Step | Hyp | Ref
| Expression |
1 | | itg2mulc.2 |
. . . . 5
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) |
2 | 1 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐹:ℝ⟶(0[,)+∞)) |
3 | | itg2mulc.3 |
. . . . 5
⊢ (𝜑 →
(∫2‘𝐹)
∈ ℝ) |
4 | 3 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → (∫2‘𝐹) ∈
ℝ) |
5 | | itg2mulc.4 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ (0[,)+∞)) |
6 | | elrege0 12928 |
. . . . . . . 8
⊢ (𝐴 ∈ (0[,)+∞) ↔
(𝐴 ∈ ℝ ∧ 0
≤ 𝐴)) |
7 | 5, 6 | sylib 221 |
. . . . . . 7
⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
8 | 7 | simpld 498 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℝ) |
9 | 8 | anim1i 618 |
. . . . 5
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴 ∈ ℝ ∧ 0 < 𝐴)) |
10 | | elrp 12474 |
. . . . 5
⊢ (𝐴 ∈ ℝ+
↔ (𝐴 ∈ ℝ
∧ 0 < 𝐴)) |
11 | 9, 10 | sylibr 237 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈
ℝ+) |
12 | 2, 4, 11 | itg2mulclem 24499 |
. . 3
⊢ ((𝜑 ∧ 0 < 𝐴) → (∫2‘((ℝ
× {𝐴})
∘f · 𝐹)) ≤ (𝐴 · (∫2‘𝐹))) |
13 | | ge0mulcl 12935 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (0[,)+∞) ∧
𝑦 ∈ (0[,)+∞))
→ (𝑥 · 𝑦) ∈
(0[,)+∞)) |
14 | 13 | adantl 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) →
(𝑥 · 𝑦) ∈
(0[,)+∞)) |
15 | | fconst6g 6567 |
. . . . . . . . 9
⊢ (𝐴 ∈ (0[,)+∞) →
(ℝ × {𝐴}):ℝ⟶(0[,)+∞)) |
16 | 5, 15 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (ℝ × {𝐴}):ℝ⟶(0[,)+∞)) |
17 | | reex 10706 |
. . . . . . . . 9
⊢ ℝ
∈ V |
18 | 17 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ℝ ∈
V) |
19 | | inidm 4109 |
. . . . . . . 8
⊢ (ℝ
∩ ℝ) = ℝ |
20 | 14, 16, 1, 18, 18, 19 | off 7442 |
. . . . . . 7
⊢ (𝜑 → ((ℝ × {𝐴}) ∘f ·
𝐹):ℝ⟶(0[,)+∞)) |
21 | 20 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < 𝐴) → ((ℝ × {𝐴}) ∘f ·
𝐹):ℝ⟶(0[,)+∞)) |
22 | | icossicc 12910 |
. . . . . . . . 9
⊢
(0[,)+∞) ⊆ (0[,]+∞) |
23 | | fss 6521 |
. . . . . . . . 9
⊢
((((ℝ × {𝐴}) ∘f · 𝐹):ℝ⟶(0[,)+∞)
∧ (0[,)+∞) ⊆ (0[,]+∞)) → ((ℝ × {𝐴}) ∘f ·
𝐹):ℝ⟶(0[,]+∞)) |
24 | 20, 22, 23 | sylancl 589 |
. . . . . . . 8
⊢ (𝜑 → ((ℝ × {𝐴}) ∘f ·
𝐹):ℝ⟶(0[,]+∞)) |
25 | 24 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < 𝐴) → ((ℝ × {𝐴}) ∘f ·
𝐹):ℝ⟶(0[,]+∞)) |
26 | 8, 3 | remulcld 10749 |
. . . . . . . 8
⊢ (𝜑 → (𝐴 · (∫2‘𝐹)) ∈
ℝ) |
27 | 26 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴 · (∫2‘𝐹)) ∈
ℝ) |
28 | | itg2lecl 24491 |
. . . . . . 7
⊢
((((ℝ × {𝐴}) ∘f · 𝐹):ℝ⟶(0[,]+∞)
∧ (𝐴 ·
(∫2‘𝐹)) ∈ ℝ ∧
(∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) ≤ (𝐴 · (∫2‘𝐹))) →
(∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) ∈
ℝ) |
29 | 25, 27, 12, 28 | syl3anc 1372 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < 𝐴) → (∫2‘((ℝ
