| Step | Hyp | Ref
| Expression |
| 1 | | simprr 773 |
. . . . . . 7
⊢ ((𝑓 ∈ dom ∫1
∧ (∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 = (∫1‘𝑓)) |
| 2 | | itg1cl 25720 |
. . . . . . . 8
⊢ (𝑓 ∈ dom ∫1
→ (∫1‘𝑓) ∈ ℝ) |
| 3 | 2 | adantr 480 |
. . . . . . 7
⊢ ((𝑓 ∈ dom ∫1
∧ (∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) →
(∫1‘𝑓)
∈ ℝ) |
| 4 | 1, 3 | eqeltrd 2841 |
. . . . . 6
⊢ ((𝑓 ∈ dom ∫1
∧ (∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 ∈ ℝ) |
| 5 | 4 | rexlimiva 3147 |
. . . . 5
⊢
(∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) → 𝑥 ∈ ℝ) |
| 6 | 5 | abssi 4070 |
. . . 4
⊢ {𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆
ℝ |
| 7 | 6 | a1i 11 |
. . 3
⊢ (𝜑 → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆
ℝ) |
| 8 | | i1f0 25722 |
. . . . . 6
⊢ (ℝ
× {0}) ∈ dom ∫1 |
| 9 | | 3nn 12345 |
. . . . . . . 8
⊢ 3 ∈
ℕ |
| 10 | | nnrp 13046 |
. . . . . . . 8
⊢ (3 ∈
ℕ → 3 ∈ ℝ+) |
| 11 | | ne0i 4341 |
. . . . . . . 8
⊢ (3 ∈
ℝ+ → ℝ+ ≠ ∅) |
| 12 | 9, 10, 11 | mp2b 10 |
. . . . . . 7
⊢
ℝ+ ≠ ∅ |
| 13 | | itg2addnc.f2 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) |
| 14 | 13 | ffvelcdmda 7104 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ (0[,)+∞)) |
| 15 | | elrege0 13494 |
. . . . . . . . . . . 12
⊢ ((𝐹‘𝑧) ∈ (0[,)+∞) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑧))) |
| 16 | 14, 15 | sylib 218 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑧))) |
| 17 | 16 | simprd 495 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐹‘𝑧)) |
| 18 | 17 | ralrimiva 3146 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐹‘𝑧)) |
| 19 | | reex 11246 |
. . . . . . . . . . 11
⊢ ℝ
∈ V |
| 20 | 19 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℝ ∈
V) |
| 21 | | c0ex 11255 |
. . . . . . . . . . 11
⊢ 0 ∈
V |
| 22 | 21 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 0 ∈
V) |
| 23 | | eqidd 2738 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦
0)) |
| 24 | 13 | feqmptd 6977 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹‘𝑧))) |
| 25 | 20, 22, 14, 23, 24 | ofrfval2 7718 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑧 ∈ ℝ ↦ 0)
∘r ≤ 𝐹
↔ ∀𝑧 ∈
ℝ 0 ≤ (𝐹‘𝑧))) |
| 26 | 18, 25 | mpbird 257 |
. . . . . . . 8
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ 0)
∘r ≤ 𝐹) |
| 27 | 26 | ralrimivw 3150 |
. . . . . . 7
⊢ (𝜑 → ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0)
∘r ≤ 𝐹) |
| 28 | | r19.2z 4495 |
. . . . . . 7
⊢
((ℝ+ ≠ ∅ ∧ ∀𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0)
∘r ≤ 𝐹)
→ ∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ 0) ∘r ≤ 𝐹) |
| 29 | 12, 27, 28 | sylancr 587 |
. . . . . 6
⊢ (𝜑 → ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0)
∘r ≤ 𝐹) |
| 30 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑓 = (ℝ × {0}) →
(∫1‘𝑓)
= (∫1‘(ℝ × {0}))) |
| 31 | | itg10 25723 |
. . . . . . . . . 10
⊢
(∫1‘(ℝ × {0})) = 0 |
| 32 | 30, 31 | eqtr2di 2794 |
. . . . . . . . 9
⊢ (𝑓 = (ℝ × {0}) →
0 = (∫1‘𝑓)) |
| 33 | 32 | biantrud 531 |
. . . . . . . 8
⊢ (𝑓 = (ℝ × {0}) →
(∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ↔ (∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 =
(∫1‘𝑓)))) |
| 34 | | fveq1 6905 |
. . . . . . . . . . . . 13
⊢ (𝑓 = (ℝ × {0}) →
(𝑓‘𝑧) = ((ℝ × {0})‘𝑧)) |
| 35 | 21 | fvconst2 7224 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ ℝ → ((ℝ
× {0})‘𝑧) =
0) |
| 36 | 34, 35 | sylan9eq 2797 |
. . . . . . . . . . . 12
⊢ ((𝑓 = (ℝ × {0}) ∧
𝑧 ∈ ℝ) →
(𝑓‘𝑧) = 0) |
| 37 | 36 | iftrued 4533 |
. . . . . . . . . . 11
⊢ ((𝑓 = (ℝ × {0}) ∧
𝑧 ∈ ℝ) →
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) = 0) |
| 38 | 37 | mpteq2dva 5242 |
. . . . . . . . . 10
⊢ (𝑓 = (ℝ × {0}) →
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ 0)) |
| 39 | 38 | breq1d 5153 |
. . . . . . . . 9
⊢ (𝑓 = (ℝ × {0}) →
((𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ 0)
∘r ≤ 𝐹)) |
| 40 | 39 | rexbidv 3179 |
. . . . . . . 8
⊢ (𝑓 = (ℝ × {0}) →
(∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ↔ ∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
0) ∘r ≤ 𝐹)) |
| 41 | 33, 40 | bitr3d 281 |
. . . . . . 7
⊢ (𝑓 = (ℝ × {0}) →
((∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 =
(∫1‘𝑓)) ↔ ∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0)
∘r ≤ 𝐹)) |
| 42 | 41 | rspcev 3622 |
. . . . . 6
⊢
(((ℝ × {0}) ∈ dom ∫1 ∧ ∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
0) ∘r ≤ 𝐹) → ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 =
(∫1‘𝑓))) |
| 43 | 8, 29, 42 | sylancr 587 |
. . . . 5
⊢ (𝜑 → ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 =
(∫1‘𝑓))) |
| 44 | | eqeq1 2741 |
. . . . . . . 8
⊢ (𝑥 = 0 → (𝑥 = (∫1‘𝑓) ↔ 0 =
(∫1‘𝑓))) |
| 45 | 44 | anbi2d 630 |
. . . . . . 7
⊢ (𝑥 = 0 → ((∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) ↔ (∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 =
(∫1‘𝑓)))) |
| 46 | 45 | rexbidv 3179 |
. . . . . 6
⊢ (𝑥 = 0 → (∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) ↔ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 =
(∫1‘𝑓)))) |
| 47 | 21, 46 | elab 3679 |
. . . . 5
⊢ (0 ∈
{𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ↔ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 0 =
(∫1‘𝑓))) |
| 48 | 43, 47 | sylibr 234 |
. . . 4
⊢ (𝜑 → 0 ∈ {𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}) |
| 49 | 48 | ne0d 4342 |
. . 3
⊢ (𝜑 → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ≠
∅) |
| 50 | | icossicc 13476 |
. . . . . . 7
⊢
(0[,)+∞) ⊆ (0[,]+∞) |
| 51 | | fss 6752 |
. . . . . . 7
⊢ ((𝐹:ℝ⟶(0[,)+∞)
∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞)) |
| 52 | 50, 51 | mpan2 691 |
. . . . . 6
⊢ (𝐹:ℝ⟶(0[,)+∞)
→ 𝐹:ℝ⟶(0[,]+∞)) |
| 53 | | eqid 2737 |
. . . . . . 7
⊢ {𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} = {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} |
| 54 | 53 | itg2addnclem 37678 |
. . . . . 6
⊢ (𝐹:ℝ⟶(0[,]+∞)
→ (∫2‘𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, <
)) |
| 55 | 13, 52, 54 | 3syl 18 |
. . . . 5
⊢ (𝜑 →
(∫2‘𝐹)
= sup({𝑥 ∣
∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, <
)) |
| 56 | | itg2addnc.f3 |
. . . . 5
⊢ (𝜑 →
(∫2‘𝐹)
∈ ℝ) |
| 57 | 55, 56 | eqeltrrd 2842 |
. . . 4
⊢ (𝜑 → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
∈ ℝ) |
| 58 | | ressxr 11305 |
. . . . . . 7
⊢ ℝ
⊆ ℝ* |
| 59 | 6, 58 | sstri 3993 |
. . . . . 6
⊢ {𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆
ℝ* |
| 60 | | supxrub 13366 |
. . . . . 6
⊢ (({𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ*
∧ 𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}) → 𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, <
)) |
| 61 | 59, 60 | mpan 690 |
. . . . 5
⊢ (𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} → 𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, <
)) |
| 62 | 61 | rgen 3063 |
. . . 4
⊢
∀𝑏 ∈
{𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, <
) |
| 63 | | brralrspcev 5203 |
. . . 4
⊢
((sup({𝑥 ∣
∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
∈ ℝ ∧ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, <
)) → ∃𝑎 ∈
ℝ ∀𝑏 ∈
{𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 ≤ 𝑎) |
| 64 | 57, 62, 63 | sylancl 586 |
. . 3
⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 ≤ 𝑎) |
| 65 | | simprr 773 |
. . . . . . 7
⊢ ((𝑔 ∈ dom ∫1
∧ (∃𝑑 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))) → 𝑥 = (∫1‘𝑔)) |
| 66 | | itg1cl 25720 |
. . . . . . . 8
⊢ (𝑔 ∈ dom ∫1
→ (∫1‘𝑔) ∈ ℝ) |
| 67 | 66 | adantr 480 |
. . . . . . 7
⊢ ((𝑔 ∈ dom ∫1
∧ (∃𝑑 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))) →
(∫1‘𝑔)
∈ ℝ) |
| 68 | 65, 67 | eqeltrd 2841 |
