Step | Hyp | Ref
| Expression |
1 | | ovolicc2.7 |
. . . 4
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈) |
2 | | ovolicc.1 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | 2 | rexrd 10762 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
4 | | ovolicc.2 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ ℝ) |
5 | 4 | rexrd 10762 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
6 | | ovolicc.3 |
. . . . 5
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
7 | | lbicc2 12931 |
. . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
8 | 3, 5, 6, 7 | syl3anc 1372 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
9 | 1, 8 | sseldd 3876 |
. . 3
⊢ (𝜑 → 𝐴 ∈ ∪ 𝑈) |
10 | | eluni2 4797 |
. . 3
⊢ (𝐴 ∈ ∪ 𝑈
↔ ∃𝑧 ∈
𝑈 𝐴 ∈ 𝑧) |
11 | 9, 10 | sylib 221 |
. 2
⊢ (𝜑 → ∃𝑧 ∈ 𝑈 𝐴 ∈ 𝑧) |
12 | | ovolicc2.6 |
. . . . . . 7
⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin)) |
13 | 12 | elin2d 4087 |
. . . . . 6
⊢ (𝜑 → 𝑈 ∈ Fin) |
14 | | ovolicc2.10 |
. . . . . . 7
⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} |
15 | 14 | ssrab3 3969 |
. . . . . 6
⊢ 𝑇 ⊆ 𝑈 |
16 | | ssfi 8765 |
. . . . . 6
⊢ ((𝑈 ∈ Fin ∧ 𝑇 ⊆ 𝑈) → 𝑇 ∈ Fin) |
17 | 13, 15, 16 | sylancl 589 |
. . . . 5
⊢ (𝜑 → 𝑇 ∈ Fin) |
18 | 1 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐴[,]𝐵) ⊆ ∪ 𝑈) |
19 | | ovolicc2.8 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐺:𝑈⟶ℕ) |
20 | | ineq1 4094 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑢 = 𝑡 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑡 ∩ (𝐴[,]𝐵))) |
21 | 20 | neeq1d 2993 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑢 = 𝑡 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)) |
22 | 21, 14 | elrab2 3588 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 ∈ 𝑇 ↔ (𝑡 ∈ 𝑈 ∧ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)) |
23 | 22 | simplbi 501 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ 𝑇 → 𝑡 ∈ 𝑈) |
24 | | ffvelrn 6853 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺:𝑈⟶ℕ ∧ 𝑡 ∈ 𝑈) → (𝐺‘𝑡) ∈ ℕ) |
25 | 19, 23, 24 | syl2an 599 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℕ) |
26 | | ovolicc2.5 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
27 | 26 | ffvelrnda 6855 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝐺‘𝑡) ∈ ℕ) → (𝐹‘(𝐺‘𝑡)) ∈ ( ≤ ∩ (ℝ ×
ℝ))) |
28 | 25, 27 | syldan 594 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘(𝐺‘𝑡)) ∈ ( ≤ ∩ (ℝ ×
ℝ))) |
29 | 28 | elin2d 4087 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘(𝐺‘𝑡)) ∈ (ℝ ×
ℝ)) |
30 | | xp2nd 7740 |
. . . . . . . . . . . 12
⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) →
(2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) |
31 | 29, 30 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) |
32 | 4 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐵 ∈ ℝ) |
33 | 31, 32 | ifcld 4457 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ℝ) |
34 | 22 | simprbi 500 |
. . . . . . . . . . . . . 14
⊢ (𝑡 ∈ 𝑇 → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅) |
35 | 34 | adantl 485 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅) |
36 | | n0 4233 |
. . . . . . . . . . . . 13
⊢ ((𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵))) |
37 | 35, 36 | sylib 221 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵))) |
38 | 2 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ∈ ℝ) |
39 | | simprr 773 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵))) |
40 | 39 | elin2d 4087 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝐴[,]𝐵)) |
41 | 4 | adantr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐵 ∈ ℝ) |
42 | | elicc2 12879 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))) |
43 | 2, 41, 42 | syl2an2r 685 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))) |
44 | 40, 43 | mpbid 235 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)) |
45 | 44 | simp1d 1143 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ℝ) |
46 | 29 | adantrr 717 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺‘𝑡)) ∈ (ℝ ×
ℝ)) |
47 | 46, 30 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) |
48 | 44 | simp2d 1144 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ≤ 𝑦) |
49 | 39 | elin1d 4086 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ 𝑡) |
50 | 25 | adantrr 717 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐺‘𝑡) ∈ ℕ) |
51 | | fvco3 6761 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ (𝐺‘𝑡) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = ((,)‘(𝐹‘(𝐺‘𝑡)))) |
52 | 26, 50, 51 | syl2an2r 685 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = ((,)‘(𝐹‘(𝐺‘𝑡)))) |
53 | | ovolicc2.9 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) |
54 | 23, 53 | sylan2 596 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) |
55 | 54 | adantrr 717 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) |
56 | | 1st2nd2 7746 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) →
(𝐹‘(𝐺‘𝑡)) = 〈(1st ‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))〉) |
