| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | inss1 4237 | . . . 4
⊢ (𝐸 ∩ 𝐵) ⊆ 𝐸 | 
| 2 |  | ioombl1.e | . . . 4
⊢ (𝜑 → 𝐸 ⊆ ℝ) | 
| 3 |  | ioombl1.v | . . . 4
⊢ (𝜑 → (vol*‘𝐸) ∈
ℝ) | 
| 4 |  | ovolsscl 25521 | . . . 4
⊢ (((𝐸 ∩ 𝐵) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∩ 𝐵)) ∈
ℝ) | 
| 5 | 1, 2, 3, 4 | mp3an2i 1468 | . . 3
⊢ (𝜑 → (vol*‘(𝐸 ∩ 𝐵)) ∈ ℝ) | 
| 6 |  | difss 4136 | . . . 4
⊢ (𝐸 ∖ 𝐵) ⊆ 𝐸 | 
| 7 |  | ovolsscl 25521 | . . . 4
⊢ (((𝐸 ∖ 𝐵) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∖
𝐵)) ∈
ℝ) | 
| 8 | 6, 2, 3, 7 | mp3an2i 1468 | . . 3
⊢ (𝜑 → (vol*‘(𝐸 ∖ 𝐵)) ∈ ℝ) | 
| 9 | 5, 8 | readdcld 11290 | . 2
⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐵)) + (vol*‘(𝐸 ∖ 𝐵))) ∈ ℝ) | 
| 10 |  | ioombl1.b | . . 3
⊢ 𝐵 = (𝐴(,)+∞) | 
| 11 |  | ioombl1.a | . . 3
⊢ (𝜑 → 𝐴 ∈ ℝ) | 
| 12 |  | ioombl1.c | . . 3
⊢ (𝜑 → 𝐶 ∈
ℝ+) | 
| 13 |  | ioombl1.s | . . 3
⊢ 𝑆 = seq1( + , ((abs ∘
− ) ∘ 𝐹)) | 
| 14 |  | ioombl1.t | . . 3
⊢ 𝑇 = seq1( + , ((abs ∘
− ) ∘ 𝐺)) | 
| 15 |  | ioombl1.u | . . 3
⊢ 𝑈 = seq1( + , ((abs ∘
− ) ∘ 𝐻)) | 
| 16 |  | ioombl1.f1 | . . 3
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) | 
| 17 |  | ioombl1.f2 | . . 3
⊢ (𝜑 → 𝐸 ⊆ ∪ ran
((,) ∘ 𝐹)) | 
| 18 |  | ioombl1.f3 | . . 3
⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤
((vol*‘𝐸) + 𝐶)) | 
| 19 |  | ioombl1.p | . . 3
⊢ 𝑃 = (1st ‘(𝐹‘𝑛)) | 
| 20 |  | ioombl1.q | . . 3
⊢ 𝑄 = (2nd ‘(𝐹‘𝑛)) | 
| 21 |  | ioombl1.g | . . 3
⊢ 𝐺 = (𝑛 ∈ ℕ ↦ 〈if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) | 
| 22 |  | ioombl1.h | . . 3
⊢ 𝐻 = (𝑛 ∈ ℕ ↦ 〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) | 
| 23 | 10, 11, 2, 3, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 | ioombl1lem2 25594 | . 2
⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈
ℝ) | 
| 24 | 12 | rpred 13077 | . . 3
⊢ (𝜑 → 𝐶 ∈ ℝ) | 
| 25 | 3, 24 | readdcld 11290 | . 2
⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ) | 
| 26 | 10, 11, 2, 3, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 | ioombl1lem1 25593 | . . . . . . . . 9
⊢ (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ
× ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ
× ℝ)))) | 
| 27 | 26 | simpld 494 | . . . . . . . 8
⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) | 
| 28 |  | eqid 2737 | . . . . . . . . 9
⊢ ((abs
∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺) | 
| 29 | 28, 14 | ovolsf 25507 | . . . . . . . 8
⊢ (𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞)) | 
| 30 | 27, 29 | syl 17 | . . . . . . 7
⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞)) | 
| 31 | 30 | frnd 6744 | . . . . . 6
⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞)) | 
| 32 |  | rge0ssre 13496 | . . . . . 6
⊢
(0[,)+∞) ⊆ ℝ | 
| 33 | 31, 32 | sstrdi 3996 | . . . . 5
⊢ (𝜑 → ran 𝑇 ⊆ ℝ) | 
| 34 |  | 1nn 12277 | . . . . . . . 8
⊢ 1 ∈
ℕ | 
| 35 | 30 | fdmd 6746 | . . . . . . . 8
⊢ (𝜑 → dom 𝑇 = ℕ) | 
| 36 | 34, 35 | eleqtrrid 2848 | . . . . . . 7
⊢ (𝜑 → 1 ∈ dom 𝑇) | 
| 37 | 36 | ne0d 4342 | . . . . . 6
⊢ (𝜑 → dom 𝑇 ≠ ∅) | 
| 38 |  | dm0rn0 5935 | . . . . . . 7
⊢ (dom
𝑇 = ∅ ↔ ran
𝑇 =
∅) | 
| 39 | 38 | necon3bii 2993 | . . . . . 6
⊢ (dom
𝑇 ≠ ∅ ↔ ran
𝑇 ≠
∅) | 
| 40 | 37, 39 | sylib 218 | . . . . 5
⊢ (𝜑 → ran 𝑇 ≠ ∅) | 
| 41 | 30 | ffvelcdmda 7104 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ∈ (0[,)+∞)) | 
| 42 | 32, 41 | sselid 3981 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ∈ ℝ) | 
| 43 |  | eqid 2737 | . . . . . . . . . . . . 13
⊢ ((abs
∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹) | 
| 44 | 43, 13 | ovolsf 25507 | . . . . . . . . . . . 12
⊢ (𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞)) | 
| 45 | 16, 44 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞)) | 
| 46 | 45 | ffvelcdmda 7104 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆‘𝑗) ∈ (0[,)+∞)) | 
| 47 | 32, 46 | sselid 3981 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆‘𝑗) ∈ ℝ) | 
| 48 | 23 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → sup(ran 𝑆, ℝ*, < )
∈ ℝ) | 
| 49 |  | simpr 484 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ) | 
| 50 |  | nnuz 12921 | . . . . . . . . . . . 12
⊢ ℕ =
(ℤ≥‘1) | 
| 51 | 49, 50 | eleqtrdi 2851 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈
(ℤ≥‘1)) | 
| 52 |  | simpl 482 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝜑) | 
| 53 |  | elfznn 13593 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ (1...𝑗) → 𝑛 ∈ ℕ) | 
| 54 | 28 | ovolfsf 25506 | . . . . . . . . . . . . . . 15
⊢ (𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞)) | 
| 55 | 27, 54 | syl 17 | . . . . . . . . . . . . . 14
