Step | Hyp | Ref
| Expression |
1 | | itg10a.1 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ dom
∫1) |
2 | | i1frn 23250 |
. . . . 5
⊢ (𝐹 ∈ dom ∫1
→ ran 𝐹 ∈
Fin) |
3 | 1, 2 | syl 17 |
. . . 4
⊢ (𝜑 → ran 𝐹 ∈ Fin) |
4 | | difss 3699 |
. . . 4
⊢ (ran
𝐹 ∖ {0}) ⊆ ran
𝐹 |
5 | | ssfi 8065 |
. . . 4
⊢ ((ran
𝐹 ∈ Fin ∧ (ran
𝐹 ∖ {0}) ⊆ ran
𝐹) → (ran 𝐹 ∖ {0}) ∈
Fin) |
6 | 3, 4, 5 | sylancl 693 |
. . 3
⊢ (𝜑 → (ran 𝐹 ∖ {0}) ∈ Fin) |
7 | | i1ff 23249 |
. . . . . . . 8
⊢ (𝐹 ∈ dom ∫1
→ 𝐹:ℝ⟶ℝ) |
8 | 1, 7 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
9 | | frn 5966 |
. . . . . . 7
⊢ (𝐹:ℝ⟶ℝ →
ran 𝐹 ⊆
ℝ) |
10 | 8, 9 | syl 17 |
. . . . . 6
⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
11 | 10 | ssdifssd 3710 |
. . . . 5
⊢ (𝜑 → (ran 𝐹 ∖ {0}) ⊆
ℝ) |
12 | 11 | sselda 3568 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℝ) |
13 | | i1fima2sn 23253 |
. . . . 5
⊢ ((𝐹 ∈ dom ∫1
∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) →
(vol‘(◡𝐹 “ {𝑘})) ∈ ℝ) |
14 | 1, 13 | sylan 487 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) ∈ ℝ) |
15 | 12, 14 | remulcld 9949 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · (vol‘(◡𝐹 “ {𝑘}))) ∈ ℝ) |
16 | | 0le0 10987 |
. . . . 5
⊢ 0 ≤
0 |
17 | | i1fima 23251 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ dom ∫1
→ (◡𝐹 “ {𝑘}) ∈ dom vol) |
18 | 1, 17 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (◡𝐹 “ {𝑘}) ∈ dom vol) |
19 | | mblvol 23105 |
. . . . . . . . . 10
⊢ ((◡𝐹 “ {𝑘}) ∈ dom vol → (vol‘(◡𝐹 “ {𝑘})) = (vol*‘(◡𝐹 “ {𝑘}))) |
20 | 18, 19 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (vol‘(◡𝐹 “ {𝑘})) = (vol*‘(◡𝐹 “ {𝑘}))) |
21 | 20 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑘 < 0) → (vol‘(◡𝐹 “ {𝑘})) = (vol*‘(◡𝐹 “ {𝑘}))) |
22 | | ffn 5958 |
. . . . . . . . . . . . . 14
⊢ (𝐹:ℝ⟶ℝ →
𝐹 Fn
ℝ) |
23 | 8, 22 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹 Fn ℝ) |
24 | | fniniseg 6246 |
. . . . . . . . . . . . 13
⊢ (𝐹 Fn ℝ → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘))) |
25 | 23, 24 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘))) |
26 | 25 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑘 < 0) → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘))) |
27 | | simprl 790 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → 𝑥 ∈ ℝ) |
28 | | eldif 3550 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (ℝ ∖ 𝐴) ↔ (𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ 𝐴)) |
29 | | itg1ge0a.4 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ (𝐹‘𝑥)) |
30 | 29 | ex 449 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑥 ∈ (ℝ ∖ 𝐴) → 0 ≤ (𝐹‘𝑥))) |
31 | 30 | ad2antrr 758 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (𝑥 ∈ (ℝ ∖ 𝐴) → 0 ≤ (𝐹‘𝑥))) |
32 | | simprr 792 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (𝐹‘𝑥) = 𝑘) |
33 | 32 | breq2d 4595 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (0 ≤ (𝐹‘𝑥) ↔ 0 ≤ 𝑘)) |
34 | | 0red 9920 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → 0 ∈ ℝ) |
35 | 12 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → 𝑘 ∈ ℝ) |
36 | 34, 35 | lenltd 10062 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (0 ≤ 𝑘 ↔ ¬ 𝑘 < 0)) |
37 | 33, 36 | bitrd 267 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (0 ≤ (𝐹‘𝑥) ↔ ¬ 𝑘 < 0)) |
38 | 31, 37 | sylibd 228 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑘 < 0)) |
39 | 28, 38 | syl5bir 232 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → ((𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ 𝐴) → ¬ 𝑘 < 0)) |
40 | 27, 39 | mpand 707 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (¬ 𝑥 ∈ 𝐴 → ¬ 𝑘 < 0)) |