. . . . . 6
⊢ ((𝑔 ∈ dom ∫1
∧ (∃𝑑 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))) → 𝑥 ∈ ℝ) |
| 69 | 68 | rexlimiva 3147 |
. . . . 5
⊢
(∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔)) → 𝑥 ∈ ℝ) |
| 70 | 69 | abssi 4070 |
. . . 4
⊢ {𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ⊆
ℝ |
| 71 | 70 | a1i 11 |
. . 3
⊢ (𝜑 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ⊆
ℝ) |
| 72 | | itg2addnc.g2 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞)) |
| 73 | 72 | ffvelcdmda 7104 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ (0[,)+∞)) |
| 74 | | elrege0 13494 |
. . . . . . . . . . . 12
⊢ ((𝐺‘𝑧) ∈ (0[,)+∞) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ 0 ≤ (𝐺‘𝑧))) |
| 75 | 73, 74 | sylib 218 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ((𝐺‘𝑧) ∈ ℝ ∧ 0 ≤ (𝐺‘𝑧))) |
| 76 | 75 | simprd 495 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐺‘𝑧)) |
| 77 | 76 | ralrimiva 3146 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑧 ∈ ℝ 0 ≤ (𝐺‘𝑧)) |
| 78 | 72 | feqmptd 6977 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℝ ↦ (𝐺‘𝑧))) |
| 79 | 20, 22, 73, 23, 78 | ofrfval2 7718 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑧 ∈ ℝ ↦ 0)
∘r ≤ 𝐺
↔ ∀𝑧 ∈
ℝ 0 ≤ (𝐺‘𝑧))) |
| 80 | 77, 79 | mpbird 257 |
. . . . . . . 8
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ 0)
∘r ≤ 𝐺) |
| 81 | 80 | ralrimivw 3150 |
. . . . . . 7
⊢ (𝜑 → ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0)
∘r ≤ 𝐺) |
| 82 | | r19.2z 4495 |
. . . . . . 7
⊢
((ℝ+ ≠ ∅ ∧ ∀𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0)
∘r ≤ 𝐺)
→ ∃𝑑 ∈
ℝ+ (𝑧
∈ ℝ ↦ 0) ∘r ≤ 𝐺) |
| 83 | 12, 81, 82 | sylancr 587 |
. . . . . 6
⊢ (𝜑 → ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0)
∘r ≤ 𝐺) |
| 84 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑔 = (ℝ × {0}) →
(∫1‘𝑔)
= (∫1‘(ℝ × {0}))) |
| 85 | 84, 31 | eqtr2di 2794 |
. . . . . . . . 9
⊢ (𝑔 = (ℝ × {0}) →
0 = (∫1‘𝑔)) |
| 86 | 85 | biantrud 531 |
. . . . . . . 8
⊢ (𝑔 = (ℝ × {0}) →
(∃𝑑 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ↔ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 =
(∫1‘𝑔)))) |
| 87 | | fveq1 6905 |
. . . . . . . . . . . . 13
⊢ (𝑔 = (ℝ × {0}) →
(𝑔‘𝑧) = ((ℝ × {0})‘𝑧)) |
| 88 | 87, 35 | sylan9eq 2797 |
. . . . . . . . . . . 12
⊢ ((𝑔 = (ℝ × {0}) ∧
𝑧 ∈ ℝ) →
(𝑔‘𝑧) = 0) |
| 89 | 88 | iftrued 4533 |
. . . . . . . . . . 11
⊢ ((𝑔 = (ℝ × {0}) ∧
𝑧 ∈ ℝ) →
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) = 0) |
| 90 | 89 | mpteq2dva 5242 |
. . . . . . . . . 10
⊢ (𝑔 = (ℝ × {0}) →
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ 0)) |
| 91 | 90 | breq1d 5153 |
. . . . . . . . 9
⊢ (𝑔 = (ℝ × {0}) →
((𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ↔ (𝑧 ∈ ℝ ↦ 0)
∘r ≤ 𝐺)) |
| 92 | 91 | rexbidv 3179 |
. . . . . . . 8
⊢ (𝑔 = (ℝ × {0}) →
(∃𝑑 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ↔ ∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
0) ∘r ≤ 𝐺)) |
| 93 | 86, 92 | bitr3d 281 |
. . . . . . 7
⊢ (𝑔 = (ℝ × {0}) →
((∃𝑑 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 =
(∫1‘𝑔)) ↔ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0)
∘r ≤ 𝐺)) |
| 94 | 93 | rspcev 3622 |
. . . . . 6
⊢
(((ℝ × {0}) ∈ dom ∫1 ∧ ∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
0) ∘r ≤ 𝐺) → ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 =
(∫1‘𝑔))) |
| 95 | 8, 83, 94 | sylancr 587 |
. . . . 5
⊢ (𝜑 → ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 =
(∫1‘𝑔))) |
| 96 | | eqeq1 2741 |
. . . . . . . 8
⊢ (𝑥 = 0 → (𝑥 = (∫1‘𝑔) ↔ 0 =
(∫1‘𝑔))) |
| 97 | 96 | anbi2d 630 |
. . . . . . 7
⊢ (𝑥 = 0 → ((∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔)) ↔ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 =
(∫1‘𝑔)))) |
| 98 | 97 | rexbidv 3179 |
. . . . . 6
⊢ (𝑥 = 0 → (∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 =
(∫1‘𝑔)))) |
| 99 | 21, 98 | elab 3679 |
. . . . 5
⊢ (0 ∈
{𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ↔ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 0 =
(∫1‘𝑔))) |
| 100 | 95, 99 | sylibr 234 |
. . . 4
⊢ (𝜑 → 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) |
| 101 | 100 | ne0d 4342 |
. . 3
⊢ (𝜑 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ≠
∅) |
| 102 | | fss 6752 |
. . . . . . 7
⊢ ((𝐺:ℝ⟶(0[,)+∞)
∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐺:ℝ⟶(0[,]+∞)) |
| 103 | 50, 102 | mpan2 691 |
. . . . . 6
⊢ (𝐺:ℝ⟶(0[,)+∞)
→ 𝐺:ℝ⟶(0[,]+∞)) |
| 104 | | eqid 2737 |
. . . . . . 7
⊢ {𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} |
| 105 | 104 | itg2addnclem 37678 |
. . . . . 6
⊢ (𝐺:ℝ⟶(0[,]+∞)
→ (∫2‘𝐺) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) |
| 106 | 72, 103, 105 | 3syl 18 |
. . . . 5
⊢ (𝜑 →
(∫2‘𝐺)
= sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) |
| 107 | | itg2addnc.g3 |
. . . . 5
⊢ (𝜑 →
(∫2‘𝐺)
∈ ℝ) |
| 108 | 106, 107 | eqeltrrd 2842 |
. . . 4
⊢ (𝜑 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )
∈ ℝ) |
| 109 | 70, 58 | sstri 3993 |
. . . . . 6
⊢ {𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ⊆
ℝ* |
| 110 | | supxrub 13366 |
. . . . . 6
⊢ (({𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ⊆ ℝ*
∧ 𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) → 𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) |
| 111 | 109, 110 | mpan 690 |
. . . . 5
⊢ (𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} → 𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) |
| 112 | 111 | rgen 3063 |
. . . 4
⊢
∀𝑏 ∈
{𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
) |
| 113 | | brralrspcev 5203 |
. . . 4
⊢
((sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )
∈ ℝ ∧ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) → ∃𝑎 ∈
ℝ ∀𝑏 ∈
{𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 ≤ 𝑎) |
| 114 | 108, 112,
113 | sylancl 586 |
. . 3
⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 ≤ 𝑎) |
| 115 | | eqid 2737 |
. . 3
⊢ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} = {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} |
| 116 | 7, 49, 64, 71, 101, 114, 115 | supadd 12236 |
. 2
⊢ (𝜑 → (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ, < ) +
sup({𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ, < )) =
sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < )) |
| 117 | | supxrre 13369 |
. . . . 5
⊢ (({𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ ∧ {𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ≠ ∅ ∧
∃𝑎 ∈ ℝ
∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 ≤ 𝑎) → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
= sup({𝑥 ∣
∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ, <
)) |
| 118 | 7, 49, 64, 117 | syl3anc 1373 |
. . . 4
⊢ (𝜑 → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
= sup({𝑥 ∣
∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ, <
)) |
| 119 | 55, 118 | eqtrd 2777 |
. . 3
⊢ (𝜑 →
(∫2‘𝐹)
= sup({𝑥 ∣
∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ, <
)) |
| 120 | | supxrre 13369 |
. . . . 5
⊢ (({𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ⊆ ℝ ∧ {𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ≠ ∅ ∧
∃𝑎 ∈ ℝ
∀𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 ≤ 𝑎) → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )
= sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ, <
)) |
| 121 | 71, 101, 114, 120 | syl3anc 1373 |
. . . 4
⊢ (𝜑 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )
= sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ, <
)) |
| 122 | 106, 121 | eqtrd 2777 |
. . 3
⊢ (𝜑 →
(∫2‘𝐺)
= sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ, <
)) |
