Step |
Hyp |
Ref |
Expression |
1 |
|
ioombl1.b |
⊢ 𝐵 = ( 𝐴 (,) +∞ ) |
2 |
|
ioombl1.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
3 |
|
ioombl1.e |
⊢ ( 𝜑 → 𝐸 ⊆ ℝ ) |
4 |
|
ioombl1.v |
⊢ ( 𝜑 → ( vol* ‘ 𝐸 ) ∈ ℝ ) |
5 |
|
ioombl1.c |
⊢ ( 𝜑 → 𝐶 ∈ ℝ+ ) |
6 |
|
ioombl1.s |
⊢ 𝑆 = seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐹 ) ) |
7 |
|
ioombl1.t |
⊢ 𝑇 = seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐺 ) ) |
8 |
|
ioombl1.u |
⊢ 𝑈 = seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐻 ) ) |
9 |
|
ioombl1.f1 |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
10 |
|
ioombl1.f2 |
⊢ ( 𝜑 → 𝐸 ⊆ ∪ ran ( (,) ∘ 𝐹 ) ) |
11 |
|
ioombl1.f3 |
⊢ ( 𝜑 → sup ( ran 𝑆 , ℝ* , < ) ≤ ( ( vol* ‘ 𝐸 ) + 𝐶 ) ) |
12 |
|
ioombl1.p |
⊢ 𝑃 = ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) |
13 |
|
ioombl1.q |
⊢ 𝑄 = ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) |
14 |
|
ioombl1.g |
⊢ 𝐺 = ( 𝑛 ∈ ℕ ↦ 〈 if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) , 𝑄 〉 ) |
15 |
|
ioombl1.h |
⊢ 𝐻 = ( 𝑛 ∈ ℕ ↦ 〈 𝑃 , if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) 〉 ) |
16 |
|
ovolfcl |
⊢ ( ( 𝐹 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ∧ 𝑛 ∈ ℕ ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
17 |
9 16
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
18 |
17
|
simp2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ) |
19 |
13 18
|
eqeltrid |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑄 ∈ ℝ ) |
20 |
19
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑄 ∈ ℂ ) |
21 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐴 ∈ ℝ ) |
22 |
17
|
simp1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ) |
23 |
12 22
|
eqeltrid |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑃 ∈ ℝ ) |
24 |
21 23
|
ifcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ∈ ℝ ) |
25 |
24 19
|
ifcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ∈ ℝ ) |
26 |
25
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ∈ ℂ ) |
27 |
23
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑃 ∈ ℂ ) |
28 |
20 26 27
|
npncand |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝑄 − if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ) + ( if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) − 𝑃 ) ) = ( 𝑄 − 𝑃 ) ) |
29 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
|
ioombl1lem1 |
⊢ ( 𝜑 → ( 𝐺 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ∧ 𝐻 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) ) |
30 |
29
|
simpld |
⊢ ( 𝜑 → 𝐺 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
31 |
|
eqid |
⊢ ( ( abs ∘ − ) ∘ 𝐺 ) = ( ( abs ∘ − ) ∘ 𝐺 ) |
32 |
31
|
ovolfsval |
⊢ ( ( 𝐺 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝐺 ) ‘ 𝑛 ) = ( ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) |
33 |
30 32
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝐺 ) ‘ 𝑛 ) = ( ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) |
34 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℕ ) |
35 |
|
opex |
⊢ 〈 if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) , 𝑄 〉 ∈ V |
36 |
14
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ ℕ ∧ 〈 if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) , 𝑄 〉 ∈ V ) → ( 𝐺 ‘ 𝑛 ) = 〈 if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) , 𝑄 〉 ) |
37 |
34 35 36
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐺 ‘ 𝑛 ) = 〈 if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) , 𝑄 〉 ) |
38 |
37
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) = ( 2nd ‘ 〈 if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) , 𝑄 〉 ) ) |
39 |
|
op2ndg |
⊢ ( ( if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ∈ ℝ ∧ 𝑄 ∈ ℝ ) → ( 2nd ‘ 〈 if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) , 𝑄 〉 ) = 𝑄 ) |
40 |
25 19 39
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ 〈 if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) , 𝑄 〉 ) = 𝑄 ) |
41 |
38 40
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) = 𝑄 ) |
42 |
37
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) = ( 1st ‘ 〈 if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) , 𝑄 〉 ) ) |
43 |
|
op1stg |
⊢ ( ( if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ∈ ℝ ∧ 𝑄 ∈ ℝ ) → ( 1st ‘ 〈 if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) , 𝑄 〉 ) = if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ) |