× {𝐴})
∘f · 𝐹)) ∈ ℝ) |
30 | 11 | rpreccld 12524 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < 𝐴) → (1 / 𝐴) ∈
ℝ+) |
31 | 21, 29, 30 | itg2mulclem 24499 |
. . . . 5
⊢ ((𝜑 ∧ 0 < 𝐴) → (∫2‘((ℝ
× {(1 / 𝐴)})
∘f · ((ℝ × {𝐴}) ∘f · 𝐹))) ≤ ((1 / 𝐴) ·
(∫2‘((ℝ × {𝐴}) ∘f · 𝐹)))) |
32 | 2 | feqmptd 6737 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐹 = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦))) |
33 | | rge0ssre 12930 |
. . . . . . . . . . . . . 14
⊢
(0[,)+∞) ⊆ ℝ |
34 | | ax-resscn 10672 |
. . . . . . . . . . . . . 14
⊢ ℝ
⊆ ℂ |
35 | 33, 34 | sstri 3886 |
. . . . . . . . . . . . 13
⊢
(0[,)+∞) ⊆ ℂ |
36 | | fss 6521 |
. . . . . . . . . . . . 13
⊢ ((𝐹:ℝ⟶(0[,)+∞)
∧ (0[,)+∞) ⊆ ℂ) → 𝐹:ℝ⟶ℂ) |
37 | 1, 35, 36 | sylancl 589 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹:ℝ⟶ℂ) |
38 | 37 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐹:ℝ⟶ℂ) |
39 | 38 | ffvelrnda 6861 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℂ) |
40 | 39 | mulid2d 10737 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝑦 ∈ ℝ) → (1 · (𝐹‘𝑦)) = (𝐹‘𝑦)) |
41 | 40 | mpteq2dva 5125 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝑦 ∈ ℝ ↦ (1 · (𝐹‘𝑦))) = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦))) |
42 | 32, 41 | eqtr4d 2776 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐹 = (𝑦 ∈ ℝ ↦ (1 · (𝐹‘𝑦)))) |
43 | 17 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < 𝐴) → ℝ ∈ V) |
44 | | 1red 10720 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝑦 ∈ ℝ) → 1 ∈
ℝ) |
45 | 43, 30, 11 | ofc12 7452 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 0 < 𝐴) → ((ℝ × {(1 / 𝐴)}) ∘f ·
(ℝ × {𝐴})) =
(ℝ × {((1 / 𝐴)
· 𝐴)})) |
46 | | fconstmpt 5585 |
. . . . . . . . . 10
⊢ (ℝ
× {((1 / 𝐴) ·
𝐴)}) = (𝑦 ∈ ℝ ↦ ((1 / 𝐴) · 𝐴)) |
47 | 45, 46 | eqtrdi 2789 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < 𝐴) → ((ℝ × {(1 / 𝐴)}) ∘f ·
(ℝ × {𝐴})) =
(𝑦 ∈ ℝ ↦
((1 / 𝐴) · 𝐴))) |
48 | 8 | recnd 10747 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ∈ ℂ) |
49 | 48 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈ ℂ) |
50 | 11 | rpne0d 12519 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ≠ 0) |
51 | 49, 50 | recid2d 11490 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 0 < 𝐴) → ((1 / 𝐴) · 𝐴) = 1) |
52 | 51 | mpteq2dv 5126 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝑦 ∈ ℝ ↦ ((1 / 𝐴) · 𝐴)) = (𝑦 ∈ ℝ ↦ 1)) |
53 | 47, 52 | eqtrd 2773 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < 𝐴) → ((ℝ × {(1 / 𝐴)}) ∘f ·
(ℝ × {𝐴})) =
(𝑦 ∈ ℝ ↦
1)) |