| 123 | 119, 122 | oveq12d 7449 |
. 2
⊢ (𝜑 →
((∫2‘𝐹) + (∫2‘𝐺)) = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ, < ) +
sup({𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ, <
))) |
| 124 | | ge0addcl 13500 |
. . . . . . 7
⊢ ((𝑥 ∈ (0[,)+∞) ∧
𝑦 ∈ (0[,)+∞))
→ (𝑥 + 𝑦) ∈
(0[,)+∞)) |
| 125 | 50, 124 | sselid 3981 |
. . . . . 6
⊢ ((𝑥 ∈ (0[,)+∞) ∧
𝑦 ∈ (0[,)+∞))
→ (𝑥 + 𝑦) ∈
(0[,]+∞)) |
| 126 | 125 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) →
(𝑥 + 𝑦) ∈ (0[,]+∞)) |
| 127 | | inidm 4227 |
. . . . 5
⊢ (ℝ
∩ ℝ) = ℝ |
| 128 | 126, 13, 72, 20, 20, 127 | off 7715 |
. . . 4
⊢ (𝜑 → (𝐹 ∘f + 𝐺):ℝ⟶(0[,]+∞)) |
| 129 | | eqid 2737 |
. . . . 5
⊢ {𝑠 ∣ ∃ℎ ∈ dom
∫1(∃𝑦
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))} = {𝑠 ∣ ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))} |
| 130 | 129 | itg2addnclem 37678 |
. . . 4
⊢ ((𝐹 ∘f + 𝐺):ℝ⟶(0[,]+∞)
→ (∫2‘(𝐹 ∘f + 𝐺)) = sup({𝑠 ∣ ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))}, ℝ*, <
)) |
| 131 | 128, 130 | syl 17 |
. . 3
⊢ (𝜑 →
(∫2‘(𝐹
∘f + 𝐺)) =
sup({𝑠 ∣ ∃ℎ ∈ dom
∫1(∃𝑦
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))}, ℝ*, <
)) |
| 132 | | itg2addnc.f1 |
. . . . . . . 8
⊢ (𝜑 → 𝐹 ∈ MblFn) |
| 133 | 132, 13, 56, 72, 107 | itg2addnclem3 37680 |
. . . . . . 7
⊢ (𝜑 → (∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)) → ∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom
∫1((∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))) |
| 134 | | simpl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1) → 𝑓
∈ dom ∫1) |
| 135 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1) → 𝑔
∈ dom ∫1) |
| 136 | 134, 135 | i1fadd 25730 |
. . . . . . . . . . . . 13
⊢ ((𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1) → (𝑓 ∘f + 𝑔) ∈ dom
∫1) |
| 137 | 136 | ad3antlr 731 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ ((∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → (𝑓 ∘f + 𝑔) ∈ dom
∫1) |
| 138 | | reeanv 3229 |
. . . . . . . . . . . . . . . . 17
⊢
(∃𝑐 ∈
ℝ+ ∃𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) ↔ (∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺)) |
| 139 | 138 | biimpri 228 |
. . . . . . . . . . . . . . . 16
⊢
((∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ ∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) → ∃𝑐 ∈ ℝ+
∃𝑑 ∈
ℝ+ ((𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺)) |
| 140 | 139 | ad2ant2r 747 |
. . . . . . . . . . . . . . 15
⊢
(((∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) → ∃𝑐 ∈ ℝ+
∃𝑑 ∈
ℝ+ ((𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺)) |
| 141 | | ifcl 4571 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑐 ∈ ℝ+
∧ 𝑑 ∈
ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈
ℝ+) |
| 142 | 141 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺)) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈
ℝ+) |
| 143 | | breq1 5146 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (0 =
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) → (0 ≤ (𝐹‘𝑧) ↔ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧))) |
| 144 | 143 | anbi1d 631 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (0 =
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) → ((0 ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) ↔ (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)))) |
| 145 | 144 | imbi1d 341 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (0 =
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) → (((0 ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) ↔ ((if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))) |
| 146 | | breq1 5146 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑓‘𝑧) + 𝑐) = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) → (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ↔ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧))) |
| 147 | 146 | anbi1d 631 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑓‘𝑧) + 𝑐) = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) ↔ (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)))) |
| 148 | 147 | imbi1d 341 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑓‘𝑧) + 𝑐) = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) → (((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) ↔ ((if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))) |
| 149 | | breq1 5146 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (0 =
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → (0 ≤ (𝐺‘𝑧) ↔ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧))) |
| 150 | 149 | anbi2d 630 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (0 =
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → ((0 ≤ (𝐹‘𝑧) ∧ 0 ≤ (𝐺‘𝑧)) ↔ (0 ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)))) |
| 151 | 150 | imbi1d 341 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (0 =
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → (((0 ≤ (𝐹‘𝑧) ∧ 0 ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) ↔ ((0 ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))) |
| 152 | | breq1 5146 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑔‘𝑧) + 𝑑) = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → (((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧) ↔ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧))) |
| 153 | 152 | anbi2d 630 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑔‘𝑧) + 𝑑) = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → ((0 ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) ↔ (0 ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)))) |
| 154 | 153 | imbi1d 341 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑔‘𝑧) + 𝑑) = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → (((0 ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) ↔ ((0 ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))) |
| 155 | | oveq12 7440 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝑓‘𝑧) = 0 ∧ (𝑔‘𝑧) = 0) → ((𝑓‘𝑧) + (𝑔‘𝑧)) = (0 + 0)) |
| 156 | | 00id 11436 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (0 + 0) =
0 |
| 157 | 155, 156 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝑓‘𝑧) = 0 ∧ (𝑔‘𝑧) = 0) → ((𝑓‘𝑧) + (𝑔‘𝑧)) = 0) |
| 158 | 157 | iftrued 4533 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑓‘𝑧) = 0 ∧ (𝑔‘𝑧) = 0) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) = 0) |
| 159 | 158 | adantll 714 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) ∧ (𝑔‘𝑧) = 0) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) = 0) |
| 160 | | simpll 767 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) → 𝜑) |
| 161 | 15 | simplbi 497 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝐹‘𝑧) ∈ (0[,)+∞) → (𝐹‘𝑧) ∈ ℝ) |
| 162 | 14, 161 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ ℝ) |
| 163 | 74 | simplbi 497 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝐺‘𝑧) ∈ (0[,)+∞) → (𝐺‘𝑧) ∈ ℝ) |
| 164 | 73, 163 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ ℝ) |
| 165 | 162, 164,
17, 76 | addge0d 11839 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 166 | 160, 165 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 167 | 166 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) ∧ (𝑔‘𝑧) = 0) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 168 | 159, 167 | eqbrtrd 5165 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) ∧ (𝑔‘𝑧) = 0) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 169 | 168 | a1d 25 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) ∧ (𝑔‘𝑧) = 0) → ((0 ≤ (𝐹‘𝑧) ∧ 0 ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 170 | 166 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 171 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑓‘𝑧) = 0 → ((𝑓‘𝑧) + (𝑔‘𝑧)) = (0 + (𝑔‘𝑧))) |
