Step |
Hyp |
Ref |
Expression |
1 |
|
ovolval2lem.1 |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
2 |
|
reex |
⊢ ℝ ∈ V |
3 |
2 2
|
xpex |
⊢ ( ℝ × ℝ ) ∈ V |
4 |
|
inss2 |
⊢ ( ≤ ∩ ( ℝ × ℝ ) ) ⊆ ( ℝ × ℝ ) |
5 |
|
mapss |
⊢ ( ( ( ℝ × ℝ ) ∈ V ∧ ( ≤ ∩ ( ℝ × ℝ ) ) ⊆ ( ℝ × ℝ ) ) → ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ⊆ ( ( ℝ × ℝ ) ↑m ℕ ) ) |
6 |
3 4 5
|
mp2an |
⊢ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ⊆ ( ( ℝ × ℝ ) ↑m ℕ ) |
7 |
3
|
inex2 |
⊢ ( ≤ ∩ ( ℝ × ℝ ) ) ∈ V |
8 |
7
|
a1i |
⊢ ( 𝜑 → ( ≤ ∩ ( ℝ × ℝ ) ) ∈ V ) |
9 |
|
nnex |
⊢ ℕ ∈ V |
10 |
9
|
a1i |
⊢ ( 𝜑 → ℕ ∈ V ) |
11 |
8 10
|
elmapd |
⊢ ( 𝜑 → ( 𝐹 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ↔ 𝐹 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) ) |
12 |
1 11
|
mpbird |
⊢ ( 𝜑 → 𝐹 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ) |
13 |
6 12
|
sseldi |
⊢ ( 𝜑 → 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ) |
14 |
|
1zzd |
⊢ ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → 1 ∈ ℤ ) |
15 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
16 |
|
elmapi |
⊢ ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → 𝐹 : ℕ ⟶ ( ℝ × ℝ ) ) |
17 |
16
|
adantr |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → 𝐹 : ℕ ⟶ ( ℝ × ℝ ) ) |
18 |
|
simpr |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℕ ) |
19 |
17 18
|
fvovco |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) = ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) [,) ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
20 |
19
|
fveq2d |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) = ( vol ‘ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) [,) ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
21 |
16
|
ffvelrnda |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) ∈ ( ℝ × ℝ ) ) |
22 |
|
xp1st |
⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ ( ℝ × ℝ ) → ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
23 |
21 22
|
syl |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
24 |
|
xp2nd |
⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ ( ℝ × ℝ ) → ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
25 |
21 24
|
syl |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
26 |
|
volicore |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) → ( vol ‘ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) [,) ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ∈ ℝ ) |
27 |
23 25 26
|
syl2anc |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( vol ‘ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) [,) ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ∈ ℝ ) |
28 |
20 27
|
eqeltrd |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ∈ ℝ ) |
29 |
28
|
recnd |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ∈ ℂ ) |
30 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ Σ 𝑘 ∈ ( 1 ... 𝑛 ) ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ) = ( 𝑛 ∈ ℕ ↦ Σ 𝑘 ∈ ( 1 ... 𝑛 ) ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ) |
31 |
|
eqid |
⊢ seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ) ) = seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ) ) |
32 |
14 15 29 30 31
|
fsumsermpt |
⊢ ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) → ( 𝑛 ∈ ℕ ↦ Σ 𝑘 ∈ ( 1 ... 𝑛 ) ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ) = seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ) ) ) |
33 |
13 32
|
syl |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ Σ 𝑘 ∈ ( 1 ... 𝑛 ) ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ) = seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ) ) ) |
34 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
35 |
34
|
iftrued |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → if ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) , 0 ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
36 |
13 23
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
37 |
36
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
38 |
13 25
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
39 |
38
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
40 |
|
ressxr |
⊢ ℝ ⊆ ℝ* |
41 |
40 37
|
sseldi |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ* ) |
42 |
40 39
|
sseldi |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ* ) |
43 |
|
xpss |
⊢ ( ℝ × ℝ ) ⊆ ( V × V ) |
44 |
43 21
|
sseldi |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) ∈ ( V × V ) ) |
45 |
|
1st2ndb |
⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ ( V × V ) ↔ ( 𝐹 ‘ 𝑘 ) = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) 〉 ) |
46 |
44 45
|
sylib |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) 〉 ) |
47 |
13 46
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) = 〈 ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) 〉 ) |
48 |
47
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 〈 ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) 〉 = ( 𝐹 ‘ 𝑘 ) ) |
49 |
|
inss1 |
⊢ ( ≤ ∩ ( ℝ × ℝ ) ) ⊆ ≤ |
50 |
49
|
a1i |
⊢ ( 𝜑 → ( ≤ ∩ ( ℝ × ℝ ) ) ⊆ ≤ ) |
51 |
1 50
|
fssd |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ ≤ ) |
52 |
51
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) ∈ ≤ ) |
53 |
48 52
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 〈 ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) 〉 ∈ ≤ ) |
54 |
|
df-br |
⊢ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) 〉 ∈ ≤ ) |
55 |
53 54
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
56 |
55
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
57 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
58 |
39 37
|
lenltd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
59 |
57 58
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
60 |
41 42 56 59
|
xrletrid |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
61 |
|
simp3 |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
62 |
|
simp1 |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
63 |
|
simp2 |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
64 |
62 63
|
