Step | Hyp | Ref
| Expression |
1 | | ovolicc2.7 |
. . . 4
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈) |
2 | | ovolicc.1 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | 2 | rexrd 9968 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
4 | | ovolicc.2 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ ℝ) |
5 | 4 | rexrd 9968 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
6 | | ovolicc.3 |
. . . . 5
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
7 | | lbicc2 12159 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
8 | 3, 5, 6, 7 | syl3anc 1318 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
9 | 1, 8 | sseldd 3569 |
. . 3
⊢ (𝜑 → 𝐴 ∈ ∪ 𝑈) |
10 | | eluni2 4376 |
. . 3
⊢ (𝐴 ∈ ∪ 𝑈
↔ ∃𝑧 ∈
𝑈 𝐴 ∈ 𝑧) |
11 | 9, 10 | sylib 207 |
. 2
⊢ (𝜑 → ∃𝑧 ∈ 𝑈 𝐴 ∈ 𝑧) |
12 | | ovolicc2.6 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin)) |
13 | | elin 3758 |
. . . . . . . 8
⊢ (𝑈 ∈ (𝒫 ran ((,)
∘ 𝐹) ∩ Fin)
↔ (𝑈 ∈ 𝒫
ran ((,) ∘ 𝐹) ∧
𝑈 ∈
Fin)) |
14 | 12, 13 | sylib 207 |
. . . . . . 7
⊢ (𝜑 → (𝑈 ∈ 𝒫 ran ((,) ∘ 𝐹) ∧ 𝑈 ∈ Fin)) |
15 | 14 | simprd 478 |
. . . . . 6
⊢ (𝜑 → 𝑈 ∈ Fin) |
16 | | ovolicc2.10 |
. . . . . . 7
⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} |
17 | | ssrab2 3650 |
. . . . . . 7
⊢ {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} ⊆ 𝑈 |
18 | 16, 17 | eqsstri 3598 |
. . . . . 6
⊢ 𝑇 ⊆ 𝑈 |
19 | | ssfi 8065 |
. . . . . 6
⊢ ((𝑈 ∈ Fin ∧ 𝑇 ⊆ 𝑈) → 𝑇 ∈ Fin) |
20 | 15, 18, 19 | sylancl 693 |
. . . . 5
⊢ (𝜑 → 𝑇 ∈ Fin) |
21 | 1 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐴[,]𝐵) ⊆ ∪ 𝑈) |
22 | | inss2 3796 |
. . . . . . . . . . . . 13
⊢ ( ≤
∩ (ℝ × ℝ)) ⊆ (ℝ ×
ℝ) |
23 | | ovolicc2.8 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐺:𝑈⟶ℕ) |
24 | | ineq1 3769 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑢 = 𝑡 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑡 ∩ (𝐴[,]𝐵))) |
25 | 24 | neeq1d 2841 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑢 = 𝑡 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)) |
26 | 25, 16 | elrab2 3333 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 ∈ 𝑇 ↔ (𝑡 ∈ 𝑈 ∧ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)) |
27 | 26 | simplbi 475 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ 𝑇 → 𝑡 ∈ 𝑈) |
28 | | ffvelrn 6265 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺:𝑈⟶ℕ ∧ 𝑡 ∈ 𝑈) → (𝐺‘𝑡) ∈ ℕ) |
29 | 23, 27, 28 | syl2an 493 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℕ) |
30 | | ovolicc2.5 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
31 | 30 | ffvelrnda 6267 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝐺‘𝑡) ∈ ℕ) → (𝐹‘(𝐺‘𝑡)) ∈ ( ≤ ∩ (ℝ ×
ℝ))) |
32 | 29, 31 | syldan 486 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘(𝐺‘𝑡)) ∈ ( ≤ ∩ (ℝ ×
ℝ))) |
33 | 22, 32 | sseldi 3566 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘(𝐺‘𝑡)) ∈ (ℝ ×
ℝ)) |
34 | | xp2nd 7090 |
. . . . . . . . . . . 12
⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) →
(2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) |
35 | 33, 34 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) |
36 | 4 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐵 ∈ ℝ) |
37 | 35, 36 | ifcld 4081 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ℝ) |
38 | 26 | simprbi 479 |
. . . . . . . . . . . . . 14
⊢ (𝑡 ∈ 𝑇 → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅) |
39 | 38 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅) |
40 | | n0 3890 |
. . . . . . . . . . . . 13
⊢ ((𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵))) |
41 | 39, 40 | sylib 207 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵))) |
42 | 2 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ∈ ℝ) |
43 | | simprr 792 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵))) |
44 | | elin 3758 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) ↔ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ (𝐴[,]𝐵))) |
45 | 43, 44 | sylib 207 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ (𝐴[,]𝐵))) |
46 | 45 | simprd 478 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝐴[,]𝐵)) |
47 | 4 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐵 ∈ ℝ) |
48 | | elicc2 12109 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))) |
49 | 42, 47, 48 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))) |