| 172 | | simplrr 778 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) → 𝑔 ∈ dom
∫1) |
| 173 | | i1ff 25711 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑔 ∈ dom ∫1
→ 𝑔:ℝ⟶ℝ) |
| 174 | 173 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑔 ∈ dom ∫1
∧ 𝑧 ∈ ℝ)
→ (𝑔‘𝑧) ∈
ℝ) |
| 175 | 172, 174 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) ∈ ℝ) |
| 176 | 175 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) ∈ ℂ) |
| 177 | 176 | addlidd 11462 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (0 + (𝑔‘𝑧)) = (𝑔‘𝑧)) |
| 178 | 171, 177 | sylan9eqr 2799 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) → ((𝑓‘𝑧) + (𝑔‘𝑧)) = (𝑔‘𝑧)) |
| 179 | 178 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) |
| 180 | 179 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) |
| 181 | 141 | rpred 13077 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑐 ∈ ℝ+
∧ 𝑑 ∈
ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℝ) |
| 182 | 181 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℝ) |
| 183 | 175, 182 | readdcld 11290 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ ℝ) |
| 184 | 183 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ ℝ) |
| 185 | 160, 164 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ ℝ) |
| 186 | 185 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → (𝐺‘𝑧) ∈ ℝ) |
| 187 | 160, 162 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ ℝ) |
| 188 | 187, 185 | readdcld 11290 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) + (𝐺‘𝑧)) ∈ ℝ) |
| 189 | 188 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝐹‘𝑧) + (𝐺‘𝑧)) ∈ ℝ) |
| 190 | | simplrr 778 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ+) |
| 191 | 190 | rpred 13077 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑑 ∈ ℝ) |
| 192 | | rpre 13043 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑐 ∈ ℝ+
→ 𝑐 ∈
ℝ) |
| 193 | | rpre 13043 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑑 ∈ ℝ+
→ 𝑑 ∈
ℝ) |
| 194 | | min2 13232 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑑) |
| 195 | 192, 193,
194 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑐 ∈ ℝ+
∧ 𝑑 ∈
ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑑) |
| 196 | 195 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑑) |
| 197 | 182, 191,
175, 196 | leadd2dd 11878 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝑔‘𝑧) + 𝑑)) |
| 198 | 175, 191 | readdcld 11290 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) + 𝑑) ∈ ℝ) |
| 199 | | letr 11355 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ ℝ ∧ ((𝑔‘𝑧) + 𝑑) ∈ ℝ ∧ (𝐺‘𝑧) ∈ ℝ) → ((((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝑔‘𝑧) + 𝑑) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐺‘𝑧))) |
| 200 | 183, 198,
185, 199 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝑔‘𝑧) + 𝑑) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐺‘𝑧))) |
| 201 | 197, 200 | mpand 695 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐺‘𝑧))) |
| 202 | 201 | imp 406 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐺‘𝑧)) |
| 203 | 164, 162 | addge02d 11852 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (0 ≤ (𝐹‘𝑧) ↔ (𝐺‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 204 | 17, 203 | mpbid 232 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 205 | 160, 204 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 206 | 205 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → (𝐺‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 207 | 184, 186,
189, 202, 206 | letrd 11418 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 208 | 207 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 209 | 180, 208 | eqbrtrd 5165 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 210 | | breq1 5146 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (0 =
if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) → (0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)) ↔ if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 211 | | breq1 5146 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) → ((((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)) ↔ if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 212 | 210, 211 | ifboth 4565 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((0 ≤
((𝐹‘𝑧) + (𝐺‘𝑧)) ∧ (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 213 | 170, 209,
212 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 214 | 213 | ex 412 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) → (((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 215 | 214 | adantld 490 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) → ((0 ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 216 | 215 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) ∧ ¬ (𝑔‘𝑧) = 0) → ((0 ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 217 | 151, 154,
169, 216 | ifbothda 4564 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑓‘𝑧) = 0) → ((0 ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 218 | 149 | anbi2d 630 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (0 =
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ 0 ≤ (𝐺‘𝑧)) ↔ (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)))) |
| 219 | 218 | imbi1d 341 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (0 =
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → (((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ 0 ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) ↔ ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))) |
| 220 | 152 | anbi2d 630 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑔‘𝑧) + 𝑑) = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) ↔ (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)))) |
| 221 | 220 | imbi1d 341 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑔‘𝑧) + 𝑑) = if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) → (((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) ↔ ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))))) |
| 222 | 166 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑔‘𝑧) = 0) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 223 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑔‘𝑧) = 0 → ((𝑓‘𝑧) + (𝑔‘𝑧)) = ((𝑓‘𝑧) + 0)) |
| 224 | | simplrl 777 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) → 𝑓 ∈ dom
∫1) |
| 225 | | i1ff 25711 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑓 ∈ dom ∫1
→ 𝑓:ℝ⟶ℝ) |
| 226 | 225 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑓 ∈ dom ∫1
∧ 𝑧 ∈ ℝ)
→ (𝑓‘𝑧) ∈
ℝ) |
| 227 | 224, 226 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ) |
| 228 | 227 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℂ) |
| 229 | 228 | addridd 11461 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + 0) = (𝑓‘𝑧)) |
| 230 | 223, 229 | sylan9eqr 2799 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑔‘𝑧) = 0) → ((𝑓‘𝑧) + (𝑔‘𝑧)) = (𝑓‘𝑧)) |
| 231 | 230 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑔‘𝑧) = 0) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) |
| 232 | 231 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑔‘𝑧) = 0) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) |
| 233 | 227, 182 | readdcld 11290 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ ℝ) |
| 234 | 233 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ ℝ) |
