Step |
Hyp |
Ref |
Expression |
1 |
|
ovolval4lem1.f |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ ( ℝ* × ℝ* ) ) |
2 |
|
ovolval4lem1.g |
⊢ 𝐺 = ( 𝑛 ∈ ℕ ↦ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) 〉 ) |
3 |
|
ovolval4lem1.a |
⊢ 𝐴 = { 𝑛 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) } |
4 |
|
ioof |
⊢ (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ |
5 |
4
|
a1i |
⊢ ( 𝜑 → (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ ) |
6 |
|
fco |
⊢ ( ( (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ ∧ 𝐹 : ℕ ⟶ ( ℝ* × ℝ* ) ) → ( (,) ∘ 𝐹 ) : ℕ ⟶ 𝒫 ℝ ) |
7 |
5 1 6
|
syl2anc |
⊢ ( 𝜑 → ( (,) ∘ 𝐹 ) : ℕ ⟶ 𝒫 ℝ ) |
8 |
7
|
ffnd |
⊢ ( 𝜑 → ( (,) ∘ 𝐹 ) Fn ℕ ) |
9 |
|
fniunfv |
⊢ ( ( (,) ∘ 𝐹 ) Fn ℕ → ∪ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ∪ ran ( (,) ∘ 𝐹 ) ) |
10 |
8 9
|
syl |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ∪ ran ( (,) ∘ 𝐹 ) ) |
11 |
10
|
eqcomd |
⊢ ( 𝜑 → ∪ ran ( (,) ∘ 𝐹 ) = ∪ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ) |
12 |
|
ssrab2 |
⊢ { 𝑛 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) } ⊆ ℕ |
13 |
3 12
|
eqsstri |
⊢ 𝐴 ⊆ ℕ |
14 |
|
undif |
⊢ ( 𝐴 ⊆ ℕ ↔ ( 𝐴 ∪ ( ℕ ∖ 𝐴 ) ) = ℕ ) |
15 |
13 14
|
mpbi |
⊢ ( 𝐴 ∪ ( ℕ ∖ 𝐴 ) ) = ℕ |
16 |
15
|
eqcomi |
⊢ ℕ = ( 𝐴 ∪ ( ℕ ∖ 𝐴 ) ) |
17 |
16
|
iuneq1i |
⊢ ∪ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ∪ 𝑛 ∈ ( 𝐴 ∪ ( ℕ ∖ 𝐴 ) ) ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) |
18 |
|
iunxun |
⊢ ∪ 𝑛 ∈ ( 𝐴 ∪ ( ℕ ∖ 𝐴 ) ) ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ( ∪ 𝑛 ∈ 𝐴 ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ∪ ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ) |
19 |
17 18
|
eqtri |
⊢ ∪ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ( ∪ 𝑛 ∈ 𝐴 ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ∪ ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ) |
20 |
19
|
a1i |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ( ∪ 𝑛 ∈ 𝐴 ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ∪ ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ) ) |
21 |
1
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐹 ‘ 𝑛 ) ∈ ( ℝ* × ℝ* ) ) |
22 |
|
xp1st |
⊢ ( ( 𝐹 ‘ 𝑛 ) ∈ ( ℝ* × ℝ* ) → ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ* ) |
23 |
21 22
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ* ) |
24 |
|
xp2nd |
⊢ ( ( 𝐹 ‘ 𝑛 ) ∈ ( ℝ* × ℝ* ) → ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ* ) |
25 |
21 24
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ* ) |
26 |
25 23
|
ifcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → if ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) ∈ ℝ* ) |
27 |
23 26
|
opelxpd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) 〉 ∈ ( ℝ* × ℝ* ) ) |
28 |
27 2
|
fmptd |
⊢ ( 𝜑 → 𝐺 : ℕ ⟶ ( ℝ* × ℝ* ) ) |
29 |
|
fco |
⊢ ( ( (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ ∧ 𝐺 : ℕ ⟶ ( ℝ* × ℝ* ) ) → ( (,) ∘ 𝐺 ) : ℕ ⟶ 𝒫 ℝ ) |
30 |
5 28 29
|
syl2anc |
⊢ ( 𝜑 → ( (,) ∘ 𝐺 ) : ℕ ⟶ 𝒫 ℝ ) |
31 |
30
|
ffnd |
⊢ ( 𝜑 → ( (,) ∘ 𝐺 ) Fn ℕ ) |
32 |
|
fniunfv |
⊢ ( ( (,) ∘ 𝐺 ) Fn ℕ → ∪ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ∪ ran ( (,) ∘ 𝐺 ) ) |
33 |
31 32
|
syl |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ∪ ran ( (,) ∘ 𝐺 ) ) |
34 |
33
|
eqcomd |
⊢ ( 𝜑 → ∪ ran ( (,) ∘ 𝐺 ) = ∪ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ) |
35 |
16
|
iuneq1i |
⊢ ∪ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ∪ 𝑛 ∈ ( 𝐴 ∪ ( ℕ ∖ 𝐴 ) ) ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) |
36 |
|
iunxun |
⊢ ∪ 𝑛 ∈ ( 𝐴 ∪ ( ℕ ∖ 𝐴 ) ) ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ( ∪ 𝑛 ∈ 𝐴 ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ∪ ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ) |
37 |
35 36
|
eqtri |
⊢ ∪ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ( ∪ 𝑛 ∈ 𝐴 ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ∪ ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ) |
38 |
37
|
a1i |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ( ∪ 𝑛 ∈ 𝐴 ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ∪ ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ) ) |
39 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → 𝐺 : ℕ ⟶ ( ℝ* × ℝ* ) ) |
40 |
13
|
sseli |
⊢ ( 𝑛 ∈ 𝐴 → 𝑛 ∈ ℕ ) |
41 |
40
