Step | Hyp | Ref
| Expression |
1 | | inss1 3833 |
. . . . 5
⊢ (𝐸 ∩ 𝐵) ⊆ 𝐸 |
2 | 1 | a1i 11 |
. . . 4
⊢ (𝜑 → (𝐸 ∩ 𝐵) ⊆ 𝐸) |
3 | | ioombl1.e |
. . . 4
⊢ (𝜑 → 𝐸 ⊆ ℝ) |
4 | | ioombl1.v |
. . . 4
⊢ (𝜑 → (vol*‘𝐸) ∈
ℝ) |
5 | | ovolsscl 23254 |
. . . 4
⊢ (((𝐸 ∩ 𝐵) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∩ 𝐵)) ∈
ℝ) |
6 | 2, 3, 4, 5 | syl3anc 1326 |
. . 3
⊢ (𝜑 → (vol*‘(𝐸 ∩ 𝐵)) ∈ ℝ) |
7 | | difss 3737 |
. . . . 5
⊢ (𝐸 ∖ 𝐵) ⊆ 𝐸 |
8 | 7 | a1i 11 |
. . . 4
⊢ (𝜑 → (𝐸 ∖ 𝐵) ⊆ 𝐸) |
9 | | ovolsscl 23254 |
. . . 4
⊢ (((𝐸 ∖ 𝐵) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∖
𝐵)) ∈
ℝ) |
10 | 8, 3, 4, 9 | syl3anc 1326 |
. . 3
⊢ (𝜑 → (vol*‘(𝐸 ∖ 𝐵)) ∈ ℝ) |
11 | 6, 10 | readdcld 10069 |
. 2
⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐵)) + (vol*‘(𝐸 ∖ 𝐵))) ∈ ℝ) |
12 | | ioombl1.b |
. . 3
⊢ 𝐵 = (𝐴(,)+∞) |
13 | | ioombl1.a |
. . 3
⊢ (𝜑 → 𝐴 ∈ ℝ) |
14 | | ioombl1.c |
. . 3
⊢ (𝜑 → 𝐶 ∈
ℝ+) |
15 | | ioombl1.s |
. . 3
⊢ 𝑆 = seq1( + , ((abs ∘
− ) ∘ 𝐹)) |
16 | | ioombl1.t |
. . 3
⊢ 𝑇 = seq1( + , ((abs ∘
− ) ∘ 𝐺)) |
17 | | ioombl1.u |
. . 3
⊢ 𝑈 = seq1( + , ((abs ∘
− ) ∘ 𝐻)) |
18 | | ioombl1.f1 |
. . 3
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
19 | | ioombl1.f2 |
. . 3
⊢ (𝜑 → 𝐸 ⊆ ∪ ran
((,) ∘ 𝐹)) |
20 | | ioombl1.f3 |
. . 3
⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤
((vol*‘𝐸) + 𝐶)) |
21 | | ioombl1.p |
. . 3
⊢ 𝑃 = (1st ‘(𝐹‘𝑛)) |
22 | | ioombl1.q |
. . 3
⊢ 𝑄 = (2nd ‘(𝐹‘𝑛)) |
23 | | ioombl1.g |
. . 3
⊢ 𝐺 = (𝑛 ∈ ℕ ↦ 〈if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) |
24 | | ioombl1.h |
. . 3
⊢ 𝐻 = (𝑛 ∈ ℕ ↦ 〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) |
25 | 12, 13, 3, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 | ioombl1lem2 23327 |
. 2
⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈
ℝ) |
26 | 14 | rpred 11872 |
. . 3
⊢ (𝜑 → 𝐶 ∈ ℝ) |
27 | 4, 26 | readdcld 10069 |
. 2
⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ) |
28 | 12, 13, 3, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 | ioombl1lem1 23326 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ
× ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ
× ℝ)))) |
29 | 28 | simpld 475 |
. . . . . . . 8
⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
30 | | eqid 2622 |
. . . . . . . . 9
⊢ ((abs
∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺) |
31 | 30, 16 | ovolsf 23241 |
. . . . . . . 8
⊢ (𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞)) |
32 | 29, 31 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞)) |
33 | | frn 6053 |
. . . . . . 7
⊢ (𝑇:ℕ⟶(0[,)+∞)
→ ran 𝑇 ⊆
(0[,)+∞)) |
34 | 32, 33 | syl 17 |
. . . . . 6
⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞)) |
35 | | rge0ssre 12280 |
. . . . . 6
⊢
(0[,)+∞) ⊆ ℝ |
36 | 34, 35 | syl6ss 3615 |
. . . . 5
⊢ (𝜑 → ran 𝑇 ⊆ ℝ) |
37 | | 1nn 11031 |
. . . . . . . 8
⊢ 1 ∈
ℕ |
38 | | fdm 6051 |
. . . . . . . . 9
⊢ (𝑇:ℕ⟶(0[,)+∞)
→ dom 𝑇 =
ℕ) |
39 | 32, 38 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → dom 𝑇 = ℕ) |
40 | 37, 39 | syl5eleqr 2708 |
. . . . . . 7
⊢ (𝜑 → 1 ∈ dom 𝑇) |
41 | | ne0i 3921 |
. . . . . . 7
⊢ (1 ∈
dom 𝑇 → dom 𝑇 ≠ ∅) |
42 | 40, 41 | syl 17 |
. . . . . 6
⊢ (𝜑 → dom 𝑇 ≠ ∅) |
43 | | dm0rn0 5342 |
. . . . . . 7
⊢ (dom
𝑇 = ∅ ↔ ran
𝑇 =
∅) |
44 | 43 | necon3bii 2846 |
. . . . . 6
⊢ (dom
𝑇 ≠ ∅ ↔ ran
𝑇 ≠
∅) |
45 | 42, 44 | sylib 208 |
. . . . 5
⊢ (𝜑 → ran 𝑇 ≠ ∅) |
46 | 32 | ffvelrnda 6359 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ∈ (0[,)+∞)) |
47 | 35, 46 | sseldi 3601 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ∈ ℝ) |
48 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ ((abs
∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹) |
49 | 48, 15 | ovolsf 23241 |
. . . . . . . . . . . 12
⊢ (𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞)) |
50 | 18, 49 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞)) |
51 | 50 | ffvelrnda 6359 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆‘𝑗) ∈ (0[,)+∞)) |
52 | 35, 51 | sseldi 3601 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆‘𝑗) ∈ ℝ) |
53 | 25 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → sup(ran 𝑆, ℝ*, < )
∈ ℝ) |
54 | | simpr 477 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ) |
55 | | nnuz 11723 |
. . . . . . . . . . . 12
⊢ ℕ =
(ℤ≥‘1) |
56 | 54, 55 | syl6eleq 2711 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈
(ℤ≥‘1)) |
57 | | simpl 473 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝜑) |
58 | | elfznn 12370 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ (1...𝑗) → 𝑛 ∈ ℕ) |
59 | 30 | ovolfsf 23240 |
. . . . . . . . . . . . . . 15
⊢ (𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞)) |
60 | 29, 59 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((abs ∘ − )
∘ 𝐺):ℕ⟶(0[,)+∞)) |
61 | 60 | ffvelrnda 6359 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ∈ (0[,)+∞)) |
62 | 35, 61 | sseldi 3601 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ∈ ℝ) |