⊢ (𝜑 → ((abs ∘ − )
∘ 𝐺):ℕ⟶(0[,)+∞)) | 
| 56 | 55 | ffvelcdmda 7104 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ∈ (0[,)+∞)) | 
| 57 | 32, 56 | sselid 3981 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ∈ ℝ) | 
| 58 | 52, 53, 57 | syl2an 596 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐺)‘𝑛) ∈ ℝ) | 
| 59 | 43 | ovolfsf 25506 | . . . . . . . . . . . . . . . 16
⊢ (𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞)) | 
| 60 | 16, 59 | syl 17 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → ((abs ∘ − )
∘ 𝐹):ℕ⟶(0[,)+∞)) | 
| 61 | 60 | ffvelcdmda 7104 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) ∈ (0[,)+∞)) | 
| 62 |  | elrege0 13494 | . . . . . . . . . . . . . 14
⊢ ((((abs
∘ − ) ∘ 𝐹)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs
∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐹)‘𝑛))) | 
| 63 | 61, 62 | sylib 218 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐹)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐹)‘𝑛))) | 
| 64 | 63 | simpld 494 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) ∈ ℝ) | 
| 65 | 52, 53, 64 | syl2an 596 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐹)‘𝑛) ∈ ℝ) | 
| 66 | 26 | simprd 495 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) | 
| 67 |  | eqid 2737 | . . . . . . . . . . . . . . . . . . 19
⊢ ((abs
∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻) | 
| 68 | 67 | ovolfsf 25506 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐻:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐻):ℕ⟶(0[,)+∞)) | 
| 69 | 66, 68 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((abs ∘ − )
∘ 𝐻):ℕ⟶(0[,)+∞)) | 
| 70 | 69 | ffvelcdmda 7104 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ∈ (0[,)+∞)) | 
| 71 |  | elrege0 13494 | . . . . . . . . . . . . . . . 16
⊢ ((((abs
∘ − ) ∘ 𝐻)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs
∘ − ) ∘ 𝐻)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐻)‘𝑛))) | 
| 72 | 70, 71 | sylib 218 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐻)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐻)‘𝑛))) | 
| 73 | 72 | simprd 495 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝐻)‘𝑛)) | 
| 74 | 72 | simpld 494 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ∈ ℝ) | 
| 75 | 57, 74 | addge01d 11851 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 ≤ (((abs
∘ − ) ∘ 𝐻)‘𝑛) ↔ (((abs ∘ − ) ∘
𝐺)‘𝑛) ≤ ((((abs ∘ − ) ∘
𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))) | 
| 76 | 73, 75 | mpbid 232 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ≤ ((((abs ∘ − ) ∘
𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))) | 
| 77 | 10, 11, 2, 3, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 | ioombl1lem3 25595 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛)) | 
| 78 | 76, 77 | breqtrd 5169 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)) | 
| 79 | 52, 53, 78 | syl2an 596 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐺)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)) | 
| 80 | 51, 58, 65, 79 | serle 14098 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝐺))‘𝑗) ≤ (seq1( + , ((abs ∘ − )
∘ 𝐹))‘𝑗)) | 
| 81 | 14 | fveq1i 6907 | . . . . . . . . . 10
⊢ (𝑇‘𝑗) = (seq1( + , ((abs ∘ − )
∘ 𝐺))‘𝑗) | 
| 82 | 13 | fveq1i 6907 | . . . . . . . . . 10
⊢ (𝑆‘𝑗) = (seq1( + , ((abs ∘ − )
∘ 𝐹))‘𝑗) | 
| 83 | 80, 81, 82 | 3brtr4g 5177 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ≤ (𝑆‘𝑗)) | 
| 84 |  | 1zzd 12648 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 1 ∈
ℤ) | 
| 85 |  | eqidd 2738 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) = (((abs ∘ − ) ∘ 𝐹)‘𝑛)) | 
| 86 | 63 | simprd 495 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝐹)‘𝑛)) | 
| 87 | 45 | frnd 6744 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞)) | 
| 88 |  | icossxr 13472 | . . . . . . . . . . . . . . . . . . . 20
⊢
(0[,)+∞) ⊆ ℝ* | 
| 89 | 87, 88 | sstrdi 3996 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ran 𝑆 ⊆
ℝ*) | 
| 90 | 89 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran 𝑆 ⊆
ℝ*) | 
| 91 | 45 | ffnd 6737 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑆 Fn ℕ) | 
| 92 |  | fnfvelrn 7100 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑆 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ran 𝑆) | 
| 93 | 91, 92 | sylan 580 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ran 𝑆) | 
| 94 |  | supxrub 13366 | . . . . . . . . . . . . . . . . . 18
⊢ ((ran
𝑆 ⊆
ℝ* ∧ (𝑆‘𝑘) ∈ ran 𝑆) → (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, <
)) | 
| 95 | 90, 93, 94 | syl2anc 584 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, <