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → 𝑛 ∈ ℕ ) |
42 |
|
fvco3 |
⊢ ( ( 𝐺 : ℕ ⟶ ( ℝ* × ℝ* ) ∧ 𝑛 ∈ ℕ ) → ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ( (,) ‘ ( 𝐺 ‘ 𝑛 ) ) ) |
43 |
39 41 42
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ( (,) ‘ ( 𝐺 ‘ 𝑛 ) ) ) |
44 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → 𝐹 : ℕ ⟶ ( ℝ* × ℝ* ) ) |
45 |
|
fvco3 |
⊢ ( ( 𝐹 : ℕ ⟶ ( ℝ* × ℝ* ) ∧ 𝑛 ∈ ℕ ) → ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ( (,) ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
46 |
44 41 45
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ( (,) ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
47 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → 𝜑 ) |
48 |
|
1st2nd2 |
⊢ ( ( 𝐹 ‘ 𝑛 ) ∈ ( ℝ* × ℝ* ) → ( 𝐹 ‘ 𝑛 ) = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) |
49 |
21 48
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐹 ‘ 𝑛 ) = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) |
50 |
47 41 49
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑛 ) = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) |
51 |
2
|
a1i |
⊢ ( 𝜑 → 𝐺 = ( 𝑛 ∈ ℕ ↦ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) 〉 ) ) |
52 |
27
|
elexd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) 〉 ∈ V ) |
53 |
51 52
|
fvmpt2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐺 ‘ 𝑛 ) = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) 〉 ) |
54 |
47 41 53
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑛 ) = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) 〉 ) |
55 |
3
|
eleq2i |
⊢ ( 𝑛 ∈ 𝐴 ↔ 𝑛 ∈ { 𝑛 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) } ) |
56 |
55
|
biimpi |
⊢ ( 𝑛 ∈ 𝐴 → 𝑛 ∈ { 𝑛 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) } ) |
57 |
|
rabid |
⊢ ( 𝑛 ∈ { 𝑛 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) } ↔ ( 𝑛 ∈ ℕ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
58 |
56 57
|
sylib |
⊢ ( 𝑛 ∈ 𝐴 → ( 𝑛 ∈ ℕ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
59 |
58
|
simprd |
⊢ ( 𝑛 ∈ 𝐴 → ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
60 |
59
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
61 |
60
|
iftrued |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → if ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
62 |
61
|
opeq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) 〉 = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) |
63 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) |
64 |
54 62 63
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑛 ) = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) |
65 |
50 64
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑛 ) = ( 𝐺 ‘ 𝑛 ) ) |
66 |
65
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → ( (,) ‘ ( 𝐹 ‘ 𝑛 ) ) = ( (,) ‘ ( 𝐺 ‘ 𝑛 ) ) ) |
67 |
46 66
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ( (,) ‘ ( 𝐺 ‘ 𝑛 ) ) ) |
68 |
43 67
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ) |
69 |
68
|
iuneq2dv |
⊢ ( 𝜑 → ∪ 𝑛 ∈ 𝐴 ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ∪ 𝑛 ∈ 𝐴 ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ) |
70 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → 𝐺 : ℕ ⟶ ( ℝ* × ℝ* ) ) |
71 |
|
eldifi |
⊢ ( 𝑛 ∈ ( ℕ ∖ 𝐴 ) → 𝑛 ∈ ℕ ) |
72 |
71
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → 𝑛 ∈ ℕ ) |
73 |
70 72 42
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ( (,) ‘ ( 𝐺 ‘ 𝑛 ) ) ) |
74 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → 𝜑 ) |
75 |
74 72 53
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( 𝐺 ‘ 𝑛 ) = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) 〉 ) |
76 |
71
|
anim1i |
⊢ ( ( 𝑛 ∈ ( ℕ ∖ 𝐴 ) ∧ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) → ( 𝑛 ∈ ℕ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
77 |
76 57
|
sylibr |
⊢ ( ( 𝑛 ∈ ( ℕ ∖ 𝐴 ) ∧ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) → 𝑛 ∈ { 𝑛 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) } ) |
78 |
77 55
|
sylibr |
⊢ ( ( 𝑛 ∈ ( ℕ ∖ 𝐴 ) ∧ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) → 𝑛 ∈ 𝐴 ) |
79 |
78
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) ∧ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) → 𝑛 ∈ 𝐴 ) |
80 |
|
eldifn |
⊢ ( 𝑛 ∈ ( ℕ ∖ 𝐴 ) → ¬ 𝑛 ∈ 𝐴 ) |
81 |
80