63 | 57, 58, 62 | syl2an 494 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐺)‘𝑛) ∈ ℝ) |
64 | 48 | ovolfsf 23240 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞)) |
65 | 18, 64 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((abs ∘ − )
∘ 𝐹):ℕ⟶(0[,)+∞)) |
66 | 65 | ffvelrnda 6359 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) ∈ (0[,)+∞)) |
67 | | elrege0 12278 |
. . . . . . . . . . . . . 14
⊢ ((((abs
∘ − ) ∘ 𝐹)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs
∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐹)‘𝑛))) |
68 | 66, 67 | sylib 208 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐹)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐹)‘𝑛))) |
69 | 68 | simpld 475 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) ∈ ℝ) |
70 | 57, 58, 69 | syl2an 494 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐹)‘𝑛) ∈ ℝ) |
71 | 28 | simprd 479 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
72 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((abs
∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻) |
73 | 72 | ovolfsf 23240 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐻:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐻):ℕ⟶(0[,)+∞)) |
74 | 71, 73 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((abs ∘ − )
∘ 𝐻):ℕ⟶(0[,)+∞)) |
75 | 74 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ∈ (0[,)+∞)) |
76 | | elrege0 12278 |
. . . . . . . . . . . . . . . 16
⊢ ((((abs
∘ − ) ∘ 𝐻)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs
∘ − ) ∘ 𝐻)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐻)‘𝑛))) |
77 | 75, 76 | sylib 208 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐻)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐻)‘𝑛))) |
78 | 77 | simprd 479 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝐻)‘𝑛)) |
79 | 77 | simpld 475 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ∈ ℝ) |
80 | 62, 79 | addge01d 10615 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 ≤ (((abs
∘ − ) ∘ 𝐻)‘𝑛) ↔ (((abs ∘ − ) ∘
𝐺)‘𝑛) ≤ ((((abs ∘ − ) ∘
𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))) |
81 | 78, 80 | mpbid 222 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ≤ ((((abs ∘ − ) ∘
𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))) |
82 | 12, 13, 3, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 | ioombl1lem3 23328 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛)) |
83 | 81, 82 | breqtrd 4679 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)) |
84 | 57, 58, 83 | syl2an 494 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐺)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)) |
85 | 56, 63, 70, 84 | serle 12856 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝐺))‘𝑗) ≤ (seq1( + , ((abs ∘ − )
∘ 𝐹))‘𝑗)) |
86 | 16 | fveq1i 6192 |
. . . . . . . . . 10
⊢ (𝑇‘𝑗) = (seq1( + , ((abs ∘ − )
∘ 𝐺))‘𝑗) |
87 | 15 | fveq1i 6192 |
. . . . . . . . . 10
⊢ (𝑆‘𝑗) = (seq1( + , ((abs ∘ − )
∘ 𝐹))‘𝑗) |
88 | 85, 86, 87 | 3brtr4g 4687 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ≤ (𝑆‘𝑗)) |
89 | | 1zzd 11408 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 1 ∈
ℤ) |
90 | | eqidd 2623 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) = (((abs ∘ − ) ∘ 𝐹)‘𝑛)) |
91 | 68 | simprd 479 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝐹)‘𝑛)) |
92 | | frn 6053 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑆:ℕ⟶(0[,)+∞)
→ ran 𝑆 ⊆
(0[,)+∞)) |
93 | 50, 92 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞)) |
94 | | icossxr 12258 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(0[,)+∞) ⊆ ℝ* |
95 | 93, 94 | syl6ss 3615 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ran 𝑆 ⊆
ℝ*) |
96 | 95 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran 𝑆 ⊆
ℝ*) |
97 | | ffn 6045 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑆:ℕ⟶(0[,)+∞)
→ 𝑆 Fn
ℕ) |
98 | 50, 97 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑆 Fn ℕ) |
99 | | fnfvelrn 6356 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑆 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ran 𝑆) |
100 | 98, 99 | sylan 488 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ran 𝑆) |
101 | | supxrub 12154 |
. . . . . . . . . . . . . . . . . 18
⊢ ((ran
𝑆 ⊆
ℝ* ∧ (𝑆‘𝑘) ∈ ran 𝑆) → (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, <
)) |
102 | 96, 100, 101 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, <
)) |
103 | 102 | ralrimiva 2966 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, <
)) |
104 | | breq2 4657 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = sup(ran 𝑆, ℝ*, < ) → ((𝑆‘𝑘) ≤ 𝑥 ↔ (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, <
))) |
105 | 104 | ralbidv 2986 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = sup(ran 𝑆, ℝ*, < ) →
(∀𝑘 ∈ ℕ
(𝑆‘𝑘) ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, <
))) |
106 | 105 | rspcev 3309 |
. . . . . . . . . . . . . . . 16