)) | 
| 96 | 95 | ralrimiva 3146 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, <
)) | 
| 97 |  | brralrspcev 5203 | . . . . . . . . . . . . . . . 16
⊢ ((sup(ran
𝑆, ℝ*,
< ) ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, < )) →
∃𝑥 ∈ ℝ
∀𝑘 ∈ ℕ
(𝑆‘𝑘) ≤ 𝑥) | 
| 98 | 23, 96, 97 | syl2anc 584 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥) | 
| 99 | 50, 13, 84, 85, 64, 86, 98 | isumsup2 15882 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑆 ⇝ sup(ran 𝑆, ℝ, < )) | 
| 100 | 87, 32 | sstrdi 3996 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → ran 𝑆 ⊆ ℝ) | 
| 101 | 45 | fdmd 6746 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → dom 𝑆 = ℕ) | 
| 102 | 34, 101 | eleqtrrid 2848 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 1 ∈ dom 𝑆) | 
| 103 | 102 | ne0d 4342 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → dom 𝑆 ≠ ∅) | 
| 104 |  | dm0rn0 5935 | . . . . . . . . . . . . . . . . 17
⊢ (dom
𝑆 = ∅ ↔ ran
𝑆 =
∅) | 
| 105 | 104 | necon3bii 2993 | . . . . . . . . . . . . . . . 16
⊢ (dom
𝑆 ≠ ∅ ↔ ran
𝑆 ≠
∅) | 
| 106 | 103, 105 | sylib 218 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → ran 𝑆 ≠ ∅) | 
| 107 |  | breq1 5146 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = (𝑆‘𝑘) → (𝑧 ≤ 𝑥 ↔ (𝑆‘𝑘) ≤ 𝑥)) | 
| 108 | 107 | ralrn 7108 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑆 Fn ℕ →
(∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥)) | 
| 109 | 91, 108 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥)) | 
| 110 | 109 | rexbidv 3179 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥)) | 
| 111 | 98, 110 | mpbird 257 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥) | 
| 112 |  | supxrre 13369 | . . . . . . . . . . . . . . 15
⊢ ((ran
𝑆 ⊆ ℝ ∧ ran
𝑆 ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥) → sup(ran 𝑆, ℝ*, < ) = sup(ran
𝑆, ℝ, <
)) | 
| 113 | 100, 106,
111, 112 | syl3anc 1373 | . . . . . . . . . . . . . 14
⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran
𝑆, ℝ, <
)) | 
| 114 | 99, 113 | breqtrrd 5171 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝑆 ⇝ sup(ran 𝑆, ℝ*, <
)) | 
| 115 | 114 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ⇝ sup(ran 𝑆, ℝ*, <
)) | 
| 116 | 13, 115 | eqbrtrrid 5179 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → seq1( + , ((abs
∘ − ) ∘ 𝐹)) ⇝ sup(ran 𝑆, ℝ*, <
)) | 
| 117 | 64 | adantlr 715 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) ∈ ℝ) | 
| 118 | 86 | adantlr 715 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝐹)‘𝑛)) | 
| 119 | 50, 49, 116, 117, 118 | climserle 15699 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝐹))‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) | 
| 120 | 82, 119 | eqbrtrid 5178 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) | 
| 121 | 42, 47, 48, 83, 120 | letrd 11418 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) | 
| 122 | 121 | ralrimiva 3146 | . . . . . . 7
⊢ (𝜑 → ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) | 
| 123 |  | brralrspcev 5203 | . . . . . . 7
⊢ ((sup(ran
𝑆, ℝ*,
< ) ∈ ℝ ∧ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ sup(ran 𝑆, ℝ*, < )) →
∃𝑥 ∈ ℝ
∀𝑗 ∈ ℕ
(𝑇‘𝑗) ≤ 𝑥) | 
| 124 | 23, 122, 123 | syl2anc 584 | . . . . . 6
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ 𝑥) | 
| 125 | 30 | ffnd 6737 | . . . . . . . 8
⊢ (𝜑 → 𝑇 Fn ℕ) | 
| 126 |  | breq1 5146 | . . . . . . . . 9
⊢ (𝑧 = (𝑇‘𝑗) → (𝑧 ≤ 𝑥 ↔ (𝑇‘𝑗) ≤ 𝑥)) | 
| 127 | 126 | ralrn 7108 | . . . . . . . 8
⊢ (𝑇 Fn ℕ →
(∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ 𝑥)) | 
| 128 | 125, 127 | syl 17 | . . . . . . 7
⊢ (𝜑 → (∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ 𝑥)) | 
| 129 | 128 | rexbidv 3179 | . . . . . 6
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ 𝑥)) | 
| 130 | 124, 129 | mpbird 257 | . . . . 5
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥) | 
| 131 | 33, 40, 130 | suprcld 12231 | . . . 4
⊢ (𝜑 → sup(ran 𝑇, ℝ, < ) ∈
ℝ) | 
| 132 | 67, 15 | ovolsf 25507 | . . . . . . . 8
⊢ (𝐻:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞)) | 
| 133 | 66, 132 | syl 17 | . . . . . . 7
⊢ (𝜑 → 𝑈:ℕ⟶(0[,)+∞)) | 
| 134 | 133 | frnd 6744 | . . . . . 6
⊢ (𝜑 → ran 𝑈 ⊆ (0[,)+∞)) | 
| 135 | 134, 32 | sstrdi 3996 | . . . . 5
⊢ (𝜑 → ran 𝑈 ⊆ ℝ) | 
| 136 | 133 | fdmd 6746 | . . . . . . . 8
⊢ (𝜑 → dom 𝑈 = ℕ) | 
| 137 | 34, 136 | eleqtrrid 2848 | . . . . . . 7
⊢ (𝜑 → 1 ∈ dom 𝑈) | 
| 138 | 137 | ne0d 4342 | . . . . . 6
⊢ (𝜑 → dom 𝑈 ≠ ∅) | 
| 139 |  | dm0rn0 5935 | . . . . . . 7
⊢ (dom
𝑈 = ∅ ↔ ran
𝑈 =
∅) | 
| 140 | 139 | necon3bii 2993 | . . . . . 6
⊢ (dom
𝑈 ≠ ∅ ↔ ran
𝑈 ≠
∅) | 