57 | 46, 56 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺‘𝑡)) = 〈(1st ‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))〉) |
58 | 57 | fveq2d 6672 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺‘𝑡))) = ((,)‘〈(1st
‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))〉)) |
59 | | df-ov 7167 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))) = ((,)‘〈(1st
‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))〉) |
60 | 58, 59 | eqtr4di 2791 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺‘𝑡))) = ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡))))) |
61 | 52, 55, 60 | 3eqtr3d 2781 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑡 = ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡))))) |
62 | 49, 61 | eleqtrd 2835 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡))))) |
63 | | xp1st 7739 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) →
(1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) |
64 | 46, 63 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) |
65 | | rexr 10758 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ → (1st
‘(𝐹‘(𝐺‘𝑡))) ∈
ℝ*) |
66 | | rexr 10758 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ → (2nd
‘(𝐹‘(𝐺‘𝑡))) ∈
ℝ*) |
67 | | elioo2 12855 |
. . . . . . . . . . . . . . . . . . . 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)))))) |
68 | 65, 66, 67 | syl2an 599 |
. . . . . . . . . . . . . . . . . . 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)))))) |
69 | 64, 47, 68 | syl2anc 587 |
. . . . . . . . . . . . . . . . . 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)))))) |
70 | 62, 69 | mpbid 235 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ (1st
‘(𝐹‘(𝐺‘𝑡))) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡))))) |
71 | 70 | simp3d 1145 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡)))) |
72 | 45, 47, 71 | ltled 10859 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡)))) |
73 | 38, 45, 47, 48, 72 | letrd 10868 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡)))) |
74 | 73 | expr 460 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))))) |
75 | 74 | exlimdv 1939 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))))) |
76 | 37, 75 | mpd 15 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡)))) |
77 | 6 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ 𝐵) |
78 | | breq2 5031 |
. . . . . . . . . . . 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))) |
79 | | breq2 5031 |
. . . . . . . . . . . 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))) |
80 | 78, 79 | ifboth 4450 |
. . . . . . . . . . 11
⊢ ((𝐴 ≤ (2nd
‘(𝐹‘(𝐺‘𝑡))) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵)) |
81 | 76, 77, 80 | syl2anc 587 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵)) |
82 | | min2 12659 |
. . . . . . . . . . 11
⊢
(((2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((2nd
‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵) |
83 | 31, 32, 82 | syl2anc 587 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵) |
84 | | elicc2 12879 |
. . . . . . . . . . . 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))) |
85 | 2, 4, 84 | syl2anc 587 |
. . . . . . . . . . 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))) |
86 | 85 | adantr 484 |
. . . . . . . . . 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))) |
87 | 33, 81, 83, 86 | mpbir3and 1343 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵)) |
88 | 18, 87 | sseldd 3876 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ∪ 𝑈) |
89 | | eluni2 4797 |
. . . . . . . 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) |
90 | 88, 89 | sylib 221 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑥 ∈ 𝑈 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) |
91 | | simprl 771 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → 𝑥 ∈ 𝑈) |
92 | | simprr 773 |
. . . . . . . . 9
⊢ (((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) |
93 | 87 | adantr 484 |
. . . . . . . . 9
⊢ (((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)) |
94 | | inelcm 4351 |
. . . . . . . . 9
⊢
((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)) ≠ ∅) |
95 | 92, 93, 94 | syl2anc 587 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅) |
96 | | ineq1 4094 |
. . . . . . . . . 10
⊢ (𝑢 = 𝑥 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑥 ∩ (𝐴[,]𝐵))) |
97 | 96 | neeq1d 2993 |
. . . . . . . . 9
⊢ (𝑢 = 𝑥 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)) |
98 | 97, 14 | elrab2 3588 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝑈 ∧ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)) |
99 | 91, 95, 98 | sylanbrc 586 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → 𝑥 ∈ 𝑇) |
100 | 90, 99, 92 | reximssdv 3185 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) |
101 | 100 | ralrimiva 3096 |
. . . . 5
⊢ (𝜑 → ∀𝑡 ∈ 𝑇 ∃𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) |
102 | | eleq2 2821 |
. . . . . 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))) |
103 | 102 | ac6sfi 8829 |