⊢ ((sup(ran
𝑆, ℝ*,
< ) ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, < )) →
∃𝑥 ∈ ℝ
∀𝑘 ∈ ℕ
(𝑆‘𝑘) ≤ 𝑥) |
107 | 25, 103, 106 | syl2anc 693 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥) |
108 | 55, 15, 89, 90, 69, 91, 107 | isumsup2 14578 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑆 ⇝ sup(ran 𝑆, ℝ, < )) |
109 | 93, 35 | syl6ss 3615 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ran 𝑆 ⊆ ℝ) |
110 | | fdm 6051 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑆:ℕ⟶(0[,)+∞)
→ dom 𝑆 =
ℕ) |
111 | 50, 110 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → dom 𝑆 = ℕ) |
112 | 37, 111 | syl5eleqr 2708 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 1 ∈ dom 𝑆) |
113 | | ne0i 3921 |
. . . . . . . . . . . . . . . . 17
⊢ (1 ∈
dom 𝑆 → dom 𝑆 ≠ ∅) |
114 | 112, 113 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → dom 𝑆 ≠ ∅) |
115 | | dm0rn0 5342 |
. . . . . . . . . . . . . . . . 17
⊢ (dom
𝑆 = ∅ ↔ ran
𝑆 =
∅) |
116 | 115 | necon3bii 2846 |
. . . . . . . . . . . . . . . 16
⊢ (dom
𝑆 ≠ ∅ ↔ ran
𝑆 ≠
∅) |
117 | 114, 116 | sylib 208 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ran 𝑆 ≠ ∅) |
118 | | breq1 4656 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = (𝑆‘𝑘) → (𝑧 ≤ 𝑥 ↔ (𝑆‘𝑘) ≤ 𝑥)) |
119 | 118 | ralrn 6362 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑆 Fn ℕ →
(∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥)) |
120 | 98, 119 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥)) |
121 | 120 | rexbidv 3052 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥)) |
122 | 107, 121 | mpbird 247 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥) |
123 | | supxrre 12157 |
. . . . . . . . . . . . . . 15
⊢ ((ran
𝑆 ⊆ ℝ ∧ ran
𝑆 ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥) → sup(ran 𝑆, ℝ*, < ) = sup(ran
𝑆, ℝ, <
)) |
124 | 109, 117,
122, 123 | syl3anc 1326 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran
𝑆, ℝ, <
)) |
125 | 108, 124 | breqtrrd 4681 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑆 ⇝ sup(ran 𝑆, ℝ*, <
)) |
126 | 125 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ⇝ sup(ran 𝑆, ℝ*, <
)) |
127 | 15, 126 | syl5eqbrr 4689 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → seq1( + , ((abs
∘ − ) ∘ 𝐹)) ⇝ sup(ran 𝑆, ℝ*, <
)) |
128 | 69 | adantlr 751 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) ∈ ℝ) |
129 | 91 | adantlr 751 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝐹)‘𝑛)) |
130 | 55, 54, 127, 128, 129 | climserle 14393 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝐹))‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) |
131 | 87, 130 | syl5eqbr 4688 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) |
132 | 47, 52, 53, 88, 131 | letrd 10194 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) |
133 | 132 | ralrimiva 2966 |
. . . . . . 7
⊢ (𝜑 → ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) |
134 | | breq2 4657 |
. . . . . . . . 9
⊢ (𝑥 = sup(ran 𝑆, ℝ*, < ) → ((𝑇‘𝑗) ≤ 𝑥 ↔ (𝑇‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
))) |
135 | 134 | ralbidv 2986 |
. . . . . . . 8
⊢ (𝑥 = sup(ran 𝑆, ℝ*, < ) →
(∀𝑗 ∈ ℕ
(𝑇‘𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
))) |
136 | 135 | rspcev 3309 |
. . . . . . 7
⊢ ((sup(ran
𝑆, ℝ*,
< ) ∈ ℝ ∧ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ sup(ran 𝑆, ℝ*, < )) →
∃𝑥 ∈ ℝ
∀𝑗 ∈ ℕ
(𝑇‘𝑗) ≤ 𝑥) |
137 | 25, 133, 136 | syl2anc 693 |
. . . . . 6
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ 𝑥) |
138 | | ffn 6045 |
. . . . . . . . 9
⊢ (𝑇:ℕ⟶(0[,)+∞)
→ 𝑇 Fn
ℕ) |
139 | 32, 138 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 Fn ℕ) |
140 | | breq1 4656 |
. . . . . . . . 9
⊢ (𝑧 = (𝑇‘𝑗) → (𝑧 ≤ 𝑥 ↔ (𝑇‘𝑗) ≤ 𝑥)) |
141 | 140 | ralrn 6362 |
. . . . . . . 8
⊢ (𝑇 Fn ℕ →
(∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ 𝑥)) |
142 | 139, 141 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ 𝑥)) |
143 | 142 | rexbidv 3052 |
. . . . . 6
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ 𝑥)) |
144 | 137, 143 | mpbird 247 |
. . . . 5
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥) |
145 | | suprcl 10983 |
. . . . 5
⊢ ((ran
𝑇 ⊆ ℝ ∧ ran
𝑇 ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥) → sup(ran 𝑇, ℝ, < ) ∈
ℝ) |
146 | 36, 45, 144, 145 | syl3anc 1326 |
. . . 4
⊢ (𝜑 → sup(ran 𝑇, ℝ, < ) ∈
ℝ) |
147 | 72, 17 | ovolsf 23241 |
. . . . . . . 8
⊢ (𝐻:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞)) |
148 | 71, 147 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑈:ℕ⟶(0[,)+∞)) |
149 | | frn 6053 |
. . . . . . 7
⊢ (𝑈:ℕ⟶(0[,)+∞)
→ ran 𝑈 ⊆
(0[,)+∞)) |
150 | 148, 149 | syl 17 |
. . . . . 6
⊢ (𝜑 → ran 𝑈 ⊆ (0[,)+∞)) |
151 | 150, 35 | syl6ss 3615 |
. . . . 5
⊢ (𝜑 → ran 𝑈 ⊆ ℝ) |
152 | | fdm 6051 |
. . . . . . . . 9
⊢ (𝑈:ℕ⟶(0[,)+∞)
→ dom 𝑈 =
ℕ) |
153 | 148, 152 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → dom 𝑈 = ℕ) |
154 | 37, 153 | syl5eleqr 2708 |
. . . . . . 7
⊢ (𝜑 → 1 ∈ dom 𝑈) |