54 | 43, 44, 39, 53, 32 | offval2 7444 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < 𝐴) → (((ℝ × {(1 / 𝐴)}) ∘f ·
(ℝ × {𝐴}))
∘f · 𝐹) = (𝑦 ∈ ℝ ↦ (1 · (𝐹‘𝑦)))) |
55 | 30 | rpcnd 12516 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < 𝐴) → (1 / 𝐴) ∈ ℂ) |
56 | | fconst6g 6567 |
. . . . . . . . 9
⊢ ((1 /
𝐴) ∈ ℂ →
(ℝ × {(1 / 𝐴)}):ℝ⟶ℂ) |
57 | 55, 56 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < 𝐴) → (ℝ × {(1 / 𝐴)}):ℝ⟶ℂ) |
58 | | fconst6g 6567 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℂ → (ℝ
× {𝐴}):ℝ⟶ℂ) |
59 | 49, 58 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < 𝐴) → (ℝ × {𝐴}):ℝ⟶ℂ) |
60 | | mulass 10703 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
61 | 60 | adantl 485 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
62 | 43, 57, 59, 38, 61 | caofass 7461 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < 𝐴) → (((ℝ × {(1 / 𝐴)}) ∘f ·
(ℝ × {𝐴}))
∘f · 𝐹) = ((ℝ × {(1 / 𝐴)}) ∘f ·
((ℝ × {𝐴})
∘f · 𝐹))) |
63 | 42, 54, 62 | 3eqtr2d 2779 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐹 = ((ℝ × {(1 / 𝐴)}) ∘f · ((ℝ
× {𝐴})
∘f · 𝐹))) |
64 | 63 | fveq2d 6678 |
. . . . 5
⊢ ((𝜑 ∧ 0 < 𝐴) → (∫2‘𝐹) =
(∫2‘((ℝ × {(1 / 𝐴)}) ∘f · ((ℝ
× {𝐴})
∘f · 𝐹)))) |
65 | 29 | recnd 10747 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < 𝐴) → (∫2‘((ℝ
× {𝐴})
∘f · 𝐹)) ∈ ℂ) |
66 | 65, 49, 50 | divrec2d 11498 |
. . . . 5
⊢ ((𝜑 ∧ 0 < 𝐴) → ((∫2‘((ℝ
× {𝐴})
∘f · 𝐹)) / 𝐴) = ((1 / 𝐴) ·
(∫2‘((ℝ × {𝐴}) ∘f · 𝐹)))) |
67 | 31, 64, 66 | 3brtr4d 5062 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → (∫2‘𝐹) ≤
((∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) / 𝐴)) |
68 | 4, 29, 11 | lemuldiv2d 12564 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → ((𝐴 · (∫2‘𝐹)) ≤
(∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) ↔
(∫2‘𝐹)
≤ ((∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) / 𝐴))) |
69 | 67, 68 | mpbird 260 |
. . 3
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴 · (∫2‘𝐹)) ≤
(∫2‘((ℝ × {𝐴}) ∘f · 𝐹))) |
70 | | itg2cl 24485 |
. . . . . 6
⊢
(((ℝ × {𝐴}) ∘f · 𝐹):ℝ⟶(0[,]+∞)
→ (∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) ∈
ℝ*) |
71 | 24, 70 | syl 17 |
. . . . 5
⊢ (𝜑 →
(∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) ∈
ℝ*) |
72 | 26 | rexrd 10769 |
. . . . 5
⊢ (𝜑 → (𝐴 · (∫2‘𝐹)) ∈
ℝ*) |
73 | | xrletri3 12630 |
. . . . 5
⊢
(((∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) ∈ ℝ*
∧ (𝐴 ·
(∫2‘𝐹)) ∈ ℝ*) →
((∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) = (𝐴 · (∫2‘𝐹)) ↔
((∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) ≤ (𝐴 · (∫2‘𝐹)) ∧ (𝐴 · (∫2‘𝐹)) ≤
(∫2‘((ℝ × {𝐴}) ∘f · 𝐹))))) |