. . . . 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))) |
104 | 17, 101, 103 | syl2anc 587 |
. . . 4
⊢ (𝜑 → ∃ℎ(ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡))) |
105 | 104 | adantr 484 |
. . 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))) |
106 | | 2fveq3 6673 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑡 → (𝐹‘(𝐺‘𝑥)) = (𝐹‘(𝐺‘𝑡))) |
107 | 106 | fveq2d 6672 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑡 → (2nd ‘(𝐹‘(𝐺‘𝑥))) = (2nd ‘(𝐹‘(𝐺‘𝑡)))) |
108 | 107 | breq1d 5037 |
. . . . . . . . 9
⊢ (𝑥 = 𝑡 → ((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵)) |
109 | 108, 107 | ifbieq1d 4435 |
. . . . . . . 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)) |
110 | | fveq2 6668 |
. . . . . . . 8
⊢ (𝑥 = 𝑡 → (ℎ‘𝑥) = (ℎ‘𝑡)) |
111 | 109, 110 | eleq12d 2827 |
. . . . . . 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))) |
112 | 111 | cbvralvw 3348 |
. . . . . 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)) |
113 | 2 | adantr 484 |
. . . . . . . . 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) |
114 | 4 | adantr 484 |
. . . . . . . . 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) |
115 | 6 | adantr 484 |
. . . . . . . . 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) |
116 | | ovolicc2.4 |
. . . . . . . . 9
⊢ 𝑆 = seq1( + , ((abs ∘
− ) ∘ 𝐹)) |
117 | 26 | adantr 484 |
. . . . . . . . 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))) |
118 | 12 | adantr 484 |
. . . . . . . . 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)) |
119 | 1 | adantr 484 |
. . . . . . . . 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) |
120 | 19 | adantr 484 |
. . . . . . . . 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) |
121 | 53 | adantlr 715 |
. . . . . . . . 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) |
122 | | simprrl 781 |
. . . . . . . . 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) |
123 | | simprrr 782 |
. . . . . . . . . 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)) |
124 | 111 | rspccva 3523 |
. . . . . . . . . 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)) |
125 | 123, 124 | sylan 583 |
. . . . . . . . 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)) |
126 | | simprlr 780 |
. . . . . . . . 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) |
127 | | simprll 779 |
. . . . . . . . . 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) |
128 | 8 | adantr 484 |
. . . . . . . . . . 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)) |
129 | | inelcm 4351 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝑧 ∧ 𝐴 ∈ (𝐴[,]𝐵)) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅) |
130 | 126, 128,
129 | syl2anc 587 |
. . . . . . . . . 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)) ≠ ∅) |
131 | | ineq1 4094 |
. . . . . . . . . . . 12
⊢ (𝑢 = 𝑧 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑧 ∩ (𝐴[,]𝐵))) |
132 | 131 | neeq1d 2993 |
. . . . . . . . . . 11
⊢ (𝑢 = 𝑧 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)) |
133 | 132, 14 | elrab2 3588 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝑇 ↔ (𝑧 ∈ 𝑈 ∧ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)) |
134 | 127, 130,
133 | sylanbrc 586 |
. . . . . . . . 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) |
135 | | eqid 2738 |
. . . . . . . . 9
⊢
seq1((ℎ ∘
1st ), (ℕ × {𝑧})) = seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧})) |
136 | | fveq2 6668 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑛 → (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑚) = (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑛)) |
137 | 136 | eleq2d 2818 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑛 → (𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑚) ↔ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑛))) |
138 | 137 | cbvrabv 3392 |
. . . . . . . . 9
⊢ {𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ),
(ℕ × {𝑧}))‘𝑚)} = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑛)} |
139 | | eqid 2738 |
. . . . . . . . 9
⊢
inf({𝑚 ∈
ℕ ∣ 𝐵 ∈
(seq1((ℎ ∘
1st ), (ℕ × {𝑧}))‘𝑚)}, ℝ, < ) = inf({𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑚)}, ℝ, <
) |
140 | 113, 114,
115, 116, 117, 118, 119, 120, 121, 14, 122, 125, 126, 134, 135, 138, 139 | ovolicc2lem4 24265 |
. . . . . . . 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*, <
)) |
141 | 140 | anassrs 471 |
. . . . . . 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*, <
)) |
142 | 141 | expr 460 |
. . . . . 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*, <
))) |
143 | 112, 142 | syl5bir 246 |
. . . . 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*, <
))) |
144 | 143 | expimpd 457 |
. . . 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*, <
))) |
145 | 144 | exlimdv 1939 |
. . 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*, <
))) |
146 | 105, 145 | mpd 15 |
. 2
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
)) |
147 | 11, 146 | rexlimddv 3200 |
1
⊢ (𝜑 → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
)) |