155 | | ne0i 3921 |
. . . . . . 7
⊢ (1 ∈
dom 𝑈 → dom 𝑈 ≠ ∅) |
156 | 154, 155 | syl 17 |
. . . . . 6
⊢ (𝜑 → dom 𝑈 ≠ ∅) |
157 | | dm0rn0 5342 |
. . . . . . 7
⊢ (dom
𝑈 = ∅ ↔ ran
𝑈 =
∅) |
158 | 157 | necon3bii 2846 |
. . . . . 6
⊢ (dom
𝑈 ≠ ∅ ↔ ran
𝑈 ≠
∅) |
159 | 156, 158 | sylib 208 |
. . . . 5
⊢ (𝜑 → ran 𝑈 ≠ ∅) |
160 | 148 | ffvelrnda 6359 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ∈ (0[,)+∞)) |
161 | 35, 160 | sseldi 3601 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ∈ ℝ) |
162 | 57, 58, 79 | syl2an 494 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐻)‘𝑛) ∈ ℝ) |
163 | | elrege0 12278 |
. . . . . . . . . . . . . . . 16
⊢ ((((abs
∘ − ) ∘ 𝐺)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs
∘ − ) ∘ 𝐺)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐺)‘𝑛))) |
164 | 61, 163 | sylib 208 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐺)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐺)‘𝑛))) |
165 | 164 | simprd 479 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝐺)‘𝑛)) |
166 | 79, 62 | addge02d 10616 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 ≤ (((abs
∘ − ) ∘ 𝐺)‘𝑛) ↔ (((abs ∘ − ) ∘
𝐻)‘𝑛) ≤ ((((abs ∘ − ) ∘
𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))) |
167 | 165, 166 | mpbid 222 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ≤ ((((abs ∘ − ) ∘
𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))) |
168 | 167, 82 | breqtrd 4679 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)) |
169 | 57, 58, 168 | syl2an 494 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐻)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)) |
170 | 56, 162, 70, 169 | serle 12856 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝐻))‘𝑗) ≤ (seq1( + , ((abs ∘ − )
∘ 𝐹))‘𝑗)) |
171 | 17 | fveq1i 6192 |
. . . . . . . . . 10
⊢ (𝑈‘𝑗) = (seq1( + , ((abs ∘ − )
∘ 𝐻))‘𝑗) |
172 | 170, 171,
87 | 3brtr4g 4687 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ≤ (𝑆‘𝑗)) |
173 | 161, 52, 53, 172, 131 | letrd 10194 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) |
174 | 173 | ralrimiva 2966 |
. . . . . . 7
⊢ (𝜑 → ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) |
175 | | breq2 4657 |
. . . . . . . . 9
⊢ (𝑥 = sup(ran 𝑆, ℝ*, < ) → ((𝑈‘𝑗) ≤ 𝑥 ↔ (𝑈‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
))) |
176 | 175 | ralbidv 2986 |
. . . . . . . 8
⊢ (𝑥 = sup(ran 𝑆, ℝ*, < ) →
(∀𝑗 ∈ ℕ
(𝑈‘𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
))) |
177 | 176 | rspcev 3309 |
. . . . . . 7
⊢ ((sup(ran
𝑆, ℝ*,
< ) ∈ ℝ ∧ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ sup(ran 𝑆, ℝ*, < )) →
∃𝑥 ∈ ℝ
∀𝑗 ∈ ℕ
(𝑈‘𝑗) ≤ 𝑥) |
178 | 25, 174, 177 | syl2anc 693 |
. . . . . 6
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ 𝑥) |
179 | | ffn 6045 |
. . . . . . . . 9
⊢ (𝑈:ℕ⟶(0[,)+∞)
→ 𝑈 Fn
ℕ) |
180 | 148, 179 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 Fn ℕ) |
181 | | breq1 4656 |
. . . . . . . . 9
⊢ (𝑧 = (𝑈‘𝑗) → (𝑧 ≤ 𝑥 ↔ (𝑈‘𝑗) ≤ 𝑥)) |
182 | 181 | ralrn 6362 |
. . . . . . . 8
⊢ (𝑈 Fn ℕ →
(∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ 𝑥)) |
183 | 180, 182 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ 𝑥)) |
184 | 183 | rexbidv 3052 |
. . . . . 6
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ 𝑥)) |
185 | 178, 184 | mpbird 247 |
. . . . 5
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥) |
186 | | suprcl 10983 |
. . . . 5
⊢ ((ran
𝑈 ⊆ ℝ ∧ ran
𝑈 ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥) → sup(ran 𝑈, ℝ, < ) ∈
ℝ) |
187 | 151, 159,
185, 186 | syl3anc 1326 |
. . . 4
⊢ (𝜑 → sup(ran 𝑈, ℝ, < ) ∈
ℝ) |
188 | | ssralv 3666 |
. . . . . . . . . 10
⊢ ((𝐸 ∩ 𝐵) ⊆ 𝐸 → (∀𝑥 ∈ 𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))) |
189 | 1, 188 | ax-mp 5 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛)))) |
190 | 21 | breq1i 4660 |
. . . . . . . . . . . . 13
⊢ (𝑃 < 𝑥 ↔ (1st ‘(𝐹‘𝑛)) < 𝑥) |
191 | | ovolfcl 23235 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) ∈ ℝ ∧
(2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st
‘(𝐹‘𝑛)) ≤ (2nd
‘(𝐹‘𝑛)))) |
192 | 18, 191 | sylan 488 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) ∈ ℝ ∧
(2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st
‘(𝐹‘𝑛)) ≤ (2nd
‘(𝐹‘𝑛)))) |
193 | 192 | simp1d 1073 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐹‘𝑛)) ∈
ℝ) |
194 | 21, 193 | syl5eqel 2705 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ) |
195 | 194 | adantlr 751 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ) |
196 | 1, 3 | syl5ss 3614 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐸 ∩ 𝐵) ⊆ ℝ) |
197 | 196 | sselda 3603 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) → 𝑥 ∈ ℝ) |
198 | 197 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ) |
199 | | ltle 10126 |
. . . . . . . . . . . . . . 15
⊢ ((𝑃 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑃 < 𝑥 → 𝑃 ≤ 𝑥)) |
200 | 195, 198,
199 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥 → 𝑃 ≤ 𝑥)) |