| 235 | 187 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → (𝐹‘𝑧) ∈ ℝ) |
| 236 | 188 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → ((𝐹‘𝑧) + (𝐺‘𝑧)) ∈ ℝ) |
| 237 | | simplrl 777 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ+) |
| 238 | 237 | rpred 13077 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → 𝑐 ∈ ℝ) |
| 239 | | min1 13231 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑐) |
| 240 | 192, 193,
239 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑐 ∈ ℝ+
∧ 𝑑 ∈
ℝ+) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑐) |
| 241 | 240 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ≤ 𝑐) |
| 242 | 182, 238,
227, 241 | leadd2dd 11878 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝑓‘𝑧) + 𝑐)) |
| 243 | 227, 238 | readdcld 11290 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + 𝑐) ∈ ℝ) |
| 244 | | letr 11355 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ ℝ ∧ ((𝑓‘𝑧) + 𝑐) ∈ ℝ ∧ (𝐹‘𝑧) ∈ ℝ) → ((((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝑓‘𝑧) + 𝑐) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐹‘𝑧))) |
| 245 | 233, 243,
187, 244 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝑓‘𝑧) + 𝑐) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐹‘𝑧))) |
| 246 | 242, 245 | mpand 695 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐹‘𝑧))) |
| 247 | 246 | imp 406 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐹‘𝑧)) |
| 248 | 162, 164 | addge01d 11851 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (0 ≤ (𝐺‘𝑧) ↔ (𝐹‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 249 | 76, 248 | mpbid 232 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 250 | 160, 249 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 251 | 250 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → (𝐹‘𝑧) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 252 | 234, 235,
236, 247, 251 | letrd 11418 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 253 | 252 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑔‘𝑧) = 0) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → ((𝑓‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 254 | 232, 253 | eqbrtrd 5165 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑔‘𝑧) = 0) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 255 | 222, 254,
212 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑔‘𝑧) = 0) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 256 | 255 | ex 412 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (𝑔‘𝑧) = 0) → (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 257 | 256 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ¬ (𝑓‘𝑧) = 0) ∧ (𝑔‘𝑧) = 0) → (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 258 | 257 | adantrd 491 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ¬ (𝑓‘𝑧) = 0) ∧ (𝑔‘𝑧) = 0) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ 0 ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 259 | 166 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧))) → 0 ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 260 | 182 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℂ) |
| 261 | 228, 176,
260 | addassd 11283 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = ((𝑓‘𝑧) + ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) |
| 262 | 261 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧))) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = ((𝑓‘𝑧) + ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) |
| 263 | 227, 237 | ltaddrpd 13110 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) < ((𝑓‘𝑧) + 𝑐)) |
| 264 | 227, 243,
263 | ltled 11409 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑐)) |
| 265 | | letr 11355 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝑓‘𝑧) ∈ ℝ ∧ ((𝑓‘𝑧) + 𝑐) ∈ ℝ ∧ (𝐹‘𝑧) ∈ ℝ) → (((𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑐) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → (𝑓‘𝑧) ≤ (𝐹‘𝑧))) |
| 266 | 227, 243,
187, 265 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑐) ∧ ((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧)) → (𝑓‘𝑧) ≤ (𝐹‘𝑧))) |
| 267 | 264, 266 | mpand 695 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) → (𝑓‘𝑧) ≤ (𝐹‘𝑧))) |
| 268 | | le2add 11745 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝑓‘𝑧) ∈ ℝ ∧ ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ ℝ) ∧ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐺‘𝑧) ∈ ℝ)) → (((𝑓‘𝑧) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐺‘𝑧)) → ((𝑓‘𝑧) + ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 269 | 227, 183,
187, 185, 268 | syl22anc 839 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ (𝐺‘𝑧)) → ((𝑓‘𝑧) + ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 270 | 267, 201,
269 | syl2and 608 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → ((𝑓‘𝑧) + ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 271 | 270 | imp 406 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧))) → ((𝑓‘𝑧) + ((𝑔‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 272 | 262, 271 | eqbrtrd 5165 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧))) → (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 273 | 259, 272,
212 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ (((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧))) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 274 | 273 | ex 412 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 275 | 274 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ¬ (𝑓‘𝑧) = 0) ∧ ¬ (𝑔‘𝑧) = 0) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ ((𝑔‘𝑧) + 𝑑) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 276 | 219, 221,
258, 275 | ifbothda 4564 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1)) ∧ (𝑐 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+))
∧ 𝑧 ∈ ℝ)
∧ ¬ (𝑓‘𝑧) = 0) → ((((𝑓‘𝑧) + 𝑐) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 277 | 145, 148,
217, 276 | ifbothda 4564 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 278 | 277 | ralimdva 3167 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) → (∀𝑧 ∈ ℝ (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) → ∀𝑧 ∈ ℝ if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 279 | | ovex 7464 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑓‘𝑧) + 𝑐) ∈ V |
| 280 | 21, 279 | ifex 4576 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ∈ V |
| 281 | 280 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ∈ V) |
| 282 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) = (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)))) |
| 283 | 20, 281, 14, 282, 24 | ofrfval2 7718 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧))) |
| 284 | | ovex 7464 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑔‘𝑧) + 𝑑) ∈ V |
| 285 | 21, 284 | ifex 4576 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ∈ V |
| 286 | 285 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ∈ V) |
| 287 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) = (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)))) |
| 288 | 20, 286, 73, 287, 78 | ofrfval2 7718 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ↔ ∀𝑧 ∈ ℝ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧))) |
| 289 | 283, 288 | anbi12d 632 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) ↔ (∀𝑧 ∈ ℝ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ ∀𝑧 ∈ ℝ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)))) |
| 290 | | r19.26 3111 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∀𝑧 ∈