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) ∧ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) → ¬ 𝑛 ∈ 𝐴 ) |
82 |
79 81
|
pm2.65da |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ¬ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
83 |
82
|
iffalsed |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → if ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) = ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
84 |
83
|
opeq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , if ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) 〉 = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) |
85 |
75 84
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( 𝐺 ‘ 𝑛 ) = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) |
86 |
85
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( (,) ‘ ( 𝐺 ‘ 𝑛 ) ) = ( (,) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) ) |
87 |
|
iooid |
⊢ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) (,) ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) = ∅ |
88 |
87
|
eqcomi |
⊢ ∅ = ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) (,) ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
89 |
|
df-ov |
⊢ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) (,) ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) = ( (,) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) |
90 |
88 89
|
eqtr2i |
⊢ ( (,) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) = ∅ |
91 |
90
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( (,) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) = ∅ ) |
92 |
73 86 91
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ∅ ) |
93 |
92
|
iuneq2dv |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ∅ ) |
94 |
|
iun0 |
⊢ ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ∅ = ∅ |
95 |
94
|
a1i |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ∅ = ∅ ) |
96 |
93 95
|
eqtrd |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ∅ ) |
97 |
74 1
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → 𝐹 : ℕ ⟶ ( ℝ* × ℝ* ) ) |
98 |
97 72 45
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ( (,) ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
99 |
74 72 49
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( 𝐹 ‘ 𝑛 ) = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) |
100 |
99
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( (,) ‘ ( 𝐹 ‘ 𝑛 ) ) = ( (,) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) ) |
101 |
|
df-ov |
⊢ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) = ( (,) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) |
102 |
101
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) = ( (,) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) ) |
103 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) ∧ ¬ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) → 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) |
104 |
72 23
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ* ) |
105 |
104
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) ∧ ¬ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) → ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ* ) |
106 |
72 25
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ* ) |
107 |
106
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) ∧ ¬ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) → ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ* ) |
108 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) ∧ ¬ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) → ¬ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
109 |
105 107
|
xrltnled |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) ∧ ¬ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ↔ ¬ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
110 |
108 109
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) ∧ ¬ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) → ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
111 |
105 107 110
|
xrltled |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) ∧ ¬ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) → ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
112 |
103 111 78