201 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) |
202 | | opex 4932 |
. . . . . . . . . . . . . . . . . . . 20
⊢
〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉 ∈ V |
203 | 23 | fvmpt2 6291 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((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〉) |
204 | 201, 202,
203 | sylancl 694 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = 〈if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) |
205 | 204 | fveq2d 6195 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐺‘𝑛)) = (1st
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉)) |
206 | 13 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
207 | 206, 194 | ifcld 4131 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ) |
208 | 192 | simp2d 1074 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐹‘𝑛)) ∈
ℝ) |
209 | 22, 208 | syl5eqel 2705 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ) |
210 | 207, 209 | ifcld 4131 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) |
211 | | op1stg 7180 |
. . . . . . . . . . . . . . . . . . 19
⊢
((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)) |
212 | 210, 209,
211 | syl2anc 693 |
. . . . . . . . . . . . . . . . . 18
⊢ ((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)) |
213 | 205, 212 | eqtrd 2656 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐺‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
214 | 213 | ad2ant2r 783 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → (1st ‘(𝐺‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
215 | 210 | ad2ant2r 783 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) |
216 | 207 | ad2ant2r 783 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ) |
217 | 196 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → (𝐸 ∩ 𝐵) ⊆ ℝ) |
218 | | simplr 792 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑥 ∈ (𝐸 ∩ 𝐵)) |
219 | 217, 218 | sseldd 3604 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑥 ∈ ℝ) |
220 | 209 | ad2ant2r 783 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑄 ∈ ℝ) |
221 | | min1 12020 |
. . . . . . . . . . . . . . . . . 18
⊢
((if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ ∧ 𝑄 ∈ ℝ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃)) |
222 | 216, 220,
221 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃)) |
223 | 13 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝐴 ∈ ℝ) |
224 | | inss2 3834 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐸 ∩ 𝐵) ⊆ 𝐵 |
225 | 224 | sseli 3599 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (𝐸 ∩ 𝐵) → 𝑥 ∈ 𝐵) |
226 | 225 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑥 ∈ 𝐵) |
227 | 13 | rexrd 10089 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
228 | | pnfxr 10092 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ +∞
∈ ℝ* |
229 | | elioo2 12216 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐴 ∈ ℝ*
∧ +∞ ∈ ℝ*) → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞))) |
230 | 227, 228,
229 | sylancl 694 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞))) |
231 | 12 | eleq2i 2693 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝐴(,)+∞)) |
232 | | ltpnf 11954 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) |
233 | 232 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) → 𝑥 < +∞) |
234 | 233 | pm4.71i 664 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ↔ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ∧ 𝑥 < +∞)) |
235 | | df-3an 1039 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞) ↔ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ∧ 𝑥 < +∞)) |
236 | 234, 235 | bitr4i 267 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞)) |
237 | 230, 231,
236 | 3bitr4g 303 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥))) |
238 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) → 𝐴 < 𝑥) |
239 | 237, 238 | syl6bi 243 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝐴 < 𝑥)) |
240 | 239 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → (𝑥 ∈ 𝐵 → 𝐴 < 𝑥)) |
241 | 226, 240 | mpd 15 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝐴 < 𝑥) |
242 | 223, 219,
241 | ltled 10185 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝐴 ≤ 𝑥) |
243 | | simprr 796 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑃 ≤ 𝑥) |
244 | | breq1 4656 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 = if(𝑃 ≤ 𝐴, 𝐴, 𝑃) → (𝐴 ≤ 𝑥 ↔ if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑥)) |
245 | | breq1 4656 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑃 = if(𝑃 ≤ 𝐴, 𝐴, 𝑃) → (𝑃 ≤ 𝑥 ↔ if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑥)) |
246 | 244, 245 | ifboth 4124 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ≤ 𝑥 ∧ 𝑃 ≤ 𝑥) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑥) |
247 | 242, 243,
246 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑥) |
248 | 215, 216,
219, 222, 247 | letrd 10194 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ≤ 𝑥) |
249 | 214, 248 | eqbrtrd 4675 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → (1st ‘(𝐺‘𝑛)) ≤ 𝑥) |
250 | 249 | expr 643 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 ≤ 𝑥 → (1st ‘(𝐺‘𝑛)) ≤ 𝑥)) |
251 | 200, 250 | syld 47 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥 → (1st ‘(𝐺‘𝑛)) ≤ 𝑥)) |
252 | 190, 251 | syl5bir 233 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) < 𝑥 → (1st ‘(𝐺‘𝑛)) ≤ 𝑥)) |
253 | 22 | breq2i 4661 |
. . . . . . . . . . . . . 14
⊢ (𝑥 < 𝑄 ↔ 𝑥 < (2nd ‘(𝐹‘𝑛))) |
254 | 209 | adantlr 751 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ) |
255 | | ltle 10126 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑄 ∈ ℝ) → (𝑥 < 𝑄 → 𝑥 ≤ 𝑄)) |
256 | 198, 254,
255 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄 → 𝑥 ≤ 𝑄)) |
257 | 253, 256 | syl5bir 233 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹‘𝑛)) → 𝑥 ≤ 𝑄)) |
258 | 204 | fveq2d 6195 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐺‘𝑛)) = (2nd
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉)) |
259 | | op2ndg 7181 |
. . . . . . . . . . . . . . . . 17
⊢
((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) |
260 | 210, 209,
259 | syl2anc 693 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) = 𝑄) |
261 | 258, 260 | eqtrd 2656 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐺‘𝑛)) = 𝑄) |
262 | 261 | adantlr 751 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐺‘𝑛)) = 𝑄) |
263 | 262 | breq2d 4665 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 ≤ (2nd ‘(𝐺‘𝑛)) ↔ 𝑥 ≤ 𝑄)) |
264 | 257, 263 | sylibrd 249 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹‘𝑛)) → 𝑥 ≤ (2nd ‘(𝐺‘𝑛)))) |
265 | 252, 264 | anim12d 586 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ((1st ‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐺‘𝑛))))) |
266 | 265 | reximdva 3017 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) → (∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∃𝑛 ∈ ℕ ((1st
‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐺‘𝑛))))) |
267 | 266 | ralimdva 2962 |
. . . . . . . . 9
⊢ (𝜑 → (∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐺‘𝑛))))) |
268 | 189, 267 | syl5 34 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑥 ∈ 𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐺‘𝑛))))) |
269 | | ovolfioo 23236 |
. . . . . . . . 9
⊢ ((𝐸 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ))) → (𝐸 ⊆ ∪ ran
((,) ∘ 𝐹) ↔
∀𝑥 ∈ 𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))) |
270 | 3, 18, 269 | syl2anc 693 |
. . . . . . . 8
⊢ (𝜑 → (𝐸 ⊆ ∪ ran
((,) ∘ 𝐹) ↔
∀𝑥 ∈ 𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))) |
271 | | ovolficc 23237 |
. . . . . . . . 9
⊢ (((𝐸 ∩ 𝐵) ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) → ((𝐸 ∩ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐺) ↔
∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐺‘𝑛))))) |
272 | 196, 29, 271 | syl2anc 693 |
. . . . . . . 8
⊢ (𝜑 → ((𝐸 ∩ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐺) ↔
∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐺‘𝑛))))) |
273 | 268, 270,
272 | 3imtr4d 283 |
. . . . . . 7
⊢ (𝜑 → (𝐸 ⊆ ∪ ran
((,) ∘ 𝐹) →
(𝐸 ∩ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐺))) |
274 | 19, 273 | mpd 15 |
. . . . . 6
⊢ (𝜑 → (𝐸 ∩ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐺)) |
275 | 16 | ovollb2 23257 |
. . . . . 6
⊢ ((𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ (𝐸 ∩ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐺)) →
(vol*‘(𝐸 ∩ 𝐵)) ≤ sup(ran 𝑇, ℝ*, <
)) |
276 | 29, 274, 275 | syl2anc 693 |
. . . . 5
⊢ (𝜑 → (vol*‘(𝐸 ∩ 𝐵)) ≤ sup(ran 𝑇, ℝ*, <
)) |
277 | | supxrre 12157 |
. . . . . 6
⊢ ((ran
𝑇 ⊆ ℝ ∧ ran
𝑇 ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥) → sup(ran 𝑇, ℝ*, < ) = sup(ran
𝑇, ℝ, <
)) |
278 | 36, 45, 144, 277 | syl3anc 1326 |
. . . . 5
⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran
𝑇, ℝ, <
)) |
279 | 276, 278 | breqtrd 4679 |
. . . 4
⊢ (𝜑 → (vol*‘(𝐸 ∩ 𝐵)) ≤ sup(ran 𝑇, ℝ, < )) |
280 | | ssralv 3666 |
. . . . . . . . . 10
⊢ ((𝐸 ∖ 𝐵) ⊆ 𝐸 → (∀𝑥 ∈ 𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))) |
281 | 7, 280 | ax-mp 5 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛)))) |
282 | 194 | adantlr 751 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ) |
283 | 7, 3 | syl5ss 3614 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐸 ∖ 𝐵) ⊆ ℝ) |
284 | 283 | sselda 3603 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) → 𝑥 ∈ ℝ) |
285 | 284 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ) |
286 | 282, 285,
199 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥 → 𝑃 ≤ 𝑥)) |
287 | 190, 286 | syl5bir 233 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) < 𝑥 → 𝑃 ≤ 𝑥)) |
288 | | opex 4932 |
. . . . . . . . . . . . . . . . . 18
⊢
〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉 ∈ V |
289 | 24 | fvmpt2 6291 |
. . . . . . . . . . . . . . . . . 18
⊢ ((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)〉) |
290 | 201, 288,
289 | sylancl 694 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘𝑛) = 〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) |
291 | 290 | fveq2d 6195 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐻‘𝑛)) = (1st
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉)) |
292 | | op1stg 7180 |
. . . . . . . . . . . . . . . . 17
⊢ ((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) |
293 | 194, 210,
292 | syl2anc 693 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) = 𝑃) |
294 | 291, 293 | eqtrd 2656 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐻‘𝑛)) = 𝑃) |
295 | 294 | adantlr 751 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐻‘𝑛)) = 𝑃) |
296 | 295 | breq1d 4663 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐻‘𝑛)) ≤ 𝑥 ↔ 𝑃 ≤ 𝑥)) |
297 | 287, 296 | sylibrd 249 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) < 𝑥 → (1st ‘(𝐻‘𝑛)) ≤ 𝑥)) |
298 | 209 | adantlr 751 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ) |
299 | 285, 298,
255 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄 → 𝑥 ≤ 𝑄)) |
300 | 283 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (𝐸 ∖ 𝐵) ⊆ ℝ) |
301 | | simplr 792 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ∈ (𝐸 ∖ 𝐵)) |
302 | 300, 301 | sseldd 3604 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ∈ ℝ) |
303 | 13 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝐴 ∈ ℝ) |
304 | 194 | ad2ant2r 783 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑃 ∈ ℝ) |
305 | 303, 304 | ifcld 4131 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ) |
306 | | eldifn 3733 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (𝐸 ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵) |
307 | 306 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → ¬ 𝑥 ∈ 𝐵) |
308 | 302 | biantrurd 529 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (𝐴 < 𝑥 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥))) |
309 | 237 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥))) |
310 | 308, 309 | bitr4d 271 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (𝐴 < 𝑥 ↔ 𝑥 ∈ 𝐵)) |
311 | 307, 310 | mtbird 315 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → ¬ 𝐴 < 𝑥) |
312 | 302, 303 | lenltd 10183 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (𝑥 ≤ 𝐴 ↔ ¬ 𝐴 < 𝑥)) |
313 | 311, 312 | mpbird 247 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ 𝐴) |
314 | | max2 12018 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑃 ∈ ℝ ∧ 𝐴 ∈ ℝ) → 𝐴 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃)) |
315 | 304, 303,
314 | syl2anc 693 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝐴 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃)) |
316 | 302, 303,
305, 313, 315 | letrd 10194 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃)) |
317 | | simprr 796 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ 𝑄) |
318 | | breq2 4657 |
. . . . . . . . . . . . . . . . . 18
⊢ (if(𝑃 ≤ 𝐴, 𝐴, 𝑃) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) → (𝑥 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ↔ 𝑥 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))) |
319 | | breq2 4657 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑄 = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) → (𝑥 ≤ 𝑄 ↔ 𝑥 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))) |
320 | 318, 319 | ifboth 4124 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∧ 𝑥 ≤ 𝑄) → 𝑥 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
321 | 316, 317,
320 | syl2anc 693 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
322 | 290 | fveq2d 6195 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐻‘𝑛)) = (2nd
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉)) |
323 | | op2ndg 7181 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((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)) |
324 | 194, 210,
323 | syl2anc 693 |
. . . . . . . . . . . . . . . . . 18
⊢ ((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)) |
325 | 322, 324 | eqtrd 2656 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐻‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
326 | 325 | ad2ant2r 783 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (2nd ‘(𝐻‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
327 | 321, 326 | breqtrrd 4681 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ (2nd ‘(𝐻‘𝑛))) |
328 | 327 | expr 643 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 ≤ 𝑄 → 𝑥 ≤ (2nd ‘(𝐻‘𝑛)))) |
329 | 299, 328 | syld 47 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄 → 𝑥 ≤ (2nd ‘(𝐻‘𝑛)))) |
330 | 253, 329 | syl5bir 233 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹‘𝑛)) → 𝑥 ≤ (2nd ‘(𝐻‘𝑛)))) |