50 | 46, 49 | mpbid 221 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)) |
51 | 50 | simp1d 1066 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ℝ) |
52 | 33 | adantrr 749 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺‘𝑡)) ∈ (ℝ ×
ℝ)) |
53 | 52, 34 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) |
54 | 50 | simp2d 1067 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ≤ 𝑦) |
55 | 45 | simpld 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ 𝑡) |
56 | 29 | adantrr 749 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐺‘𝑡) ∈ ℕ) |
57 | | fvco3 6185 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ (𝐺‘𝑡) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = ((,)‘(𝐹‘(𝐺‘𝑡)))) |
58 | 30, 57 | sylan 487 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝐺‘𝑡) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = ((,)‘(𝐹‘(𝐺‘𝑡)))) |
59 | 56, 58 | syldan 486 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = ((,)‘(𝐹‘(𝐺‘𝑡)))) |
60 | | ovolicc2.9 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) |
61 | 27, 60 | sylan2 490 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) |
62 | 61 | adantrr 749 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) |
63 | | 1st2nd2 7096 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) →
(𝐹‘(𝐺‘𝑡)) = 〈(1st ‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))〉) |
64 | 52, 63 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺‘𝑡)) = 〈(1st ‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))〉) |
65 | 64 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺‘𝑡))) = ((,)‘〈(1st
‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))〉)) |
66 | | df-ov 6552 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))) = ((,)‘〈(1st
‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))〉) |
67 | 65, 66 | syl6eqr 2662 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺‘𝑡))) = ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡))))) |
68 | 59, 62, 67 | 3eqtr3d 2652 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑡 = ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡))))) |
69 | 55, 68 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡))))) |
70 | | xp1st 7089 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) →
(1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) |
71 | 52, 70 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) |
72 | | rexr 9964 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ → (1st
‘(𝐹‘(𝐺‘𝑡))) ∈
ℝ*) |
73 | | rexr 9964 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ → (2nd
‘(𝐹‘(𝐺‘𝑡))) ∈
ℝ*) |
74 | | elioo2 12087 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ* ∧
(2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ*) → (𝑦 ∈ ((1st
‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st
‘(𝐹‘(𝐺‘𝑡))) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡)))))) |
75 | 72, 73, 74 | syl2an 493 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ ∧ (2nd
‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) → (𝑦 ∈ ((1st
‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st
‘(𝐹‘(𝐺‘𝑡))) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡)))))) |
76 | 71, 53, 75 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st
‘(𝐹‘(𝐺‘𝑡))) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡)))))) |
77 | 69, 76 | mpbid 221 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ (1st
‘(𝐹‘(𝐺‘𝑡))) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡))))) |
78 | 77 | simp3d 1068 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡)))) |
79 | 51, 53, 78 | ltled 10064 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡)))) |
80 | 42, 51, 53, 54, 79 | letrd 10073 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡)))) |
81 | 80 | expr 641 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))))) |
82 | 81 | exlimdv 1848 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))))) |
83 | 41, 82 | mpd 15 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡)))) |
84 | 6 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ 𝐵) |
85 | | breq2 4587 |
. . . . . . . . . . . 12
⊢
((2nd ‘(𝐹‘(𝐺‘𝑡))) = if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) → (𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))) ↔ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵))) |
86 | | breq2 4587 |
. . . . . . . . . . . 12
⊢ (𝐵 = if((2nd
‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵))) |
87 | 85, 86 | ifboth 4074 |
. . . . . . . . . . 11
⊢ ((𝐴 ≤ (2nd
‘(𝐹‘(𝐺‘𝑡))) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵)) |
88 | 83, 84, 87 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵)) |
89 | | min2 11895 |
. . . . . . . . . . 11
⊢
(((2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((2nd
‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵) |
90 | 35, 36, 89 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵) |
91 | | elicc2 12109 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) →
(if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵))) |
92 | 2, 4, 91 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (𝜑 → (if((2nd
‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵))) |
93 | 92 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵))) |
94 | 37, 88, 90, 93 | mpbir3and 1238 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵)) |
95 | 21, 94 | sseldd 3569 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ∪ 𝑈) |
96 | | eluni2 4376 |
. . . . . . . 8
⊢
(if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ∪ 𝑈 ↔ ∃𝑥 ∈ 𝑈 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) |
97 | 95, 96 | sylib 207 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑥 ∈ 𝑈 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) |
98 | | simprl 790 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → 𝑥 ∈ 𝑈) |
99 | | simprr 792 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) |
100 | 94 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵)) |
101 | | inelcm 3984 |
. . . . . . . . . . . 12
⊢
((if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵)) → (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅) |
102 | 99, 100, 101 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅) |
103 | | ineq1 3769 |
. . . . . . . . . . . . 13
⊢ (𝑢 = 𝑥 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑥 ∩ (𝐴[,]𝐵))) |
104 | 103 | neeq1d 2841 |
. . . . . . . . . . . 12
⊢ (𝑢 = 𝑥 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)) |
105 | 104, 16 | elrab2 3333 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝑈 ∧ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)) |
106 | 98, 102, 105 | sylanbrc 695 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → 𝑥 ∈ 𝑇) |
107 | 106, 99 | jca 553 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → (𝑥 ∈ 𝑇 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) |
108 | 107 | ex 449 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) → (𝑥 ∈ 𝑇 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥))) |
109 | 108 | reximdv2 2997 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (∃𝑥 ∈ 𝑈 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥 → ∃𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) |
110 | 97, 109 | mpd 15 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) |
111 | 110 | ralrimiva 2949 |
. . . . 5
⊢ (𝜑 → ∀𝑡 ∈ 𝑇 ∃𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) |
112 | | eleq2 2677 |
. . . . . 6
⊢ (𝑥 = (ℎ‘𝑡) → (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥 ↔ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡))) |
113 | 112 | ac6sfi 8089 |
. . . . 5
⊢ ((𝑇 ∈ Fin ∧ ∀𝑡 ∈ 𝑇 ∃𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) → ∃ℎ(ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡))) |
114 | 20, 111, 113 | syl2anc 691 |
. . . 4
⊢ (𝜑 → ∃ℎ(ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡))) |
115 | 114 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) → ∃ℎ(ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡))) |
116 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑡 → (𝐺‘𝑥) = (𝐺‘𝑡)) |
117 | 116 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑡 → (𝐹‘(𝐺‘𝑥)) = (𝐹‘(𝐺‘𝑡))) |
118 | 117 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑡 → (2nd ‘(𝐹‘(𝐺‘𝑥))) = (2nd ‘(𝐹‘(𝐺‘𝑡)))) |
119 | 118 | breq1d 4593 |
. . . . . . . . 9
⊢ (𝑥 = 𝑡 → ((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵)) |
120 | 119, 118 | ifbieq1d 4059 |
. . . . . . . 8
⊢ (𝑥 = 𝑡 → if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) = if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵)) |
121 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑥 = 𝑡 → (ℎ‘𝑥) = (ℎ‘𝑡)) |
122 | 120, 121 | eleq12d 2682 |
. . . . . . 7
