Step | Hyp | Ref
| Expression |
1 | | i1frn 23250 |
. . . . 5
⊢ (𝐹 ∈ dom ∫1
→ ran 𝐹 ∈
Fin) |
2 | | difss 3699 |
. . . . 5
⊢ (ran
𝐹 ∖ {0}) ⊆ ran
𝐹 |
3 | | ssfi 8065 |
. . . . 5
⊢ ((ran
𝐹 ∈ Fin ∧ (ran
𝐹 ∖ {0}) ⊆ ran
𝐹) → (ran 𝐹 ∖ {0}) ∈
Fin) |
4 | 1, 2, 3 | sylancl 693 |
. . . 4
⊢ (𝐹 ∈ dom ∫1
→ (ran 𝐹 ∖ {0})
∈ Fin) |
5 | 4 | adantr 480 |
. . 3
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → (ran 𝐹 ∖ {0}) ∈ Fin) |
6 | | i1ff 23249 |
. . . . . . . 8
⊢ (𝐹 ∈ dom ∫1
→ 𝐹:ℝ⟶ℝ) |
7 | 6 | adantr 480 |
. . . . . . 7
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → 𝐹:ℝ⟶ℝ) |
8 | | frn 5966 |
. . . . . . 7
⊢ (𝐹:ℝ⟶ℝ →
ran 𝐹 ⊆
ℝ) |
9 | 7, 8 | syl 17 |
. . . . . 6
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → ran 𝐹 ⊆ ℝ) |
10 | 9 | ssdifssd 3710 |
. . . . 5
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → (ran 𝐹 ∖ {0}) ⊆
ℝ) |
11 | 10 | sselda 3568 |
. . . 4
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → 𝑥 ∈ ℝ) |
12 | | i1fima2sn 23253 |
. . . . 5
⊢ ((𝐹 ∈ dom ∫1
∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) →
(vol‘(◡𝐹 “ {𝑥})) ∈ ℝ) |
13 | 12 | adantlr 747 |
. . . 4
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ) |
14 | 11, 13 | remulcld 9949 |
. . 3
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (𝑥 · (vol‘(◡𝐹 “ {𝑥}))) ∈ ℝ) |
15 | | eldifi 3694 |
. . . . 5
⊢ (𝑥 ∈ (ran 𝐹 ∖ {0}) → 𝑥 ∈ ran 𝐹) |
16 | | 0cn 9911 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℂ |
17 | | fnconstg 6006 |
. . . . . . . . . . . 12
⊢ (0 ∈
ℂ → (ℂ × {0}) Fn ℂ) |
18 | 16, 17 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (ℂ
× {0}) Fn ℂ |
19 | | df-0p 23243 |
. . . . . . . . . . . 12
⊢
0𝑝 = (ℂ × {0}) |
20 | 19 | fneq1i 5899 |
. . . . . . . . . . 11
⊢
(0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn
ℂ) |
21 | 18, 20 | mpbir 220 |
. . . . . . . . . 10
⊢
0𝑝 Fn ℂ |
22 | 21 | a1i 11 |
. . . . . . . . 9
⊢ (𝐹 ∈ dom ∫1
→ 0𝑝 Fn ℂ) |
23 | | ffn 5958 |
. . . . . . . . . 10
⊢ (𝐹:ℝ⟶ℝ →
𝐹 Fn
ℝ) |
24 | 6, 23 | syl 17 |
. . . . . . . . 9
⊢ (𝐹 ∈ dom ∫1
→ 𝐹 Fn
ℝ) |
25 | | cnex 9896 |
. . . . . . . . . 10
⊢ ℂ
∈ V |
26 | 25 | a1i 11 |
. . . . . . . . 9
⊢ (𝐹 ∈ dom ∫1
→ ℂ ∈ V) |
27 | | reex 9906 |
. . . . . . . . . 10
⊢ ℝ
∈ V |
28 | 27 | a1i 11 |
. . . . . . . . 9
⊢ (𝐹 ∈ dom ∫1
→ ℝ ∈ V) |
29 | | ax-resscn 9872 |
. . . . . . . . . 10
⊢ ℝ
⊆ ℂ |
30 | | sseqin2 3779 |
. . . . . . . . . 10
⊢ (ℝ
⊆ ℂ ↔ (ℂ ∩ ℝ) = ℝ) |
31 | 29, 30 | mpbi 219 |
. . . . . . . . 9
⊢ (ℂ
∩ ℝ) = ℝ |
32 | | 0pval 23244 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℂ →
(0𝑝‘𝑦) = 0) |
33 | 32 | adantl 481 |
. . . . . . . . 9