| 141 | 138, 140 | sylib 218 | . . . . 5
⊢ (𝜑 → ran 𝑈 ≠ ∅) | 
| 142 | 133 | ffvelcdmda 7104 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ∈ (0[,)+∞)) | 
| 143 | 32, 142 | sselid 3981 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ∈ ℝ) | 
| 144 | 52, 53, 74 | syl2an 596 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐻)‘𝑛) ∈ ℝ) | 
| 145 |  | elrege0 13494 | . . . . . . . . . . . . . . . 16
⊢ ((((abs
∘ − ) ∘ 𝐺)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs
∘ − ) ∘ 𝐺)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐺)‘𝑛))) | 
| 146 | 56, 145 | sylib 218 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐺)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐺)‘𝑛))) | 
| 147 | 146 | simprd 495 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝐺)‘𝑛)) | 
| 148 | 74, 57 | addge02d 11852 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 ≤ (((abs
∘ − ) ∘ 𝐺)‘𝑛) ↔ (((abs ∘ − ) ∘
𝐻)‘𝑛) ≤ ((((abs ∘ − ) ∘
𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))) | 
| 149 | 147, 148 | mpbid 232 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ≤ ((((abs ∘ − ) ∘
𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))) | 
| 150 | 149, 77 | breqtrd 5169 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)) | 
| 151 | 52, 53, 150 | syl2an 596 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐻)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)) | 
| 152 | 51, 144, 65, 151 | serle 14098 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝐻))‘𝑗) ≤ (seq1( + , ((abs ∘ − )
∘ 𝐹))‘𝑗)) | 
| 153 | 15 | fveq1i 6907 | . . . . . . . . . 10
⊢ (𝑈‘𝑗) = (seq1( + , ((abs ∘ − )
∘ 𝐻))‘𝑗) | 
| 154 | 152, 153,
82 | 3brtr4g 5177 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ≤ (𝑆‘𝑗)) | 
| 155 | 143, 47, 48, 154, 120 | letrd 11418 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) | 
| 156 | 155 | ralrimiva 3146 | . . . . . . 7
⊢ (𝜑 → ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) | 
| 157 |  | brralrspcev 5203 | . . . . . . 7
⊢ ((sup(ran
𝑆, ℝ*,
< ) ∈ ℝ ∧ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ sup(ran 𝑆, ℝ*, < )) →
∃𝑥 ∈ ℝ
∀𝑗 ∈ ℕ
(𝑈‘𝑗) ≤ 𝑥) | 
| 158 | 23, 156, 157 | syl2anc 584 | . . . . . 6
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ 𝑥) | 
| 159 | 133 | ffnd 6737 | . . . . . . . 8
⊢ (𝜑 → 𝑈 Fn ℕ) | 
| 160 |  | breq1 5146 | . . . . . . . . 9
⊢ (𝑧 = (𝑈‘𝑗) → (𝑧 ≤ 𝑥 ↔ (𝑈‘𝑗) ≤ 𝑥)) | 
| 161 | 160 | ralrn 7108 | . . . . . . . 8
⊢ (𝑈 Fn ℕ →
(∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ 𝑥)) | 
| 162 | 159, 161 | syl 17 | . . . . . . 7
⊢ (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ 𝑥)) | 
| 163 | 162 | rexbidv 3179 | . . . . . 6
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ 𝑥)) | 
| 164 | 158, 163 | mpbird 257 | . . . . 5
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥) | 
| 165 | 135, 141,
164 | suprcld 12231 | . . . 4
⊢ (𝜑 → sup(ran 𝑈, ℝ, < ) ∈
ℝ) | 
| 166 |  | ssralv 4052 | . . . . . . . . . 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))))) | 
| 167 | 1, 166 | 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)))) | 
| 168 | 19 | breq1i 5150 | . . . . . . . . . . . . 13
⊢ (𝑃 < 𝑥 ↔ (1st ‘(𝐹‘𝑛)) < 𝑥) | 
| 169 |  | ovolfcl 25501 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) ∈ ℝ ∧
(2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st
‘(𝐹‘𝑛)) ≤ (2nd
‘(𝐹‘𝑛)))) | 
| 170 | 16, 169 | sylan 580 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) ∈ ℝ ∧
(2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st
‘(𝐹‘𝑛)) ≤ (2nd
‘(𝐹‘𝑛)))) | 
| 171 | 170 | simp1d 1143 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐹‘𝑛)) ∈
ℝ) | 
| 172 | 19, 171 | eqeltrid 2845 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ) | 
| 173 | 172 | adantlr 715 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ) | 
| 174 | 1, 2 | sstrid 3995 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐸 ∩ 𝐵) ⊆ ℝ) | 
| 175 | 174 | sselda 3983 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) → 𝑥 ∈ ℝ) | 
| 176 | 175 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ) | 
| 177 |  | ltle 11349 | . . . . . . . . . . . . . . 15
⊢ ((𝑃 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑃 < 𝑥 → 𝑃 ≤ 𝑥)) | 
| 178 | 173, 176,
177 | syl2anc 584 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥 → 𝑃 ≤ 𝑥)) | 
| 179 |  | simpr 484 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) | 
| 180 |  | opex 5469 | . . . . . . . . . . . . . . . . . . . 20
⊢
〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉 ∈ V | 
| 181 | 21 | fvmpt2 7027 | . . . . . . . . . . . . . . . . . . . 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〉) | 