eqleltd |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∧ ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
65 |
61 64
|
mpbid |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∧ ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
66 |
65
|
simprd |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
67 |
66
|
iffalsed |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → if ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) , 0 ) = 0 ) |
68 |
63
|
recnd |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ) |
69 |
61
|
eqcomd |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
70 |
68 69
|
subeq0bd |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) = 0 ) |
71 |
67 70
|
eqtr4d |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → if ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) , 0 ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
72 |
37 39 60 71
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) ∧ ¬ ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) → if ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) , 0 ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
73 |
35 72
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → if ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) , 0 ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
74 |
|
volico |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) → ( vol ‘ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) [,) ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) = if ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) , 0 ) ) |
75 |
36 38 74
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( vol ‘ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) [,) ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) = if ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) , 0 ) ) |
76 |
36 38 55
|
abssuble0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( abs ‘ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
77 |
73 75 76
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( vol ‘ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) [,) ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( abs ‘ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
78 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ) |
79 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℕ ) |
80 |
78 79 20
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) = ( vol ‘ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) [,) ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
81 |
46
|
fveq2d |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( ( abs ∘ − ) ‘ ( 𝐹 ‘ 𝑘 ) ) = ( ( abs ∘ − ) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) 〉 ) ) |
82 |
|
df-ov |
⊢ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ( abs ∘ − ) ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) = ( ( abs ∘ − ) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) 〉 ) |
83 |
82
|
eqcomi |
⊢ ( ( abs ∘ − ) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) 〉 ) = ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ( abs ∘ − ) ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
84 |
83
|
a1i |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( ( abs ∘ − ) ‘ 〈 ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) , ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) 〉 ) = ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ( abs ∘ − ) ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
85 |
23
|
recnd |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ) |
86 |
25
|
recnd |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ) |
87 |
|
eqid |
⊢ ( abs ∘ − ) = ( abs ∘ − ) |
88 |
87
|
cnmetdval |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℂ ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ( abs ∘ − ) ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) = ( abs ‘ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
89 |
85 86 88
|
syl2anc |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) ( abs ∘ − ) ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) = ( abs ‘ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
90 |
81 84 89
|
3eqtrd |
⊢ ( ( 𝐹 ∈ ( ( ℝ × ℝ ) ↑m ℕ ) ∧ 𝑘 ∈ ℕ ) → ( ( abs ∘ − ) ‘ ( 𝐹 ‘ 𝑘 ) ) = ( abs ‘ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
91 |
78 79 90
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( abs ∘ − ) ‘ ( 𝐹 ‘ 𝑘 ) ) = ( abs ‘ ( ( 1st ‘ ( 𝐹 ‘ 𝑘 ) ) − ( 2nd ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
92 |
77 80 91
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) = ( ( abs ∘ − ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
93 |
92
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑘 ∈ ℕ ↦ ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ) = ( 𝑘 ∈ ℕ ↦ ( ( abs ∘ − ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
94 |
13 16
|
syl |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ ( ℝ × ℝ ) ) |
95 |
|
rr2sscn2 |
⊢ ( ℝ × ℝ ) ⊆ ( ℂ × ℂ ) |
96 |
95
|
a1i |
⊢ ( 𝜑 → ( ℝ × ℝ ) ⊆ ( ℂ × ℂ ) ) |
97 |
|
absf |
⊢ abs : ℂ ⟶ ℝ |
98 |
|
subf |
⊢ − : ( ℂ × ℂ ) ⟶ ℂ |
99 |
|
fco |
⊢ ( ( abs : ℂ ⟶ ℝ ∧ − : ( ℂ × ℂ ) ⟶ ℂ ) → ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ ) |
100 |
97 98 99
|
mp2an |
⊢ ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ |
101 |
100
|
a1i |
⊢ ( 𝜑 → ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ ) |
102 |
94 96 101
|
fcomptss |
⊢ ( 𝜑 → ( ( abs ∘ − ) ∘ 𝐹 ) = ( 𝑘 ∈ ℕ ↦ ( ( abs ∘ − ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
103 |
93 102
|
eqtr4d |
⊢ ( 𝜑 → ( 𝑘 ∈ ℕ ↦ ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ) = ( ( abs ∘ − ) ∘ 𝐹 ) ) |
104 |
103
|
seqeq3d |
⊢ ( 𝜑 → seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ) ) = seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐹 ) ) ) |
105 |
33 104
|
eqtr2d |
⊢ ( 𝜑 → seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐹 ) ) = ( 𝑛 ∈ ℕ ↦ Σ 𝑘 ∈ ( 1 ... 𝑛 ) ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ) ) |
106 |
105
|
rneqd |
⊢ ( 𝜑 → ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐹 ) ) = ran ( 𝑛 ∈ ℕ ↦ Σ 𝑘 ∈ ( 1 ... 𝑛 ) ( vol ‘ ( ( [,) ∘ 𝐹 ) ‘ 𝑘 ) ) ) ) |