) ) ) |
72 |
68 71
|
anbi12d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ↔ ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝐺 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) ) ) ) |
73 |
72
|
rspcev |
⊢ ( ( 𝐺 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝐺 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝐺 ) ) ) ) → ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) |
74 |
39 64 73
|
syl2anc |
⊢ ( ( 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ) → ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) |
75 |
74
|
rexlimiva |
⊢ ( ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) → ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) |
76 |
|
inss2 |
⊢ ( ≤ ∩ ( ℝ × ℝ ) ) ⊆ ( ℝ × ℝ ) |
77 |
|
mapss |
⊢ ( ( ( ℝ × ℝ ) ∈ V ∧ ( ≤ ∩ ( ℝ × ℝ ) ) ⊆ ( ℝ × ℝ ) ) → ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ⊆ ( ( ℝ × ℝ ) ↑m ℕ ) ) |
78 |
32 76 77
|
mp2an |
⊢ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ⊆ ( ( ℝ × ℝ ) ↑m ℕ ) |
79 |
78
|
sseli |
⊢ ( 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) → 𝑔 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ) |
80 |
79
|
adantr |
⊢ ( ( 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) → 𝑔 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ) |
81 |
|
simpr |
⊢ ( ( 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) → ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) |
82 |
|
coeq2 |
⊢ ( 𝑓 = 𝑔 → ( (,) ∘ 𝑓 ) = ( (,) ∘ 𝑔 ) ) |
83 |
82
|
rneqd |
⊢ ( 𝑓 = 𝑔 → ran ( (,) ∘ 𝑓 ) = ran ( (,) ∘ 𝑔 ) ) |
84 |
83
|
unieqd |
⊢ ( 𝑓 = 𝑔 → ∪ ran ( (,) ∘ 𝑓 ) = ∪ ran ( (,) ∘ 𝑔 ) ) |
85 |
84
|
sseq2d |
⊢ ( 𝑓 = 𝑔 → ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ↔ 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ) ) |
86 |
|
coeq2 |
⊢ ( 𝑓 = 𝑔 → ( ( vol ∘ (,) ) ∘ 𝑓 ) = ( ( vol ∘ (,) ) ∘ 𝑔 ) ) |
87 |
86
|
fveq2d |
⊢ ( 𝑓 = 𝑔 → ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) |
88 |
87
|
eqeq2d |
⊢ ( 𝑓 = 𝑔 → ( 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ↔ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) |
89 |
85 88
|
anbi12d |
⊢ ( 𝑓 = 𝑔 → ( ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ↔ ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) ) |
90 |
89
|
rspcev |
⊢ ( ( 𝑔 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) → ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ) |
91 |
80 81 90
|
syl2anc |
⊢ ( ( 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ∧ ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) → ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ) |
92 |
91
|
rexlimiva |
⊢ ( ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) → ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ) |
93 |
75 92
|
impbii |
⊢ ( ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) ↔ ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) ) |
94 |
93
|
rabbii |
⊢ { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑓 ) ) ) } = { 𝑦 ∈ ℝ* ∣ ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) } |
95 |
2 94
|
eqtri |
⊢ 𝑀 = { 𝑦 ∈ ℝ* ∣ ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑦 = ( Σ^ ‘ ( ( vol ∘ (,) ) ∘ 𝑔 ) ) ) } |
96 |
1 95
|
ovolval3 |
⊢ ( 𝜑 → ( vol* ‘ 𝐴 ) = inf ( 𝑀 , ℝ* , < ) ) |