| 182 | 179, 180,
181 | sylancl 586 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = 〈if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) | 
| 183 | 182 | fveq2d 6910 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐺‘𝑛)) = (1st
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉)) | 
| 184 | 11 | adantr 480 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) | 
| 185 | 184, 172 | ifcld 4572 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ) | 
| 186 | 170 | simp2d 1144 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐹‘𝑛)) ∈
ℝ) | 
| 187 | 20, 186 | eqeltrid 2845 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ) | 
| 188 | 185, 187 | ifcld 4572 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) | 
| 189 |  | op1stg 8026 | . . . . . . . . . . . . . . . . . . 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)) | 
| 190 | 188, 187,
189 | syl2anc 584 | . . . . . . . . . . . . . . . . . 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)) | 
| 191 | 183, 190 | eqtrd 2777 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐺‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) | 
| 192 | 191 | ad2ant2r 747 | . . . . . . . . . . . . . . . 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)) | 
| 193 | 188 | ad2ant2r 747 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) | 
| 194 | 185 | ad2ant2r 747 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ) | 
| 195 | 174 | ad2antrr 726 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → (𝐸 ∩ 𝐵) ⊆ ℝ) | 
| 196 |  | simplr 769 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑥 ∈ (𝐸 ∩ 𝐵)) | 
| 197 | 195, 196 | sseldd 3984 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑥 ∈ ℝ) | 
| 198 | 187 | ad2ant2r 747 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑄 ∈ ℝ) | 
| 199 |  | min1 13231 | . . . . . . . . . . . . . . . . . 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)) | 
| 200 | 194, 198,
199 | syl2anc 584 | . . . . . . . . . . . . . . . . 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)) | 
| 201 | 11 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝐴 ∈ ℝ) | 
| 202 |  | elinel2 4202 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (𝐸 ∩ 𝐵) → 𝑥 ∈ 𝐵) | 
| 203 | 202 | ad2antlr 727 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑥 ∈ 𝐵) | 
| 204 | 11 | rexrd 11311 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝐴 ∈
ℝ*) | 
| 205 |  | pnfxr 11315 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ +∞
∈ ℝ* | 
| 206 |  | elioo2 13428 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐴 ∈ ℝ*
∧ +∞ ∈ ℝ*) → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞))) | 
| 207 | 204, 205,
206 | sylancl 586 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞))) | 
| 208 | 10 | eleq2i 2833 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝐴(,)+∞)) | 
| 209 |  | ltpnf 13162 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | 
| 210 | 209 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) → 𝑥 < +∞) | 
| 211 | 210 | pm4.71i 559 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ↔ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ∧ 𝑥 < +∞)) | 
| 212 |  | df-3an 1089 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞) ↔ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ∧ 𝑥 < +∞)) | 
| 213 | 211, 212 | bitr4i 278 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞)) | 
| 214 | 207, 208,
213 | 3bitr4g 314 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥))) | 
| 215 |  | simpr 484 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) → 𝐴 < 𝑥) | 
| 216 | 214, 215 | biimtrdi 253 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝐴 < 𝑥)) | 
| 217 | 216 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → (𝑥 ∈ 𝐵 → 𝐴 < 𝑥)) | 
| 218 | 203, 217 | mpd 15 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝐴 < 𝑥) | 
| 219 | 201, 197,
218 | ltled 11409 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝐴 ≤ 𝑥) | 
| 220 |  | simprr 773 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑃 ≤ 𝑥) | 
| 221 |  | breq1 5146 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 = if(𝑃 ≤ 𝐴, 𝐴, 𝑃) → (𝐴 ≤ 𝑥 ↔ if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑥)) | 
| 222 |  | breq1 5146 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑃 = if(𝑃 ≤ 𝐴, 𝐴, 𝑃) → (𝑃 ≤ 𝑥 ↔ if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑥)) | 
| 223 | 221, 222 | ifboth 4565 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ≤ 𝑥 ∧ 𝑃 ≤ 𝑥) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑥) | 
| 224 | 219, 220,
223 | syl2anc 584 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑥) | 
| 225 | 193, 194,
197, 200, 224 | letrd 11418 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ≤ 𝑥) | 
| 226 | 192, 225 | eqbrtrd 5165 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → (1st ‘(𝐺‘𝑛)) ≤ 𝑥) | 
| 227 | 226 | expr 456 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 ≤ 𝑥 → (1st ‘(𝐺‘𝑛)) ≤ 𝑥)) | 
| 228 | 178, 227 | syld 47 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥 → (1st ‘(𝐺‘𝑛)) ≤ 𝑥)) | 
| 229 | 168, 228 | biimtrrid 243 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) < 𝑥 → (1st ‘(𝐺‘𝑛)) ≤ 𝑥)) | 
| 230 | 20 | breq2i 5151 | . . . . . . . . . . . . . 14
⊢ (𝑥 < 𝑄 ↔ 𝑥 < (2nd ‘(𝐹‘𝑛))) | 
| 231 | 187 | adantlr 715 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ) | 
| 232 |  | ltle 11349 | . . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑄 ∈ ℝ) → (𝑥 < 𝑄 → 𝑥 ≤ 𝑄)) | 
| 233 | 176, 231,
232 | syl2anc 584 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄 → 𝑥 ≤ 𝑄)) | 
| 234 | 230, 233 | biimtrrid 243 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹‘𝑛)) → 𝑥 ≤ 𝑄)) | 
| 235 | 182 | fveq2d 6910 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐺‘𝑛)) = (2nd
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉)) | 
| 236 |  | op2ndg 8027 | . . . . . . . . . . . . . . . . 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) | 
| 237 | 188, 187,
236 | syl2anc 584 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) = 𝑄) | 
| 238 | 235, 237 | eqtrd 2777 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐺‘𝑛)) = 𝑄) | 
| 239 | 238 | adantlr 715 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐺‘𝑛)) = 𝑄) | 
| 240 | 239 | breq2d 5155 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 ≤ (2nd ‘(𝐺‘𝑛)) ↔ 𝑥 ≤ 𝑄)) | 
| 241 | 234, 240 | sylibrd 259 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹‘𝑛)) → 𝑥 ≤ (2nd ‘(𝐺‘𝑛)))) | 
| 242 | 229, 241 | anim12d 609 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ((1st ‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐺‘𝑛))))) | 
| 243 | 242 | reximdva 3168 | . . . . . . . . . 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))))) | 
| 244 | 243 | ralimdva 3167 | . . . . . . . . 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))))) | 
| 245 | 167, 244 | 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))))) | 
| 246 |  | ovolfioo 25502 | . . . . . . . . 9
⊢ ((𝐸 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ))) → (𝐸 ⊆ ∪ ran
((,) ∘ 𝐹) ↔
∀𝑥 ∈ 𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))) | 
| 247 | 2, 16, 246 | syl2anc 584 | . . . . . . . 8
⊢ (𝜑 → (𝐸 ⊆ ∪ ran
((,) ∘ 𝐹) ↔
∀𝑥 ∈ 𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))) | 
| 248 |  | ovolficc 25503 | . . . . . . . . 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))))) | 
| 249 | 174, 27, 248 | syl2anc 584 | . . . . . . . 8
⊢ (𝜑 → ((𝐸 ∩ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐺) ↔
∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐺‘𝑛))))) | 
| 250 | 245, 247,
249 | 3imtr4d 294 | . . . . . . 7
⊢ (𝜑 → (𝐸 ⊆ ∪ ran
((,) ∘ 𝐹) →
(𝐸 ∩ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐺))) | 
| 251 | 17, 250 | mpd 15 | . . . . . 6
⊢ (𝜑 → (𝐸 ∩ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐺)) | 
| 252 | 14 | ovollb2 25524 | . . . . . 6
⊢ ((𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ (𝐸 ∩ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐺)) →
(vol*‘(𝐸 ∩ 𝐵)) ≤ sup(ran 𝑇, ℝ*, <
)) | 
| 253 | 27, 251, 252 | syl2anc 584 | . . . . 5
⊢ (𝜑 → (vol*‘(𝐸 ∩ 𝐵)) ≤ sup(ran 𝑇, ℝ*, <
)) | 
| 254 |  | supxrre 13369 | . . . . . 6
⊢ ((ran
𝑇 ⊆ ℝ ∧ ran
𝑇 ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥) → sup(ran 𝑇, ℝ*, < ) = sup(ran
𝑇, ℝ, <
)) | 
| 255 | 33, 40, 130, 254 | syl3anc 1373 | . . . . 5
⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran
𝑇, ℝ, <
)) | 
| 256 | 253, 255 | breqtrd 5169 | . . . 4
⊢ (𝜑 → (vol*‘(𝐸 ∩ 𝐵)) ≤ sup(ran 𝑇, ℝ, < )) | 
| 257 |  | ssralv 4052 | . . . . . . . . . 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))))) | 
| 258 | 6, 257 | 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)))) | 
| 259 | 172 | adantlr 715 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ) | 
| 260 | 6, 2 | sstrid 3995 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐸 ∖ 𝐵) ⊆ ℝ) | 
| 261 | 260 | sselda 3983 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) → 𝑥 ∈ ℝ) | 
| 262 | 261 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ) | 
| 263 | 259, 262,
177 | syl2anc 584 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥 → 𝑃 ≤ 𝑥)) | 
| 264 | 168, 263 | biimtrrid 243 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) < 𝑥 → 𝑃 ≤ 𝑥)) | 