⊢ (𝑥 = 𝑡 → (if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥) ↔ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡))) |
123 | 122 | cbvralv 3147 |
. . . . . 6
⊢
(∀𝑥 ∈
𝑇 if((2nd
‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥) ↔ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)) |
124 | 2 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐴 ∈ ℝ) |
125 | 4 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐵 ∈ ℝ) |
126 | 6 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐴 ≤ 𝐵) |
127 | | ovolicc2.4 |
. . . . . . . . 9
⊢ 𝑆 = seq1( + , ((abs ∘
− ) ∘ 𝐹)) |
128 | 30 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐹:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
129 | 12 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin)) |
130 | 1 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → (𝐴[,]𝐵) ⊆ ∪ 𝑈) |
131 | 23 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐺:𝑈⟶ℕ) |
132 | 60 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) |
133 | | simprrl 800 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → ℎ:𝑇⟶𝑇) |
134 | | simprrr 801 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)) |
135 | 122 | rspccva 3281 |
. . . . . . . . . 10
⊢
((∀𝑥 ∈
𝑇 if((2nd
‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥) ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)) |
136 | 134, 135 | sylan 487 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)) |
137 | | simprlr 799 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐴 ∈ 𝑧) |
138 | | simprll 798 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝑧 ∈ 𝑈) |
139 | 8 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐴 ∈ (𝐴[,]𝐵)) |
140 | | inelcm 3984 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝑧 ∧ 𝐴 ∈ (𝐴[,]𝐵)) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅) |
141 | 137, 139,
140 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅) |
142 | | ineq1 3769 |
. . . . . . . . . . . 12
⊢ (𝑢 = 𝑧 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑧 ∩ (𝐴[,]𝐵))) |
143 | 142 | neeq1d 2841 |
. . . . . . . . . . 11
⊢ (𝑢 = 𝑧 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)) |
144 | 143, 16 | elrab2 3333 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝑇 ↔ (𝑧 ∈ 𝑈 ∧ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)) |
145 | 138, 141,
144 | sylanbrc 695 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝑧 ∈ 𝑇) |
146 | | eqid 2610 |
. . . . . . . . 9
⊢
seq1((ℎ ∘
1st ), (ℕ × {𝑧})) = seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧})) |
147 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑛 → (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑚) = (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑛)) |
148 | 147 | eleq2d 2673 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑛 → (𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑚) ↔ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑛))) |
149 | 148 | cbvrabv 3172 |
. . . . . . . . 9
⊢ {𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ),
(ℕ × {𝑧}))‘𝑚)} = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑛)} |
150 | | eqid 2610 |
. . . . . . . . 9
⊢
inf({𝑚 ∈
ℕ ∣ 𝐵 ∈
(seq1((ℎ ∘
1st ), (ℕ × {𝑧}))‘𝑚)}, ℝ, < ) = inf({𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑚)}, ℝ, <
) |
151 | 124, 125,
126, 127, 128, 129, 130, 131, 132, 16, 133, 136, 137, 145, 146, 149, 150 | ovolicc2lem4 23095 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
)) |
152 | 151 | anassrs 678 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥))) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
)) |
153 | 152 | expr 641 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) ∧ ℎ:𝑇⟶𝑇) → (∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
))) |
154 | 123, 153 | syl5bir 232 |
. . . . 5
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) ∧ ℎ:𝑇⟶𝑇) → (∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
))) |
155 | 154 | expimpd 627 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) → ((ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
))) |
156 | 155 | exlimdv 1848 |
. . 3
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) → (∃ℎ(ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
))) |
157 | 115, 156 | mpd 15 |
. 2
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
)) |
158 | 11, 157 | rexlimddv 3017 |
1
⊢ (𝜑 → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
)) |