⊢ ((𝐹 ∈ dom ∫1
∧ 𝑦 ∈ ℂ)
→ (0𝑝‘𝑦) = 0) |
34 | | eqidd 2611 |
. . . . . . . . 9
⊢ ((𝐹 ∈ dom ∫1
∧ 𝑦 ∈ ℝ)
→ (𝐹‘𝑦) = (𝐹‘𝑦)) |
35 | 22, 24, 26, 28, 31, 33, 34 | ofrfval 6803 |
. . . . . . . 8
⊢ (𝐹 ∈ dom ∫1
→ (0𝑝 ∘𝑟 ≤ 𝐹 ↔ ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦))) |
36 | 35 | biimpa 500 |
. . . . . . 7
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦)) |
37 | 24 | adantr 480 |
. . . . . . . 8
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → 𝐹 Fn ℝ) |
38 | | breq2 4587 |
. . . . . . . . 9
⊢ (𝑥 = (𝐹‘𝑦) → (0 ≤ 𝑥 ↔ 0 ≤ (𝐹‘𝑦))) |
39 | 38 | ralrn 6270 |
. . . . . . . 8
⊢ (𝐹 Fn ℝ →
(∀𝑥 ∈ ran 𝐹0 ≤ 𝑥 ↔ ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦))) |
40 | 37, 39 | syl 17 |
. . . . . . 7
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → (∀𝑥 ∈ ran 𝐹0 ≤ 𝑥 ↔ ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦))) |
41 | 36, 40 | mpbird 246 |
. . . . . 6
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → ∀𝑥 ∈ ran 𝐹0 ≤ 𝑥) |
42 | 41 | r19.21bi 2916 |
. . . . 5
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ ran 𝐹) → 0 ≤ 𝑥) |
43 | 15, 42 | sylan2 490 |
. . . 4
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → 0 ≤ 𝑥) |
44 | | i1fima 23251 |
. . . . . 6
⊢ (𝐹 ∈ dom ∫1
→ (◡𝐹 “ {𝑥}) ∈ dom vol) |
45 | 44 | ad2antrr 758 |
. . . . 5
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑥}) ∈ dom vol) |
46 | | mblss 23106 |
. . . . . . 7
⊢ ((◡𝐹 “ {𝑥}) ∈ dom vol → (◡𝐹 “ {𝑥}) ⊆ ℝ) |
47 | | ovolge0 23056 |
. . . . . . 7
⊢ ((◡𝐹 “ {𝑥}) ⊆ ℝ → 0 ≤
(vol*‘(◡𝐹 “ {𝑥}))) |
48 | 46, 47 | syl 17 |
. . . . . 6
⊢ ((◡𝐹 “ {𝑥}) ∈ dom vol → 0 ≤
(vol*‘(◡𝐹 “ {𝑥}))) |
49 | | mblvol 23105 |
. . . . . 6
⊢ ((◡𝐹 “ {𝑥}) ∈ dom vol → (vol‘(◡𝐹 “ {𝑥})) = (vol*‘(◡𝐹 “ {𝑥}))) |
50 | 48, 49 | breqtrrd 4611 |
. . . . 5
⊢ ((◡𝐹 “ {𝑥}) ∈ dom vol → 0 ≤
(vol‘(◡𝐹 “ {𝑥}))) |
51 | 45, 50 | syl 17 |
. . . 4
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → 0 ≤
(vol‘(◡𝐹 “ {𝑥}))) |
52 | 11, 13, 43, 51 | mulge0d 10483 |
. . 3
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → 0 ≤ (𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
53 | 5, 14, 52 | fsumge0 14368 |
. 2
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → 0 ≤ Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
54 | | itg1val 23256 |
. . 3
⊢ (𝐹 ∈ dom ∫1
→ (∫1‘𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
55 | 54 | adantr 480 |
. 2
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → (∫1‘𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
56 | 53, 55 | breqtrrd 4611 |
1
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → 0 ≤
(∫1‘𝐹)) |