Step | Hyp | Ref
| Expression |
1 | | itg10a.1 |
. . 3
⊢ (𝜑 → 𝐹 ∈ dom
∫1) |
2 | | itg1val 23256 |
. . 3
⊢ (𝐹 ∈ dom ∫1
→ (∫1‘𝐹) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘})))) |
3 | 1, 2 | syl 17 |
. 2
⊢ (𝜑 →
(∫1‘𝐹)
= Σ𝑘 ∈ (ran
𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘})))) |
4 | | i1ff 23249 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹 ∈ dom ∫1
→ 𝐹:ℝ⟶ℝ) |
5 | 1, 4 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
6 | | ffn 5958 |
. . . . . . . . . . . . . . 15
⊢ (𝐹:ℝ⟶ℝ →
𝐹 Fn
ℝ) |
7 | 5, 6 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐹 Fn ℝ) |
8 | 7 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐹 Fn ℝ) |
9 | | fniniseg 6246 |
. . . . . . . . . . . . 13
⊢ (𝐹 Fn ℝ → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘))) |
10 | 8, 9 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘))) |
11 | | eldifsni 4261 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (ran 𝐹 ∖ {0}) → 𝑘 ≠ 0) |
12 | 11 | ad2antlr 759 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → 𝑘 ≠ 0) |
13 | | simprl 790 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → 𝑥 ∈ ℝ) |
14 | | eldif 3550 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ (ℝ ∖ 𝐴) ↔ (𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ 𝐴)) |
15 | | simplrr 797 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = 𝑘) |
16 | | simpll 786 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → 𝜑) |
17 | | itg10a.4 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = 0) |
18 | 16, 17 | sylan 487 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = 0) |
19 | 15, 18 | eqtr3d 2646 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 𝑘 = 0) |
20 | 19 | ex 449 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑘 = 0)) |
21 | 14, 20 | syl5bir 232 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → ((𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ 𝐴) → 𝑘 = 0)) |
22 | 13, 21 | mpand 707 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (¬ 𝑥 ∈ 𝐴 → 𝑘 = 0)) |
23 | 22 | necon1ad 2799 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (𝑘 ≠ 0 → 𝑥 ∈ 𝐴)) |
24 | 12, 23 | mpd 15 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → 𝑥 ∈ 𝐴) |
25 | 24 | ex 449 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘) → 𝑥 ∈ 𝐴)) |
26 | 10, 25 | sylbid 229 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (◡𝐹 “ {𝑘}) → 𝑥 ∈ 𝐴)) |
27 | 26 | ssrdv 3574 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) ⊆ 𝐴) |
28 | | itg10a.2 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
29 | 28 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐴 ⊆ ℝ) |
30 | 27, 29 | sstrd 3578 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) ⊆ ℝ) |
31 | | itg10a.3 |
. . . . . . . . . . 11
⊢ (𝜑 → (vol*‘𝐴) = 0) |
32 | 31 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol*‘𝐴) = 0) |
33 | | ovolssnul 23062 |
. . . . . . . . . 10
⊢ (((◡𝐹 “ {𝑘}) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → (vol*‘(◡𝐹 “ {𝑘})) = 0) |
34 | 27, 29, 32, 33 | syl3anc 1318 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol*‘(◡𝐹 “ {𝑘})) = 0) |
35 | | nulmbl 23110 |
. . . . . . . . 9
⊢ (((◡𝐹 “ {𝑘}) ⊆ ℝ ∧ (vol*‘(◡𝐹 “ {𝑘})) = 0) → (◡𝐹 “ {𝑘}) ∈ dom vol) |
36 | 30, 34, 35 | syl2anc 691 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) ∈ dom vol) |
37 | | mblvol 23105 |
. . . . . . . 8
⊢ ((◡𝐹 “ {𝑘}) ∈ dom vol → (vol‘(◡𝐹 “ {𝑘})) = (vol*‘(◡𝐹 “ {𝑘}))) |
38 | 36, 37 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) = (vol*‘(◡𝐹 “ {𝑘}))) |
39 | 38, 34 | eqtrd 2644 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) = 0) |
40 | 39 | oveq2d 6565 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · (vol‘(◡𝐹 “ {𝑘}))) = (𝑘 · 0)) |
41 | | frn 5966 |
. . . . . . . . . 10
⊢ (𝐹:ℝ⟶ℝ →
ran 𝐹 ⊆
ℝ) |
42 | 5, 41 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
43 | 42 | ssdifssd 3710 |
. . . . . . . 8
⊢ (𝜑 → (ran 𝐹 ∖ {0}) ⊆
ℝ) |
44 | 43 | sselda 3568 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℝ) |
45 | 44 | recnd 9947 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℂ) |
46 | 45 | mul01d 10114 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · 0) = 0) |
47 | 40, 46 | eqtrd 2644 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · (vol‘(◡𝐹 “ {𝑘}))) = 0) |
48 | 47 | sumeq2dv 14281 |
. . 3
⊢ (𝜑 → Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘}))) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})0) |
49 | | i1frn 23250 |
. . . . . . 7
⊢ (𝐹 ∈ dom ∫1
→ ran 𝐹 ∈
Fin) |
50 | 1, 49 | syl 17 |
. . . . . 6
⊢ (𝜑 → ran 𝐹 ∈ Fin) |
51 | | difss 3699 |
. . . . . 6
⊢ (ran
𝐹 ∖ {0}) ⊆ ran
𝐹 |
52 | | ssfi 8065 |
. . . . . 6
⊢ ((ran
𝐹 ∈ Fin ∧ (ran
𝐹 ∖ {0}) ⊆ ran
𝐹) → (ran 𝐹 ∖ {0}) ∈
Fin) |
53 | 50, 51, 52 | sylancl 693 |
. . . . 5
⊢ (𝜑 → (ran 𝐹 ∖ {0}) ∈ Fin) |
54 | 53 | olcd 407 |
. . . 4
⊢ (𝜑 → ((ran 𝐹 ∖ {0}) ⊆
(ℤ≥‘0) ∨ (ran 𝐹 ∖ {0}) ∈ Fin)) |
55 | | sumz 14300 |
. . . 4
⊢ (((ran
𝐹 ∖ {0}) ⊆
(ℤ≥‘0) ∨ (ran 𝐹 ∖ {0}) ∈ Fin) →
Σ𝑘 ∈ (ran 𝐹 ∖ {0})0 =
0) |
56 | 54, 55 | syl 17 |
. . 3
⊢ (𝜑 → Σ𝑘 ∈ (ran 𝐹 ∖ {0})0 = 0) |
57 | 48, 56 | eqtrd 2644 |
. 2
⊢ (𝜑 → Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘}))) = 0) |
58 | 3, 57 | eqtrd 2644 |
1
⊢ (𝜑 →
(∫1‘𝐹)
= 0) |