| 265 |  | opex 5469 | . . . . . . . . . . . . . . . . . 18
⊢
〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉 ∈ V | 
| 266 | 22 | fvmpt2 7027 | . . . . . . . . . . . . . . . . . 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)〉) | 
| 267 | 179, 265,
266 | sylancl 586 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘𝑛) = 〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) | 
| 268 | 267 | fveq2d 6910 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐻‘𝑛)) = (1st
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉)) | 
| 269 |  | op1stg 8026 | . . . . . . . . . . . . . . . . 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) | 
| 270 | 172, 188,
269 | syl2anc 584 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) = 𝑃) | 
| 271 | 268, 270 | eqtrd 2777 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐻‘𝑛)) = 𝑃) | 
| 272 | 271 | adantlr 715 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐻‘𝑛)) = 𝑃) | 
| 273 | 272 | breq1d 5153 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐻‘𝑛)) ≤ 𝑥 ↔ 𝑃 ≤ 𝑥)) | 
| 274 | 264, 273 | sylibrd 259 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) < 𝑥 → (1st ‘(𝐻‘𝑛)) ≤ 𝑥)) | 
| 275 | 187 | adantlr 715 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ) | 
| 276 | 262, 275,
232 | syl2anc 584 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄 → 𝑥 ≤ 𝑄)) | 
| 277 | 260 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (𝐸 ∖ 𝐵) ⊆ ℝ) | 
| 278 |  | simplr 769 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ∈ (𝐸 ∖ 𝐵)) | 
| 279 | 277, 278 | sseldd 3984 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ∈ ℝ) | 
| 280 | 11 | ad2antrr 726 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝐴 ∈ ℝ) | 
| 281 | 172 | ad2ant2r 747 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑃 ∈ ℝ) | 
| 282 | 280, 281 | ifcld 4572 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ) | 
| 283 |  | eldifn 4132 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (𝐸 ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵) | 
| 284 | 283 | ad2antlr 727 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → ¬ 𝑥 ∈ 𝐵) | 
| 285 | 279 | biantrurd 532 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (𝐴 < 𝑥 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥))) | 
| 286 | 214 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥))) | 
| 287 | 285, 286 | bitr4d 282 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (𝐴 < 𝑥 ↔ 𝑥 ∈ 𝐵)) | 
| 288 | 284, 287 | mtbird 325 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → ¬ 𝐴 < 𝑥) | 
| 289 | 279, 280,
288 | nltled 11411 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ 𝐴) | 
| 290 |  | max2 13229 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑃 ∈ ℝ ∧ 𝐴 ∈ ℝ) → 𝐴 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃)) | 
| 291 | 281, 280,
290 | syl2anc 584 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝐴 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃)) | 
| 292 | 279, 280,
282, 289, 291 | letrd 11418 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃)) | 
| 293 |  | simprr 773 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ 𝑄) | 
| 294 |  | breq2 5147 | . . . . . . . . . . . . . . . . . 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))) | 
| 295 |  | breq2 5147 | . . . . . . . . . . . . . . . . . 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))) | 
| 296 | 294, 295 | ifboth 4565 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∧ 𝑥 ≤ 𝑄) → 𝑥 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) | 
| 297 | 292, 293,
296 | syl2anc 584 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) | 
| 298 | 267 | fveq2d 6910 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐻‘𝑛)) = (2nd
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉)) | 
| 299 |  | op2ndg 8027 | . . . . . . . . . . . . . . . . . . 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)) | 
| 300 | 172, 188,
299 | syl2anc 584 | . . . . . . . . . . . . . . . . . 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)) | 
| 301 | 298, 300 | eqtrd 2777 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐻‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) | 
| 302 | 301 | ad2ant2r 747 | . . . . . . . . . . . . . . . 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)) | 
| 303 | 297, 302 | breqtrrd 5171 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ (2nd ‘(𝐻‘𝑛))) | 
| 304 | 303 | expr 456 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 ≤ 𝑄 → 𝑥 ≤ (2nd ‘(𝐻‘𝑛)))) | 
| 305 | 276, 304 | syld 47 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄 → 𝑥 ≤ (2nd ‘(𝐻‘𝑛)))) | 