ℝ (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)) ↔ (∀𝑧 ∈ ℝ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ ∀𝑧 ∈ ℝ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧))) |
| 291 | 289, 290 | bitr4di 289 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) ↔ ∀𝑧 ∈ ℝ (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)))) |
| 292 | 291 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) → (((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) ↔ ∀𝑧 ∈ ℝ (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐)) ≤ (𝐹‘𝑧) ∧ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑)) ≤ (𝐺‘𝑧)))) |
| 293 | 19 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) → ℝ ∈ V) |
| 294 | | ovex 7464 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) ∈ V |
| 295 | 21, 294 | ifex 4576 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ∈ V |
| 296 | 295 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ∈ V) |
| 297 | | ovexd 7466 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) + (𝐺‘𝑧)) ∈ V) |
| 298 | 225 | ffnd 6737 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑓 ∈ dom ∫1
→ 𝑓 Fn
ℝ) |
| 299 | 298 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1) → 𝑓
Fn ℝ) |
| 300 | 299 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) → 𝑓 Fn ℝ) |
| 301 | 173 | ffnd 6737 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑔 ∈ dom ∫1
→ 𝑔 Fn
ℝ) |
| 302 | 301 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1) → 𝑔
Fn ℝ) |
| 303 | 302 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) → 𝑔 Fn ℝ) |
| 304 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) = (𝑓‘𝑧)) |
| 305 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) = (𝑔‘𝑧)) |
| 306 | 300, 303,
293, 293, 127, 304, 305 | ofval 7708 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → ((𝑓 ∘f + 𝑔)‘𝑧) = ((𝑓‘𝑧) + (𝑔‘𝑧))) |
| 307 | 306 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓 ∘f + 𝑔)‘𝑧) = 0 ↔ ((𝑓‘𝑧) + (𝑔‘𝑧)) = 0)) |
| 308 | 306 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)) = (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) |
| 309 | 307, 308 | ifbieq2d 4552 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ 𝑧 ∈ ℝ) → if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) = if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) |
| 310 | 309 | mpteq2dva 5242 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) → (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) = (𝑧 ∈ ℝ ↦ if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))))) |
| 311 | 13 | ffnd 6737 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝐹 Fn ℝ) |
| 312 | 72 | ffnd 6737 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝐺 Fn ℝ) |
| 313 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧)) |
| 314 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧)) |
| 315 | 311, 312,
20, 20, 127, 313, 314 | offval 7706 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 316 | 315 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) → (𝐹 ∘f + 𝐺) = (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 317 | 293, 296,
297, 310, 316 | ofrfval2 7718 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) → ((𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹 ∘f + 𝐺) ↔ ∀𝑧 ∈ ℝ if(((𝑓‘𝑧) + (𝑔‘𝑧)) = 0, 0, (((𝑓‘𝑧) + (𝑔‘𝑧)) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) ≤ ((𝐹‘𝑧) + (𝐺‘𝑧)))) |
| 318 | 278, 292,
317 | 3imtr4d 294 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) → (((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) → (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹 ∘f + 𝐺))) |
| 319 | 318 | imp 406 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺)) → (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹 ∘f + 𝐺)) |
| 320 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = if(𝑐 ≤ 𝑑, 𝑐, 𝑑) → (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦) = (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))) |
| 321 | 320 | ifeq2d 4546 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = if(𝑐 ≤ 𝑑, 𝑐, 𝑑) → if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦)) = if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) |
| 322 | 321 | mpteq2dv 5244 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = if(𝑐 ≤ 𝑑, 𝑐, 𝑑) → (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑))))) |
| 323 | 322 | breq1d 5153 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = if(𝑐 ≤ 𝑑, 𝑐, 𝑑) → ((𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ↔ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹 ∘f + 𝐺))) |
| 324 | 323 | rspcev 3622 |
. . . . . . . . . . . . . . . . . 18
⊢
((if(𝑐 ≤ 𝑑, 𝑐, 𝑑) ∈ ℝ+ ∧ (𝑧 ∈ ℝ ↦
if(((𝑓 ∘f
+ 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + if(𝑐 ≤ 𝑑, 𝑐, 𝑑)))) ∘r ≤ (𝐹 ∘f + 𝐺)) → ∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if(((𝑓 ∘f
+ 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺)) |
| 325 | 142, 319,
324 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) ∧ ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺)) → ∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if(((𝑓 ∘f
+ 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺)) |
| 326 | 325 | ex 412 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ (𝑐 ∈
ℝ+ ∧ 𝑑
∈ ℝ+)) → (((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) → ∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if(((𝑓 ∘f
+ 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺))) |
| 327 | 326 | rexlimdvva 3213 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
→ (∃𝑐 ∈
ℝ+ ∃𝑑 ∈ ℝ+ ((𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺) → ∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if(((𝑓 ∘f
+ 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺))) |
| 328 | 140, 327 | syl5 34 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
→ (((∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) → ∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if(((𝑓 ∘f
+ 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺))) |
| 329 | 328 | a1dd 50 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
→ (((∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦
if(((𝑓 ∘f
+ 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺)))) |
| 330 | 329 | imp31 417 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ ((∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦
if(((𝑓 ∘f
+ 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺)) |
| 331 | | oveq12 7440 |
. . . . . . . . . . . . . . 15
⊢ ((𝑡 =
(∫1‘𝑓)
∧ 𝑢 =
(∫1‘𝑔)) → (𝑡 + 𝑢) = ((∫1‘𝑓) +
(∫1‘𝑔))) |
| 332 | 331 | ad2ant2l 746 |
. . . . . . . . . . . . . 14
⊢
(((∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) → (𝑡 + 𝑢) = ((∫1‘𝑓) +
(∫1‘𝑔))) |
| 333 | 134, 135 | itg1add 25736 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1) → (∫1‘(𝑓 ∘f + 𝑔)) = ((∫1‘𝑓) +
(∫1‘𝑔))) |
| 334 | 333 | eqcomd 2743 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓 ∈ dom ∫1
∧ 𝑔 ∈ dom
∫1) → ((∫1‘𝑓) + (∫1‘𝑔)) =
(∫1‘(𝑓
∘f + 𝑔))) |
| 335 | 334 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
→ ((∫1‘𝑓) + (∫1‘𝑔)) =
(∫1‘(𝑓
∘f + 𝑔))) |
| 336 | 332, 335 | sylan9eqr 2799 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ ((∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)))) → (𝑡 + 𝑢) = (∫1‘(𝑓 ∘f + 𝑔))) |
| 337 | | eqtr 2760 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 = (𝑡 + 𝑢) ∧ (𝑡 + 𝑢) = (∫1‘(𝑓 ∘f + 𝑔))) → 𝑠 = (∫1‘(𝑓 ∘f + 𝑔))) |
| 338 | 337 | ancoms 458 |
. . . . . . . . . . . . 13
⊢ (((𝑡 + 𝑢) = (∫1‘(𝑓 ∘f + 𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (∫1‘(𝑓 ∘f + 𝑔))) |