74 | 71, 72, 73 | syl2anc 587 |
. . . 4
⊢ (𝜑 →
((∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) = (𝐴 · (∫2‘𝐹)) ↔
((∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) ≤ (𝐴 · (∫2‘𝐹)) ∧ (𝐴 · (∫2‘𝐹)) ≤
(∫2‘((ℝ × {𝐴}) ∘f · 𝐹))))) |
75 | 74 | adantr 484 |
. . 3
⊢ ((𝜑 ∧ 0 < 𝐴) → ((∫2‘((ℝ
× {𝐴})
∘f · 𝐹)) = (𝐴 · (∫2‘𝐹)) ↔
((∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) ≤ (𝐴 · (∫2‘𝐹)) ∧ (𝐴 · (∫2‘𝐹)) ≤
(∫2‘((ℝ × {𝐴}) ∘f · 𝐹))))) |
76 | 12, 69, 75 | mpbir2and 713 |
. 2
⊢ ((𝜑 ∧ 0 < 𝐴) → (∫2‘((ℝ
× {𝐴})
∘f · 𝐹)) = (𝐴 · (∫2‘𝐹))) |
77 | 17 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 0 = 𝐴) → ℝ ∈ V) |
78 | 37 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 0 = 𝐴) → 𝐹:ℝ⟶ℂ) |
79 | 8 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 0 = 𝐴) → 𝐴 ∈ ℝ) |
80 | | 0re 10721 |
. . . . . . 7
⊢ 0 ∈
ℝ |
81 | 80 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 0 = 𝐴) → 0 ∈ ℝ) |
82 | | simplr 769 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 = 𝐴) ∧ 𝑥 ∈ ℂ) → 0 = 𝐴) |
83 | 82 | oveq1d 7185 |
. . . . . . 7
⊢ (((𝜑 ∧ 0 = 𝐴) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = (𝐴 · 𝑥)) |
84 | | mul02 10896 |
. . . . . . . 8
⊢ (𝑥 ∈ ℂ → (0
· 𝑥) =
0) |
85 | 84 | adantl 485 |
. . . . . . 7
⊢ (((𝜑 ∧ 0 = 𝐴) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0) |
86 | 83, 85 | eqtr3d 2775 |
. . . . . 6
⊢ (((𝜑 ∧ 0 = 𝐴) ∧ 𝑥 ∈ ℂ) → (𝐴 · 𝑥) = 0) |
87 | 77, 78, 79, 81, 86 | caofid2 7458 |
. . . . 5
⊢ ((𝜑 ∧ 0 = 𝐴) → ((ℝ × {𝐴}) ∘f ·
𝐹) = (ℝ ×
{0})) |
88 | 87 | fveq2d 6678 |
. . . 4
⊢ ((𝜑 ∧ 0 = 𝐴) → (∫2‘((ℝ
× {𝐴})
∘f · 𝐹)) = (∫2‘(ℝ
× {0}))) |
89 | | itg20 24490 |
. . . 4
⊢
(∫2‘(ℝ × {0})) = 0 |
90 | 88, 89 | eqtrdi 2789 |
. . 3
⊢ ((𝜑 ∧ 0 = 𝐴) → (∫2‘((ℝ
× {𝐴})
∘f · 𝐹)) = 0) |
91 | 3 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 0 = 𝐴) → (∫2‘𝐹) ∈
ℝ) |
92 | 91 | recnd 10747 |
. . . 4
⊢ ((𝜑 ∧ 0 = 𝐴) → (∫2‘𝐹) ∈
ℂ) |
93 | 92 | mul02d 10916 |
. . 3
⊢ ((𝜑 ∧ 0 = 𝐴) → (0 ·
(∫2‘𝐹)) = 0) |
94 | | simpr 488 |
. . . 4
⊢ ((𝜑 ∧ 0 = 𝐴) → 0 = 𝐴) |
95 | 94 | oveq1d 7185 |
. . 3
⊢ ((𝜑 ∧ 0 = 𝐴) → (0 ·
(∫2‘𝐹)) = (𝐴 · (∫2‘𝐹))) |
96 | 90, 93, 95 | 3eqtr2d 2779 |
. 2
⊢ ((𝜑 ∧ 0 = 𝐴) → (∫2‘((ℝ
× {𝐴})
∘f · 𝐹)) = (𝐴 · (∫2‘𝐹))) |
97 | 7 | simprd 499 |
. . 3
⊢ (𝜑 → 0 ≤ 𝐴) |
98 | | leloe 10805 |
. . . 4
⊢ ((0
∈ ℝ ∧ 𝐴
∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴))) |
99 | 80, 8, 98 | sylancr 590 |
. . 3
⊢ (𝜑 → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴))) |
100 | 97, 99 | mpbid 235 |
. 2
⊢ (𝜑 → (0 < 𝐴 ∨ 0 = 𝐴)) |
101 | 76, 96, 100 | mpjaodan 958 |
1
⊢ (𝜑 →
(∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) = (𝐴 · (∫2‘𝐹))) |