| 306 | 230, 305 | biimtrrid 243 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹‘𝑛)) → 𝑥 ≤ (2nd ‘(𝐻‘𝑛)))) | 
| 307 | 274, 306 | anim12d 609 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ((1st ‘(𝐻‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐻‘𝑛))))) | 
| 308 | 307 | reximdva 3168 | . . . . . . . . . 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))))) | 
| 309 | 308 | ralimdva 3167 | . . . . . . . . 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))))) | 
| 310 | 258, 309 | 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))))) | 
| 311 |  | ovolficc 25503 | . . . . . . . . 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))))) | 
| 312 | 260, 66, 311 | syl2anc 584 | . . . . . . . 8
⊢ (𝜑 → ((𝐸 ∖ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐻) ↔
∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐻‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐻‘𝑛))))) | 
| 313 | 310, 247,
312 | 3imtr4d 294 | . . . . . . 7
⊢ (𝜑 → (𝐸 ⊆ ∪ ran
((,) ∘ 𝐹) →
(𝐸 ∖ 𝐵) ⊆ ∪ ran ([,] ∘ 𝐻))) | 
| 314 | 17, 313 | mpd 15 | . . . . . 6
⊢ (𝜑 → (𝐸 ∖ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐻)) | 
| 315 | 15 | ovollb2 25524 | . . . . . 6
⊢ ((𝐻:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ (𝐸 ∖ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐻)) →
(vol*‘(𝐸 ∖
𝐵)) ≤ sup(ran 𝑈, ℝ*, <
)) | 
| 316 | 66, 314, 315 | syl2anc 584 | . . . . 5
⊢ (𝜑 → (vol*‘(𝐸 ∖ 𝐵)) ≤ sup(ran 𝑈, ℝ*, <
)) | 
| 317 |  | supxrre 13369 | . . . . . 6
⊢ ((ran
𝑈 ⊆ ℝ ∧ ran
𝑈 ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥) → sup(ran 𝑈, ℝ*, < ) = sup(ran
𝑈, ℝ, <
)) | 
| 318 | 135, 141,
164, 317 | syl3anc 1373 | . . . . 5
⊢ (𝜑 → sup(ran 𝑈, ℝ*, < ) = sup(ran
𝑈, ℝ, <
)) | 
| 319 | 316, 318 | breqtrd 5169 | . . . 4
⊢ (𝜑 → (vol*‘(𝐸 ∖ 𝐵)) ≤ sup(ran 𝑈, ℝ, < )) | 
| 320 | 5, 8, 131, 165, 256, 319 | le2addd 11882 | . . 3
⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐵)) + (vol*‘(𝐸 ∖ 𝐵))) ≤ (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, <
))) | 
| 321 |  | eqidd 2738 | . . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) = (((abs ∘ − ) ∘ 𝐺)‘𝑛)) | 
| 322 | 50, 14, 84, 321, 57, 147, 124 | isumsup2 15882 | . . . . 5
⊢ (𝜑 → 𝑇 ⇝ sup(ran 𝑇, ℝ, < )) | 
| 323 |  | seqex 14044 | . . . . . . 7
⊢ seq1( + ,
((abs ∘ − ) ∘ 𝐹)) ∈ V | 
| 324 | 13, 323 | eqeltri 2837 | . . . . . 6
⊢ 𝑆 ∈ V | 
| 325 | 324 | a1i 11 | . . . . 5
⊢ (𝜑 → 𝑆 ∈ V) | 
| 326 |  | eqidd 2738 | . . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) = (((abs ∘ − ) ∘ 𝐻)‘𝑛)) | 
| 327 | 50, 15, 84, 326, 74, 73, 158 | isumsup2 15882 | . . . . 5
⊢ (𝜑 → 𝑈 ⇝ sup(ran 𝑈, ℝ, < )) | 
| 328 | 42 | recnd 11289 | . . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ∈ ℂ) | 
| 329 | 143 | recnd 11289 | . . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ∈ ℂ) | 
| 330 | 57 | recnd 11289 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ∈ ℂ) | 
| 331 | 52, 53, 330 | syl2an 596 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐺)‘𝑛) ∈ ℂ) | 
| 332 | 74 | recnd 11289 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ∈ ℂ) | 
| 333 | 52, 53, 332 | syl2an 596 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐻)‘𝑛) ∈ ℂ) | 
| 334 | 77 | eqcomd 2743 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) = ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))) | 
| 335 | 52, 53, 334 | syl2an 596 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐹)‘𝑛) = ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))) | 
| 336 | 51, 331, 333, 335 | seradd 14085 | . . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝐹))‘𝑗) = ((seq1( + , ((abs ∘ − )
∘ 𝐺))‘𝑗) + (seq1( + , ((abs ∘
− ) ∘ 𝐻))‘𝑗))) | 
| 337 | 81, 153 | oveq12i 7443 | . . . . . 6
⊢ ((𝑇‘𝑗) + (𝑈‘𝑗)) = ((seq1( + , ((abs ∘ − )
∘ 𝐺))‘𝑗) + (seq1( + , ((abs ∘
− ) ∘ 𝐻))‘𝑗)) | 
| 338 | 336, 82, 337 | 3eqtr4g 2802 | . . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆‘𝑗) = ((𝑇‘𝑗) + (𝑈‘𝑗))) | 
| 339 | 50, 84, 322, 325, 327, 328, 329, 338 | climadd 15668 | . . . 4
⊢ (𝜑 → 𝑆 ⇝ (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, <
))) | 
| 340 |  | climuni 15588 | . . . 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*, <
)) | 
| 341 | 339, 114,
340 | syl2anc 584 | . . 3
⊢ (𝜑 → (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, < )) = sup(ran
𝑆, ℝ*,
< )) | 
| 342 | 320, 341 | breqtrd 5169 | . 2
⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐵)) + (vol*‘(𝐸 ∖ 𝐵))) ≤ sup(ran 𝑆, ℝ*, <
)) | 
| 343 | 9, 23, 25, 342, 18 | letrd 11418 | 1
⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐵)) + (vol*‘(𝐸 ∖ 𝐵))) ≤ ((vol*‘𝐸) + 𝐶)) |