| 339 | 336, 338 | sylan 580 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ ((∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (∫1‘(𝑓 ∘f + 𝑔))) |
| 340 | | fveq1 6905 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℎ = (𝑓 ∘f + 𝑔) → (ℎ‘𝑧) = ((𝑓 ∘f + 𝑔)‘𝑧)) |
| 341 | 340 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . . 18
⊢ (ℎ = (𝑓 ∘f + 𝑔) → ((ℎ‘𝑧) = 0 ↔ ((𝑓 ∘f + 𝑔)‘𝑧) = 0)) |
| 342 | 340 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . 18
⊢ (ℎ = (𝑓 ∘f + 𝑔) → ((ℎ‘𝑧) + 𝑦) = (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦)) |
| 343 | 341, 342 | ifbieq2d 4552 |
. . . . . . . . . . . . . . . . 17
⊢ (ℎ = (𝑓 ∘f + 𝑔) → if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦)) = if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) |
| 344 | 343 | mpteq2dv 5244 |
. . . . . . . . . . . . . . . 16
⊢ (ℎ = (𝑓 ∘f + 𝑔) → (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦)))) |
| 345 | 344 | breq1d 5153 |
. . . . . . . . . . . . . . 15
⊢ (ℎ = (𝑓 ∘f + 𝑔) → ((𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ↔ (𝑧 ∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺))) |
| 346 | 345 | rexbidv 3179 |
. . . . . . . . . . . . . 14
⊢ (ℎ = (𝑓 ∘f + 𝑔) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦
if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ↔ ∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if(((𝑓 ∘f
+ 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺))) |
| 347 | | fveq2 6906 |
. . . . . . . . . . . . . . 15
⊢ (ℎ = (𝑓 ∘f + 𝑔) → (∫1‘ℎ) =
(∫1‘(𝑓
∘f + 𝑔))) |
| 348 | 347 | eqeq2d 2748 |
. . . . . . . . . . . . . 14
⊢ (ℎ = (𝑓 ∘f + 𝑔) → (𝑠 = (∫1‘ℎ) ↔ 𝑠 = (∫1‘(𝑓 ∘f + 𝑔)))) |
| 349 | 346, 348 | anbi12d 632 |
. . . . . . . . . . . . 13
⊢ (ℎ = (𝑓 ∘f + 𝑔) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦
if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)) ↔ (∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if(((𝑓 ∘f
+ 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘(𝑓 ∘f + 𝑔))))) |
| 350 | 349 | rspcev 3622 |
. . . . . . . . . . . 12
⊢ (((𝑓 ∘f + 𝑔) ∈ dom ∫1
∧ (∃𝑦 ∈
ℝ+ (𝑧
∈ ℝ ↦ if(((𝑓 ∘f + 𝑔)‘𝑧) = 0, 0, (((𝑓 ∘f + 𝑔)‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘(𝑓 ∘f + 𝑔)))) → ∃ℎ ∈ dom
∫1(∃𝑦
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))) |
| 351 | 137, 330,
339, 350 | syl12anc 837 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
∧ ((∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))) |
| 352 | 351 | exp31 419 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1))
→ (((∃𝑐 ∈
ℝ+ (𝑧
∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))))) |
| 353 | 352 | rexlimdvva 3213 |
. . . . . . . . 9
⊢ (𝜑 → (∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom
∫1((∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) → (𝑠 = (𝑡 + 𝑢) → ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))))) |
| 354 | 353 | impd 410 |
. . . . . . . 8
⊢ (𝜑 → ((∃𝑓 ∈ dom
∫1∃𝑔
∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)))) |
| 355 | 354 | exlimdvv 1934 |
. . . . . . 7
⊢ (𝜑 → (∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom
∫1((∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) → ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)))) |
| 356 | 133, 355 | impbid 212 |
. . . . . 6
⊢ (𝜑 → (∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)) ↔ ∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom
∫1((∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)))) |
| 357 | | eqeq1 2741 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑡 → (𝑥 = (∫1‘𝑓) ↔ 𝑡 = (∫1‘𝑓))) |
| 358 | 357 | anbi2d 630 |
. . . . . . . . 9
⊢ (𝑥 = 𝑡 → ((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) ↔ (∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)))) |
| 359 | 358 | rexbidv 3179 |
. . . . . . . 8
⊢ (𝑥 = 𝑡 → (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) ↔ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)))) |
| 360 | 359 | rexab 3700 |
. . . . . . 7
⊢
(∃𝑡 ∈
{𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢))) |
| 361 | | eqeq1 2741 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑢 → (𝑥 = (∫1‘𝑔) ↔ 𝑢 = (∫1‘𝑔))) |
| 362 | 361 | anbi2d 630 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑢 → ((∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔)) ↔ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)))) |
| 363 | 362 | rexbidv 3179 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑢 → (∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)))) |
| 364 | 363 | rexab 3700 |
. . . . . . . . . 10
⊢
(∃𝑢 ∈
{𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑢(∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))) |
| 365 | 364 | anbi2i 623 |
. . . . . . . . 9
⊢
((∃𝑓 ∈
dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢(∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)))) |
| 366 | | 19.42v 1953 |
. . . . . . . . 9
⊢
(∃𝑢(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢(∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)))) |
| 367 | | reeanv 3229 |
. . . . . . . . . . . 12
⊢
(∃𝑓 ∈ dom
∫1∃𝑔
∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ↔ (∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)))) |
| 368 | 367 | anbi1i 624 |
. . . . . . . . . . 11
⊢
((∃𝑓 ∈
dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) ↔ ((∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))) |
| 369 | | anass 468 |
. . . . . . . . . . 11
⊢
(((∃𝑓 ∈
dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢)) ↔ (∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢)))) |
| 370 | 368, 369 | bitr2i 276 |
. . . . . . . . . 10
⊢
((∃𝑓 ∈
dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))) ↔ (∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom
∫1((∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))) |
| 371 | 370 | exbii 1848 |
. . . . . . . . 9
⊢
(∃𝑢(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔)) ∧ 𝑠 = (𝑡 + 𝑢))) ↔ ∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom
∫1((∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))) |
| 372 | 365, 366,
371 | 3bitr2i 299 |
. . . . . . . 8
⊢
((∃𝑓 ∈
dom ∫1(∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)) ↔ ∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom
∫1((∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))) |
| 373 | 372 | exbii 1848 |
. . . . . . 7
⊢
(∃𝑡(∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ ∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)) ↔ ∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom
∫1((∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))) |
| 374 | 360, 373 | bitri 275 |
. . . . . 6
⊢
(∃𝑡 ∈
{𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom
∫1((∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))) |
| 375 | 356, 374 | bitr4di 289 |
. . . . 5
⊢ (𝜑 → (∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)) ↔ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢))) |
| 376 | 375 | abbidv 2808 |
. . . 4
⊢ (𝜑 → {𝑠 ∣ ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))} = {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}) |
| 377 | 376 | supeq1d 9486 |
. . 3
⊢ (𝜑 → sup({𝑠 ∣ ∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ))}, ℝ*, < )
= sup({𝑠 ∣
∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ*, <
)) |
| 378 | | simpr 484 |
. . . . . . . . 9
⊢ (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 = (𝑡 + 𝑢)) |
| 379 | 6 | sseli 3979 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} → 𝑡 ∈ ℝ) |
| 380 | 379 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑡 ∈ ℝ) |
| 381 | 70 | sseli 3979 |
. . . . . . . . . . 11