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) ∧ ¬ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) → 𝑛 ∈ 𝐴 ) |
113 |
80
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) ∧ ¬ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) → ¬ 𝑛 ∈ 𝐴 ) |
114 |
112 113
|
condan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
115 |
|
ioo0 |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ* ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ* ) → ( ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) = ∅ ↔ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
116 |
104 106 115
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) = ∅ ↔ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
117 |
114 116
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) = ∅ ) |
118 |
102 117
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( (,) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) = ∅ ) |
119 |
98 100 118
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ∅ ) |
120 |
119
|
iuneq2dv |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ∅ ) |
121 |
120 95
|
eqtrd |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ∅ ) |
122 |
96 121
|
eqtr4d |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ) |
123 |
69 122
|
uneq12d |
⊢ ( 𝜑 → ( ∪ 𝑛 ∈ 𝐴 ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ∪ ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ) = ( ∪ 𝑛 ∈ 𝐴 ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ∪ ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ) ) |
124 |
34 38 123
|
3eqtrrd |
⊢ ( 𝜑 → ( ∪ 𝑛 ∈ 𝐴 ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ∪ ∪ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ) = ∪ ran ( (,) ∘ 𝐺 ) ) |
125 |
11 20 124
|
3eqtrd |
⊢ ( 𝜑 → ∪ ran ( (,) ∘ 𝐹 ) = ∪ ran ( (,) ∘ 𝐺 ) ) |
126 |
|
volf |
⊢ vol : dom vol ⟶ ( 0 [,] +∞ ) |
127 |
126
|
a1i |
⊢ ( 𝜑 → vol : dom vol ⟶ ( 0 [,] +∞ ) ) |
128 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐹 : ℕ ⟶ ( ℝ* × ℝ* ) ) |
129 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℕ ) |
130 |
128 129 45
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ( (,) ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
131 |
49
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( (,) ‘ ( 𝐹 ‘ 𝑛 ) ) = ( (,) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) ) |
132 |
101
|
eqcomi |
⊢ ( (,) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) = ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
133 |
132
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( (,) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) 〉 ) = ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
134 |
130 131 133
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
135 |
|
ioombl |
⊢ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ∈ dom vol |
136 |
135
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) (,) ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ∈ dom vol ) |
137 |
134 136
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ∈ dom vol ) |
138 |
137
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ∈ dom vol ) |
139 |
8 138
|
jca |
⊢ ( 𝜑 → ( ( (,) ∘ 𝐹 ) Fn ℕ ∧ ∀ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ∈ dom vol ) ) |
140 |
|
ffnfv |
⊢ ( ( (,) ∘ 𝐹 ) : ℕ ⟶ dom vol ↔ ( ( (,) ∘ 𝐹 ) Fn ℕ ∧ ∀ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ∈ dom vol ) ) |
141 |
139 140
|
sylibr |
⊢ ( 𝜑 → ( (,) ∘ 𝐹 ) : ℕ ⟶ dom vol ) |
142 |
|
fco |
⊢ ( ( vol : dom vol ⟶ ( 0 [,] +∞ ) ∧ ( (,) ∘ 𝐹 ) : ℕ ⟶ dom vol ) → ( vol ∘ ( (,) ∘ 𝐹 ) ) : ℕ ⟶ ( 0 [,] +∞ ) ) |
143 |
127 141 142
|
syl2anc |
⊢ ( 𝜑 → ( vol ∘ ( (,) ∘ 𝐹 ) ) : ℕ ⟶ ( 0 [,] +∞ ) ) |
144 |
143
|
ffnd |
⊢ ( 𝜑 → ( vol ∘ ( (,) ∘ 𝐹 ) ) Fn ℕ ) |
145 |
68
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑛 ∈ 𝐴 ) → ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) = ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ) |
146 |
137
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑛 ∈ 𝐴 ) → ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ∈ dom vol ) |
147 |
145 146
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑛 ∈ 𝐴 ) → ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ∈ dom vol ) |
148 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ¬ 𝑛 ∈ 𝐴 ) → 𝜑 ) |
149 |
