Step | Hyp | Ref
| Expression |
1 | | itg10a.1 |
. . 3
⊢ (𝜑 → 𝐹 ∈ dom
∫1) |
2 | | itg1val 24847 |
. . 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 24840 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹 ∈ dom ∫1
→ 𝐹:ℝ⟶ℝ) |
5 | 1, 4 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
6 | 5 | ffnd 6601 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐹 Fn ℝ) |
7 | 6 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐹 Fn ℝ) |
8 | | fniniseg 6937 |
. . . . . . . . . . . . 13
⊢ (𝐹 Fn ℝ → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘))) |
9 | 7, 8 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘))) |
10 | | eldifsni 4723 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (ran 𝐹 ∖ {0}) → 𝑘 ≠ 0) |
11 | 10 | ad2antlr 724 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → 𝑘 ≠ 0) |
12 | | simprl 768 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → 𝑥 ∈ ℝ) |
13 | | eldif 3897 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ (ℝ ∖ 𝐴) ↔ (𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ 𝐴)) |
14 | | simplrr 775 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = 𝑘) |
15 | | itg10a.4 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = 0) |
16 | 15 | ad4ant14 749 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = 0) |
17 | 14, 16 | eqtr3d 2780 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 𝑘 = 0) |
18 | 17 | ex 413 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑘 = 0)) |
19 | 13, 18 | syl5bir 242 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → ((𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ 𝐴) → 𝑘 = 0)) |
20 | 12, 19 | mpand 692 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (¬ 𝑥 ∈ 𝐴 → 𝑘 = 0)) |
21 | 20 | necon1ad 2960 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (𝑘 ≠ 0 → 𝑥 ∈ 𝐴)) |
22 | 11, 21 | mpd 15 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → 𝑥 ∈ 𝐴) |
23 | 22 | ex 413 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘) → 𝑥 ∈ 𝐴)) |
24 | 9, 23 | sylbid 239 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (◡𝐹 “ {𝑘}) → 𝑥 ∈ 𝐴)) |
25 | 24 | ssrdv 3927 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) ⊆ 𝐴) |
26 | | itg10a.2 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
27 | 26 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐴 ⊆ ℝ) |
28 | 25, 27 | sstrd 3931 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) ⊆ ℝ) |
29 | | itg10a.3 |
. . . . . . . . . . 11
⊢ (𝜑 → (vol*‘𝐴) = 0) |
30 | 29 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol*‘𝐴) = 0) |
31 | | ovolssnul 24651 |
. . . . . . . . . 10
⊢ (((◡𝐹 “ {𝑘}) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → (vol*‘(◡𝐹 “ {𝑘})) = 0) |
32 | 25, 27, 30, 31 | syl3anc 1370 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol*‘(◡𝐹 “ {𝑘})) = 0) |
33 | | nulmbl 24699 |
. . . . . . . . 9
⊢ (((◡𝐹 “ {𝑘}) ⊆ ℝ ∧ (vol*‘(◡𝐹 “ {𝑘})) = 0) → (◡𝐹 “ {𝑘}) ∈ dom vol) |
34 | 28, 32, 33 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) ∈ dom vol) |
35 | | mblvol 24694 |
. . . . . . . 8
⊢ ((◡𝐹 “ {𝑘}) ∈ dom vol → (vol‘(◡𝐹 “ {𝑘})) = (vol*‘(◡𝐹 “ {𝑘}))) |
36 | 34, 35 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) = (vol*‘(◡𝐹 “ {𝑘}))) |
37 | 36, 32 | eqtrd 2778 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) = 0) |
38 | 37 | oveq2d 7291 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · (vol‘(◡𝐹 “ {𝑘}))) = (𝑘 · 0)) |
39 | 5 | frnd 6608 |
. . . . . . . . 9
⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
40 | 39 | ssdifssd 4077 |
. . . . . . . 8
⊢ (𝜑 → (ran 𝐹 ∖ {0}) ⊆
ℝ) |
41 | 40 | sselda 3921 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℝ) |
42 | 41 | recnd 11003 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℂ) |
43 | 42 | mul01d 11174 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · 0) = 0) |
44 | 38, 43 | eqtrd 2778 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · (vol‘(◡𝐹 “ {𝑘}))) = 0) |
45 | 44 | sumeq2dv 15415 |
. . 3
⊢ (𝜑 → Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘}))) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})0) |
46 | | i1frn 24841 |
. . . . . . 7
⊢ (𝐹 ∈ dom ∫1
→ ran 𝐹 ∈
Fin) |
47 | 1, 46 | syl 17 |
. . . . . 6
⊢ (𝜑 → ran 𝐹 ∈ Fin) |
48 | | difss 4066 |
. . . . . 6
⊢ (ran
𝐹 ∖ {0}) ⊆ ran
𝐹 |
49 | | ssfi 8956 |
. . . . . 6
⊢ ((ran
𝐹 ∈ Fin ∧ (ran
𝐹 ∖ {0}) ⊆ ran
𝐹) → (ran 𝐹 ∖ {0}) ∈
Fin) |
50 | 47, 48, 49 | sylancl 586 |
. . . . 5
⊢ (𝜑 → (ran 𝐹 ∖ {0}) ∈ Fin) |
51 | 50 | olcd 871 |
. . . 4
⊢ (𝜑 → ((ran 𝐹 ∖ {0}) ⊆
(ℤ≥‘0) ∨ (ran 𝐹 ∖ {0}) ∈ Fin)) |
52 | | sumz 15434 |
. . . 4
⊢ (((ran
𝐹 ∖ {0}) ⊆
(ℤ≥‘0) ∨ (ran 𝐹 ∖ {0}) ∈ Fin) →
Σ𝑘 ∈ (ran 𝐹 ∖ {0})0 =
0) |
53 | 51, 52 | syl 17 |
. . 3
⊢ (𝜑 → Σ𝑘 ∈ (ran 𝐹 ∖ {0})0 = 0) |
54 | 45, 53 | eqtrd 2778 |
. 2
⊢ (𝜑 → Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘}))) = 0) |
55 | 3, 54 | eqtrd 2778 |
1
⊢ (𝜑 →
(∫1‘𝐹)
= 0) |