Metamath Proof Explorer


Theorem ioombl1lem3

Description: Lemma for ioombl1 . (Contributed by Mario Carneiro, 18-Aug-2014)

Ref Expression
Hypotheses ioombl1.b 𝐵 = ( 𝐴 (,) +∞ )
ioombl1.a ( 𝜑𝐴 ∈ ℝ )
ioombl1.e ( 𝜑𝐸 ⊆ ℝ )
ioombl1.v ( 𝜑 → ( vol* ‘ 𝐸 ) ∈ ℝ )
ioombl1.c ( 𝜑𝐶 ∈ ℝ+ )
ioombl1.s 𝑆 = seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐹 ) )
ioombl1.t 𝑇 = seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐺 ) )
ioombl1.u 𝑈 = seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐻 ) )
ioombl1.f1 ( 𝜑𝐹 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) )
ioombl1.f2 ( 𝜑𝐸 ran ( (,) ∘ 𝐹 ) )
ioombl1.f3 ( 𝜑 → sup ( ran 𝑆 , ℝ* , < ) ≤ ( ( vol* ‘ 𝐸 ) + 𝐶 ) )
ioombl1.p 𝑃 = ( 1st ‘ ( 𝐹𝑛 ) )
ioombl1.q 𝑄 = ( 2nd ‘ ( 𝐹𝑛 ) )
ioombl1.g 𝐺 = ( 𝑛 ∈ ℕ ↦ ⟨ if ( if ( 𝑃𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) , 𝑄 ⟩ )
ioombl1.h 𝐻 = ( 𝑛 ∈ ℕ ↦ ⟨ 𝑃 , if ( if ( 𝑃𝐴 , 𝐴 , 𝑃 ) ≤ 𝑄 , if ( 𝑃𝐴 , 𝐴 , 𝑃 ) , 𝑄 ) ⟩ )
Assertion ioombl1lem3 ( ( 𝜑𝑛 ∈ ℕ ) → ( ( ( ( abs ∘ − ) ∘ 𝐺 ) ‘ 𝑛 ) + ( ( ( abs ∘ − ) ∘ 𝐻 ) ‘ 𝑛 ) ) = ( ( ( abs ∘ − ) ∘ 𝐹 ) ‘ 𝑛 ) )

Proof

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