41 | 40 | con4d 113 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (𝑘 < 0 → 𝑥 ∈ 𝐴)) |
42 | 41 | impancom 455 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑘 < 0) → ((𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘) → 𝑥 ∈ 𝐴)) |
43 | 26, 42 | sylbid 229 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑘 < 0) → (𝑥 ∈ (◡𝐹 “ {𝑘}) → 𝑥 ∈ 𝐴)) |
44 | 43 | ssrdv 3574 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑘 < 0) → (◡𝐹 “ {𝑘}) ⊆ 𝐴) |
45 | | itg10a.2 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
46 | 45 | ad2antrr 758 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑘 < 0) → 𝐴 ⊆ ℝ) |
47 | | itg10a.3 |
. . . . . . . . . 10
⊢ (𝜑 → (vol*‘𝐴) = 0) |
48 | 47 | ad2antrr 758 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑘 < 0) → (vol*‘𝐴) = 0) |
49 | | ovolssnul 23062 |
. . . . . . . . 9
⊢ (((◡𝐹 “ {𝑘}) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → (vol*‘(◡𝐹 “ {𝑘})) = 0) |
50 | 44, 46, 48, 49 | syl3anc 1318 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑘 < 0) → (vol*‘(◡𝐹 “ {𝑘})) = 0) |
51 | 21, 50 | eqtrd 2644 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑘 < 0) → (vol‘(◡𝐹 “ {𝑘})) = 0) |
52 | 51 | oveq2d 6565 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑘 < 0) → (𝑘 · (vol‘(◡𝐹 “ {𝑘}))) = (𝑘 · 0)) |
53 | 12 | recnd 9947 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℂ) |
54 | 53 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑘 < 0) → 𝑘 ∈ ℂ) |
55 | 54 | mul01d 10114 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑘 < 0) → (𝑘 · 0) = 0) |
56 | 52, 55 | eqtrd 2644 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑘 < 0) → (𝑘 · (vol‘(◡𝐹 “ {𝑘}))) = 0) |
57 | 16, 56 | syl5breqr 4621 |
. . . 4
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑘 < 0) → 0 ≤ (𝑘 · (vol‘(◡𝐹 “ {𝑘})))) |
58 | 12 | adantr 480 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 0 ≤ 𝑘) → 𝑘 ∈ ℝ) |
59 | 14 | adantr 480 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 0 ≤ 𝑘) → (vol‘(◡𝐹 “ {𝑘})) ∈ ℝ) |
60 | | simpr 476 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 0 ≤ 𝑘) → 0 ≤ 𝑘) |
61 | 18 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 0 ≤ 𝑘) → (◡𝐹 “ {𝑘}) ∈ dom vol) |
62 | | mblss 23106 |
. . . . . . . 8
⊢ ((◡𝐹 “ {𝑘}) ∈ dom vol → (◡𝐹 “ {𝑘}) ⊆ ℝ) |
63 | 61, 62 | syl 17 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 0 ≤ 𝑘) → (◡𝐹 “ {𝑘}) ⊆ ℝ) |
64 | | ovolge0 23056 |
. . . . . . 7
⊢ ((◡𝐹 “ {𝑘}) ⊆ ℝ → 0 ≤
(vol*‘(◡𝐹 “ {𝑘}))) |
65 | 63, 64 | syl 17 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 0 ≤ 𝑘) → 0 ≤
(vol*‘(◡𝐹 “ {𝑘}))) |
66 | 20 | ad2antrr 758 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 0 ≤ 𝑘) → (vol‘(◡𝐹 “ {𝑘})) = (vol*‘(◡𝐹 “ {𝑘}))) |
67 | 65, 66 | breqtrrd 4611 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 0 ≤ 𝑘) → 0 ≤
(vol‘(◡𝐹 “ {𝑘}))) |
68 | 58, 59, 60, 67 | mulge0d 10483 |
. . . 4
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 0 ≤ 𝑘) → 0 ≤ (𝑘 · (vol‘(◡𝐹 “ {𝑘})))) |
69 | | 0red 9920 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 0 ∈
ℝ) |
70 | 57, 68, 12, 69 | ltlecasei 10024 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 0 ≤ (𝑘 · (vol‘(◡𝐹 “ {𝑘})))) |
71 | 6, 15, 70 | fsumge0 14368 |
. 2
⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘})))) |
72 | | itg1val 23256 |
. . 3
⊢ (𝐹 ∈ dom ∫1
→ (∫1‘𝐹) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘})))) |
73 | 1, 72 | syl 17 |
. 2
⊢ (𝜑 →
(∫1‘𝐹)
= Σ𝑘 ∈ (ran
𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘})))) |
74 | 71, 73 | breqtrrd 4611 |
1
⊢ (𝜑 → 0 ≤
(∫1‘𝐹)) |