⊢ (𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} → 𝑢 ∈ ℝ) |
| 382 | 381 | ad2antlr 727 |
. . . . . . . . . 10
⊢ (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑢 ∈ ℝ) |
| 383 | 380, 382 | readdcld 11290 |
. . . . . . . . 9
⊢ (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → (𝑡 + 𝑢) ∈ ℝ) |
| 384 | 378, 383 | eqeltrd 2841 |
. . . . . . . 8
⊢ (((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) ∧ 𝑠 = (𝑡 + 𝑢)) → 𝑠 ∈ ℝ) |
| 385 | 384 | ex 412 |
. . . . . . 7
⊢ ((𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) → (𝑠 = (𝑡 + 𝑢) → 𝑠 ∈ ℝ)) |
| 386 | 385 | rexlimivv 3201 |
. . . . . 6
⊢
(∃𝑡 ∈
{𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢) → 𝑠 ∈ ℝ) |
| 387 | 386 | abssi 4070 |
. . . . 5
⊢ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ⊆ ℝ |
| 388 | 387 | a1i 11 |
. . . 4
⊢ (𝜑 → {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ⊆ ℝ) |
| 389 | 156 | eqcomi 2746 |
. . . . . . . 8
⊢ 0 = (0 +
0) |
| 390 | | rspceov 7480 |
. . . . . . . 8
⊢ ((0
∈ {𝑥 ∣
∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ∧ 0 = (0 + 0)) →
∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}0 = (𝑡 + 𝑢)) |
| 391 | 389, 390 | mp3an3 1452 |
. . . . . . 7
⊢ ((0
∈ {𝑥 ∣
∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 0 ∈ {𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) → ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}0 = (𝑡 + 𝑢)) |
| 392 | 48, 100, 391 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}0 = (𝑡 + 𝑢)) |
| 393 | | eqeq1 2741 |
. . . . . . . 8
⊢ (𝑠 = 0 → (𝑠 = (𝑡 + 𝑢) ↔ 0 = (𝑡 + 𝑢))) |
| 394 | 393 | 2rexbidv 3222 |
. . . . . . 7
⊢ (𝑠 = 0 → (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}0 = (𝑡 + 𝑢))) |
| 395 | 21, 394 | spcev 3606 |
. . . . . 6
⊢
(∃𝑡 ∈
{𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}0 = (𝑡 + 𝑢) → ∃𝑠∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)) |
| 396 | 392, 395 | syl 17 |
. . . . 5
⊢ (𝜑 → ∃𝑠∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)) |
| 397 | | abn0 4385 |
. . . . 5
⊢ ({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ≠ ∅ ↔ ∃𝑠∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)) |
| 398 | 396, 397 | sylibr 234 |
. . . 4
⊢ (𝜑 → {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ≠ ∅) |
| 399 | 57, 108 | readdcld 11290 |
. . . . 5
⊢ (𝜑 → (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) ∈ ℝ) |
| 400 | | simpr 484 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → 𝑏 = (𝑡 + 𝑢)) |
| 401 | 379 | ad2antrl 728 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → 𝑡 ∈ ℝ) |
| 402 | 381 | ad2antll 729 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → 𝑢 ∈ ℝ) |
| 403 | 57 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → sup({𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
∈ ℝ) |
| 404 | 108 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → sup({𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )
∈ ℝ) |
| 405 | | supxrub 13366 |
. . . . . . . . . . . . 13
⊢ (({𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ*
∧ 𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}) → 𝑡 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, <
)) |
| 406 | 59, 405 | mpan 690 |
. . . . . . . . . . . 12
⊢ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} → 𝑡 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, <
)) |
| 407 | 406 | ad2antrl 728 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → 𝑡 ≤ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, <
)) |
| 408 | | supxrub 13366 |
. . . . . . . . . . . . 13
⊢ (({𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} ⊆ ℝ*
∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}) → 𝑢 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) |
| 409 | 109, 408 | mpan 690 |
. . . . . . . . . . . 12
⊢ (𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))} → 𝑢 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) |
| 410 | 409 | ad2antll 729 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → 𝑢 ≤ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) |
| 411 | 401, 402,
403, 404, 407, 410 | le2addd 11882 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → (𝑡 + 𝑢) ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
))) |
| 412 | 411 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → (𝑡 + 𝑢) ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
))) |
| 413 | 400, 412 | eqbrtrd 5165 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) ∧ 𝑏 = (𝑡 + 𝑢)) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
))) |
| 414 | 413 | ex 412 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ∧ 𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))})) → (𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)))) |
| 415 | 414 | rexlimdvva 3213 |
. . . . . 6
⊢ (𝜑 → (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)))) |
| 416 | 415 | alrimiv 1927 |
. . . . 5
⊢ (𝜑 → ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)))) |
| 417 | | breq2 5147 |
. . . . . . . 8
⊢ (𝑎 = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) → (𝑏 ≤ 𝑎 ↔ 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)))) |
| 418 | 417 | ralbidv 3178 |
. . . . . . 7
⊢ (𝑎 = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) → (∀𝑏 ∈
{𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ 𝑎 ↔ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)))) |
| 419 | | eqeq1 2741 |
. . . . . . . . 9
⊢ (𝑠 = 𝑏 → (𝑠 = (𝑡 + 𝑢) ↔ 𝑏 = (𝑡 + 𝑢))) |
| 420 | 419 | 2rexbidv 3222 |
. . . . . . . 8
⊢ (𝑠 = 𝑏 → (∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢) ↔ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 = (𝑡 + 𝑢))) |
| 421 | 420 | ralab 3697 |
. . . . . . 7
⊢
(∀𝑏 ∈
{𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) ↔ ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)))) |
| 422 | 418, 421 | bitrdi 287 |
. . . . . 6
⊢ (𝑎 = (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) → (∀𝑏 ∈
{𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ 𝑎 ↔ ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
))))) |
| 423 | 422 | rspcev 3622 |
. . . . 5
⊢
(((sup({𝑥 ∣
∃𝑓 ∈ dom
∫1(∃𝑐
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) ∈ ℝ ∧ ∀𝑏(∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑏 = (𝑡 + 𝑢) → 𝑏 ≤ (sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < )
+ sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(∃𝑑
∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)))) → ∃𝑎 ∈
ℝ ∀𝑏 ∈
{𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ 𝑎) |
| 424 | 399, 416,
423 | syl2anc 584 |
. . . 4
⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ 𝑎) |
| 425 | | supxrre 13369 |
. . . 4
⊢ (({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ⊆ ℝ ∧ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)} ≠ ∅ ∧ ∃𝑎 ∈ ℝ ∀𝑏 ∈ {𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}𝑏 ≤ 𝑎) → sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ*, < ) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < )) |
| 426 | 388, 398,
424, 425 | syl3anc 1373 |
. . 3
⊢ (𝜑 → sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ*, < ) = sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < )) |
| 427 | 131, 377,
426 | 3eqtrd 2781 |
. 2
⊢ (𝜑 →
(∫2‘(𝐹
∘f + 𝐺)) =
sup({𝑠 ∣ ∃𝑡 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(∃𝑐 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}∃𝑢 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑑 ∈ ℝ+
(𝑧 ∈ ℝ ↦
if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑥 = (∫1‘𝑔))}𝑠 = (𝑡 + 𝑢)}, ℝ, < )) |
| 428 | 116, 123,
427 | 3eqtr4rd 2788 |
1
⊢ (𝜑 →
(∫2‘(𝐹
∘f + 𝐺)) =
((∫2‘𝐹) + (∫2‘𝐺))) |