44 |
25 19 43
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ 〈 if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) , 𝑄 〉 ) = if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ) |
45 |
42 44
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) = if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ) |
46 |
41 45
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) ) = ( 𝑄 − if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ) ) |
47 |
33 46
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝐺 ) ‘ 𝑛 ) = ( 𝑄 − if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ) ) |
48 |
29
|
simprd |
⊢ ( 𝜑 → 𝐻 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
49 |
|
eqid |
⊢ ( ( abs ∘ − ) ∘ 𝐻 ) = ( ( abs ∘ − ) ∘ 𝐻 ) |
50 |
49
|
ovolfsval |
⊢ ( ( 𝐻 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝐻 ) ‘ 𝑛 ) = ( ( 2nd ‘ ( 𝐻 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐻 ‘ 𝑛 ) ) ) ) |
51 |
48 50
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝐻 ) ‘ 𝑛 ) = ( ( 2nd ‘ ( 𝐻 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐻 ‘ 𝑛 ) ) ) ) |
52 |
|
opex |
⊢ 〈 𝑃 , if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) 〉 ∈ V |
53 |
15
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ ℕ ∧ 〈 𝑃 , if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) 〉 ∈ V ) → ( 𝐻 ‘ 𝑛 ) = 〈 𝑃 , if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) 〉 ) |
54 |
34 52 53
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐻 ‘ 𝑛 ) = 〈 𝑃 , if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) 〉 ) |
55 |
54
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ ( 𝐻 ‘ 𝑛 ) ) = ( 2nd ‘ 〈 𝑃 , if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) 〉 ) ) |
56 |
|
op2ndg |
⊢ ( ( 𝑃 ∈ ℝ ∧ if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ∈ ℝ ) → ( 2nd ‘ 〈 𝑃 , if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) 〉 ) = if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ) |
57 |
23 25 56
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ 〈 𝑃 , if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) 〉 ) = if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ) |
58 |
55 57
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ ( 𝐻 ‘ 𝑛 ) ) = if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ) |
59 |
54
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝐻 ‘ 𝑛 ) ) = ( 1st ‘ 〈 𝑃 , if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) 〉 ) ) |
60 |
|
op1stg |
⊢ ( ( 𝑃 ∈ ℝ ∧ if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ∈ ℝ ) → ( 1st ‘ 〈 𝑃 , if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) 〉 ) = 𝑃 ) |
61 |
23 25 60
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ 〈 𝑃 , if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) 〉 ) = 𝑃 ) |
62 |
59 61
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝐻 ‘ 𝑛 ) ) = 𝑃 ) |
63 |
58 62
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 2nd ‘ ( 𝐻 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐻 ‘ 𝑛 ) ) ) = ( if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) − 𝑃 ) ) |
64 |
51 63
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝐻 ) ‘ 𝑛 ) = ( if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) − 𝑃 ) ) |
65 |
47 64
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( ( abs ∘ − ) ∘ 𝐺 ) ‘ 𝑛 ) + ( ( ( abs ∘ − ) ∘ 𝐻 ) ‘ 𝑛 ) ) = ( ( 𝑄 − if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ) + ( if ( if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃 ≤ 𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) − 𝑃 ) ) ) |
66 |
|
eqid |
⊢ ( ( abs ∘ − ) ∘ 𝐹 ) = ( ( abs ∘ − ) ∘ 𝐹 ) |
67 |
66
|
ovolfsval |
⊢ ( ( 𝐹 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝐹 ) ‘ 𝑛 ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
68 |
9 67
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝐹 ) ‘ 𝑛 ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
69 |
13 12
|
oveq12i |
⊢ ( 𝑄 − 𝑃 ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
70 |
68 69
|
eqtr4di |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝐹 ) ‘ 𝑛 ) = ( 𝑄 − 𝑃 ) ) |
71 |
28 65 70
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( ( abs ∘ − ) ∘ 𝐺 ) ‘ 𝑛 ) + ( ( ( abs ∘ − ) ∘ 𝐻 ) ‘ 𝑛 ) ) = ( ( ( abs ∘ − ) ∘ 𝐹 ) ‘ 𝑛 ) ) |