331 | 297, 330 | anim12d 586 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ((1st ‘(𝐻‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐻‘𝑛))))) |
332 | 331 | reximdva 3017 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) → (∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∃𝑛 ∈ ℕ ((1st
‘(𝐻‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐻‘𝑛))))) |
333 | 332 | ralimdva 2962 |
. . . . . . . . 9
⊢ (𝜑 → (∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐻‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐻‘𝑛))))) |
334 | 281, 333 | syl5 34 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑥 ∈ 𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐻‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐻‘𝑛))))) |
335 | | ovolficc 23237 |
. . . . . . . . 9
⊢ (((𝐸 ∖ 𝐵) ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) → ((𝐸 ∖ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐻) ↔
∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐻‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐻‘𝑛))))) |
336 | 283, 71, 335 | syl2anc 693 |
. . . . . . . 8
⊢ (𝜑 → ((𝐸 ∖ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐻) ↔
∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐻‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐻‘𝑛))))) |
337 | 334, 270,
336 | 3imtr4d 283 |
. . . . . . 7
⊢ (𝜑 → (𝐸 ⊆ ∪ ran
((,) ∘ 𝐹) →
(𝐸 ∖ 𝐵) ⊆ ∪ ran ([,] ∘ 𝐻))) |
338 | 19, 337 | mpd 15 |
. . . . . 6
⊢ (𝜑 → (𝐸 ∖ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐻)) |
339 | 17 | ovollb2 23257 |
. . . . . 6
⊢ ((𝐻:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ (𝐸 ∖ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐻)) →
(vol*‘(𝐸 ∖
𝐵)) ≤ sup(ran 𝑈, ℝ*, <
)) |
340 | 71, 338, 339 | syl2anc 693 |
. . . . 5
⊢ (𝜑 → (vol*‘(𝐸 ∖ 𝐵)) ≤ sup(ran 𝑈, ℝ*, <
)) |
341 | | supxrre 12157 |
. . . . . 6
⊢ ((ran
𝑈 ⊆ ℝ ∧ ran
𝑈 ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥) → sup(ran 𝑈, ℝ*, < ) = sup(ran
𝑈, ℝ, <
)) |
342 | 151, 159,
185, 341 | syl3anc 1326 |
. . . . 5
⊢ (𝜑 → sup(ran 𝑈, ℝ*, < ) = sup(ran
𝑈, ℝ, <
)) |
343 | 340, 342 | breqtrd 4679 |
. . . 4
⊢ (𝜑 → (vol*‘(𝐸 ∖ 𝐵)) ≤ sup(ran 𝑈, ℝ, < )) |
344 | 6, 10, 146, 187, 279, 343 | le2addd 10646 |
. . 3
⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐵)) + (vol*‘(𝐸 ∖ 𝐵))) ≤ (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, <
))) |
345 | | eqidd 2623 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) = (((abs ∘ − ) ∘ 𝐺)‘𝑛)) |
346 | 55, 16, 89, 345, 62, 165, 137 | isumsup2 14578 |
. . . . 5
⊢ (𝜑 → 𝑇 ⇝ sup(ran 𝑇, ℝ, < )) |
347 | | seqex 12803 |
. . . . . . 7
⊢ seq1( + ,
((abs ∘ − ) ∘ 𝐹)) ∈ V |
348 | 15, 347 | eqeltri 2697 |
. . . . . 6
⊢ 𝑆 ∈ V |
349 | 348 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝑆 ∈ V) |
350 | | eqidd 2623 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) = (((abs ∘ − ) ∘ 𝐻)‘𝑛)) |
351 | 55, 17, 89, 350, 79, 78, 178 | isumsup2 14578 |
. . . . 5
⊢ (𝜑 → 𝑈 ⇝ sup(ran 𝑈, ℝ, < )) |
352 | 47 | recnd 10068 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ∈ ℂ) |
353 | 161 | recnd 10068 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ∈ ℂ) |
354 | 62 | recnd 10068 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ∈ ℂ) |
355 | 57, 58, 354 | syl2an 494 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐺)‘𝑛) ∈ ℂ) |
356 | 79 | recnd 10068 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ∈ ℂ) |
357 | 57, 58, 356 | syl2an 494 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐻)‘𝑛) ∈ ℂ) |
358 | 82 | eqcomd 2628 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) = ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))) |
359 | 57, 58, 358 | syl2an 494 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐹)‘𝑛) = ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))) |
360 | 56, 355, 357, 359 | seradd 12843 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝐹))‘𝑗) = ((seq1( + , ((abs ∘ − )
∘ 𝐺))‘𝑗) + (seq1( + , ((abs ∘
− ) ∘ 𝐻))‘𝑗))) |
361 | 86, 171 | oveq12i 6662 |
. . . . . 6
⊢ ((𝑇‘𝑗) + (𝑈‘𝑗)) = ((seq1( + , ((abs ∘ − )
∘ 𝐺))‘𝑗) + (seq1( + , ((abs ∘
− ) ∘ 𝐻))‘𝑗)) |
362 | 360, 87, 361 | 3eqtr4g 2681 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆‘𝑗) = ((𝑇‘𝑗) + (𝑈‘𝑗))) |
363 | 55, 89, 346, 349, 351, 352, 353, 362 | climadd 14362 |
. . . 4
⊢ (𝜑 → 𝑆 ⇝ (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, <
))) |
364 | | climuni 14283 |
. . . 4
⊢ ((𝑆 ⇝ (sup(ran 𝑇, ℝ, < ) + sup(ran
𝑈, ℝ, < )) ∧
𝑆 ⇝ sup(ran 𝑆, ℝ*, < ))
→ (sup(ran 𝑇, ℝ,
< ) + sup(ran 𝑈,
ℝ, < )) = sup(ran 𝑆, ℝ*, <
)) |
365 | 363, 125,
364 | syl2anc 693 |
. . 3
⊢ (𝜑 → (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, < )) = sup(ran
𝑆, ℝ*,
< )) |
366 | 344, 365 | breqtrd 4679 |
. 2
⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐵)) + (vol*‘(𝐸 ∖ 𝐵))) ≤ sup(ran 𝑆, ℝ*, <
)) |
367 | 11, 25, 27, 366, 20 | letrd 10194 |
1
⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐵)) + (vol*‘(𝐸 ∖ 𝐵))) ≤ ((vol*‘𝐸) + 𝐶)) |