|
eldif |
⊢ ( 𝑛 ∈ ( ℕ ∖ 𝐴 ) ↔ ( 𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐴 ) ) |
150 |
149
|
bicomi |
⊢ ( ( 𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐴 ) ↔ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) |
151 |
150
|
biimpi |
⊢ ( ( 𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐴 ) → 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) |
152 |
151
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ¬ 𝑛 ∈ 𝐴 ) → 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) |
153 |
117 135
|
eqeltrrdi |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ∅ ∈ dom vol ) |
154 |
92 153
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ∈ dom vol ) |
155 |
148 152 154
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ¬ 𝑛 ∈ 𝐴 ) → ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ∈ dom vol ) |
156 |
147 155
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ∈ dom vol ) |
157 |
156
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ∈ dom vol ) |
158 |
31 157
|
jca |
⊢ ( 𝜑 → ( ( (,) ∘ 𝐺 ) Fn ℕ ∧ ∀ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ∈ dom vol ) ) |
159 |
|
ffnfv |
⊢ ( ( (,) ∘ 𝐺 ) : ℕ ⟶ dom vol ↔ ( ( (,) ∘ 𝐺 ) Fn ℕ ∧ ∀ 𝑛 ∈ ℕ ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ∈ dom vol ) ) |
160 |
158 159
|
sylibr |
⊢ ( 𝜑 → ( (,) ∘ 𝐺 ) : ℕ ⟶ dom vol ) |
161 |
|
fco |
⊢ ( ( vol : dom vol ⟶ ( 0 [,] +∞ ) ∧ ( (,) ∘ 𝐺 ) : ℕ ⟶ dom vol ) → ( vol ∘ ( (,) ∘ 𝐺 ) ) : ℕ ⟶ ( 0 [,] +∞ ) ) |
162 |
127 160 161
|
syl2anc |
⊢ ( 𝜑 → ( vol ∘ ( (,) ∘ 𝐺 ) ) : ℕ ⟶ ( 0 [,] +∞ ) ) |
163 |
162
|
ffnd |
⊢ ( 𝜑 → ( vol ∘ ( (,) ∘ 𝐺 ) ) Fn ℕ ) |
164 |
145
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑛 ∈ 𝐴 ) → ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ) |
165 |
119 92
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℕ ∖ 𝐴 ) ) → ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ) |
166 |
148 152 165
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ ¬ 𝑛 ∈ 𝐴 ) → ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ) |
167 |
164 166
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) = ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ) |
168 |
167
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( vol ‘ ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ) = ( vol ‘ ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ) ) |
169 |
|
fnfun |
⊢ ( ( (,) ∘ 𝐹 ) Fn ℕ → Fun ( (,) ∘ 𝐹 ) ) |
170 |
8 169
|
syl |
⊢ ( 𝜑 → Fun ( (,) ∘ 𝐹 ) ) |
171 |
170
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → Fun ( (,) ∘ 𝐹 ) ) |
172 |
7
|
fdmd |
⊢ ( 𝜑 → dom ( (,) ∘ 𝐹 ) = ℕ ) |
173 |
172
|
eqcomd |
⊢ ( 𝜑 → ℕ = dom ( (,) ∘ 𝐹 ) ) |
174 |
173
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ℕ = dom ( (,) ∘ 𝐹 ) ) |
175 |
129 174
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ dom ( (,) ∘ 𝐹 ) ) |
176 |
|
fvco |
⊢ ( ( Fun ( (,) ∘ 𝐹 ) ∧ 𝑛 ∈ dom ( (,) ∘ 𝐹 ) ) → ( ( vol ∘ ( (,) ∘ 𝐹 ) ) ‘ 𝑛 ) = ( vol ‘ ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ) ) |
177 |
171 175 176
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( vol ∘ ( (,) ∘ 𝐹 ) ) ‘ 𝑛 ) = ( vol ‘ ( ( (,) ∘ 𝐹 ) ‘ 𝑛 ) ) ) |
178 |
|
fnfun |
⊢ ( ( (,) ∘ 𝐺 ) Fn ℕ → Fun ( (,) ∘ 𝐺 ) ) |
179 |
31 178
|
syl |
⊢ ( 𝜑 → Fun ( (,) ∘ 𝐺 ) ) |
180 |
179
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → Fun ( (,) ∘ 𝐺 ) ) |
181 |
30
|
fdmd |
⊢ ( 𝜑 → dom ( (,) ∘ 𝐺 ) = ℕ ) |
182 |
181
|
eqcomd |
⊢ ( 𝜑 → ℕ = dom ( (,) ∘ 𝐺 ) ) |
183 |
182
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ℕ = dom ( (,) ∘ 𝐺 ) ) |
184 |
129 183
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ dom ( (,) ∘ 𝐺 ) ) |
185 |
|
fvco |
⊢ ( ( Fun ( (,) ∘ 𝐺 ) ∧ 𝑛 ∈ dom ( (,) ∘ 𝐺 ) ) → ( ( vol ∘ ( (,) ∘ 𝐺 ) ) ‘ 𝑛 ) = ( vol ‘ ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ) ) |
186 |
180 184 185
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( vol ∘ ( (,) ∘ 𝐺 ) ) ‘ 𝑛 ) = ( vol ‘ ( ( (,) ∘ 𝐺 ) ‘ 𝑛 ) ) ) |
187 |
168 177 186
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( vol ∘ ( (,) ∘ 𝐹 ) ) ‘ 𝑛 ) = ( ( vol ∘ ( (,) ∘ 𝐺 ) ) ‘ 𝑛 ) ) |
188 |
144 163 187
|
eqfnfvd |
⊢ ( 𝜑 → ( vol ∘ ( (,) ∘ 𝐹 ) ) = ( vol ∘ ( (,) ∘ 𝐺 ) ) ) |
189 |
125 188
|
jca |
⊢ ( 𝜑 → ( ∪ ran ( (,) ∘ 𝐹 ) = ∪ ran ( (,) ∘ 𝐺 ) ∧ ( vol ∘ ( (,) ∘ 𝐹 ) ) = ( vol ∘ ( (,) ∘ 𝐺 ) ) ) ) |