Proof of Theorem itg1val2
Step | Hyp | Ref
| Expression |
1 | | itg1val 24435 |
. . 3
⊢ (𝐹 ∈ dom ∫1
→ (∫1‘𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
2 | 1 | adantr 484 |
. 2
⊢ ((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) →
(∫1‘𝐹)
= Σ𝑥 ∈ (ran
𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
3 | | simpr2 1196 |
. . 3
⊢ ((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) → (ran
𝐹 ∖ {0}) ⊆
𝐴) |
4 | 3 | sselda 3877 |
. . . 4
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → 𝑥 ∈ 𝐴) |
5 | | simpr3 1197 |
. . . . . . . 8
⊢ ((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) →
𝐴 ⊆ (ℝ ∖
{0})) |
6 | 5 | sselda 3877 |
. . . . . . 7
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (ℝ ∖
{0})) |
7 | | eldifi 4017 |
. . . . . . 7
⊢ (𝑥 ∈ (ℝ ∖ {0})
→ 𝑥 ∈
ℝ) |
8 | 6, 7 | syl 17 |
. . . . . 6
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
9 | | i1fima2sn 24432 |
. . . . . . . 8
⊢ ((𝐹 ∈ dom ∫1
∧ 𝑥 ∈ (ℝ
∖ {0})) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ) |
10 | 9 | adantlr 715 |
. . . . . . 7
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (ℝ ∖ {0}))
→ (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ) |
11 | 6, 10 | syldan 594 |
. . . . . 6
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ 𝐴) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ) |
12 | 8, 11 | remulcld 10749 |
. . . . 5
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ 𝐴) → (𝑥 · (vol‘(◡𝐹 “ {𝑥}))) ∈ ℝ) |
13 | 12 | recnd 10747 |
. . . 4
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ 𝐴) → (𝑥 · (vol‘(◡𝐹 “ {𝑥}))) ∈ ℂ) |
14 | 4, 13 | syldan 594 |
. . 3
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (𝑥 · (vol‘(◡𝐹 “ {𝑥}))) ∈ ℂ) |
15 | | i1ff 24428 |
. . . . . . . . . 10
⊢ (𝐹 ∈ dom ∫1
→ 𝐹:ℝ⟶ℝ) |
16 | 15 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → 𝐹:ℝ⟶ℝ) |
17 | | ffrn 6518 |
. . . . . . . . 9
⊢ (𝐹:ℝ⟶ℝ →
𝐹:ℝ⟶ran 𝐹) |
18 | 16, 17 | syl 17 |
. . . . . . . 8
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → 𝐹:ℝ⟶ran 𝐹) |
19 | | eldifn 4018 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0})) → ¬ 𝑥 ∈ (ran 𝐹 ∖ {0})) |
20 | 19 | adantl 485 |
. . . . . . . . . 10
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → ¬ 𝑥 ∈ (ran 𝐹 ∖ {0})) |
21 | | eldif 3853 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (ran 𝐹 ∖ {0}) ↔ (𝑥 ∈ ran 𝐹 ∧ ¬ 𝑥 ∈ {0})) |
22 | | simplr3 1218 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → 𝐴 ⊆ (ℝ ∖
{0})) |
23 | 22 | ssdifssd 4033 |
. . . . . . . . . . . . . 14
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → (𝐴 ∖ (ran 𝐹 ∖ {0})) ⊆ (ℝ ∖
{0})) |
24 | | simpr 488 |
. . . . . . . . . . . . . 14
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) |
25 | 23, 24 | sseldd 3878 |
. . . . . . . . . . . . 13
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → 𝑥 ∈ (ℝ ∖
{0})) |
26 | | eldifn 4018 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (ℝ ∖ {0})
→ ¬ 𝑥 ∈
{0}) |
27 | 25, 26 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → ¬ 𝑥 ∈ {0}) |
28 | 27 | biantrud 535 |
. . . . . . . . . . 11
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → (𝑥 ∈ ran 𝐹 ↔ (𝑥 ∈ ran 𝐹 ∧ ¬ 𝑥 ∈ {0}))) |
29 | 21, 28 | bitr4id 293 |
. . . . . . . . . 10
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → (𝑥 ∈ (ran 𝐹 ∖ {0}) ↔ 𝑥 ∈ ran 𝐹)) |
30 | 20, 29 | mtbid 327 |
. . . . . . . . 9
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → ¬ 𝑥 ∈ ran 𝐹) |
31 | | disjsn 4602 |
. . . . . . . . 9
⊢ ((ran
𝐹 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝐹) |
32 | 30, 31 | sylibr 237 |
. . . . . . . 8
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → (ran 𝐹 ∩ {𝑥}) = ∅) |
33 | | fimacnvdisj 6556 |
. . . . . . . 8
⊢ ((𝐹:ℝ⟶ran 𝐹 ∧ (ran 𝐹 ∩ {𝑥}) = ∅) → (◡𝐹 “ {𝑥}) = ∅) |
34 | 18, 32, 33 | syl2anc 587 |
. . . . . . 7
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → (◡𝐹 “ {𝑥}) = ∅) |
35 | 34 | fveq2d 6678 |
. . . . . 6
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → (vol‘(◡𝐹 “ {𝑥})) = (vol‘∅)) |
36 | | 0mbl 24291 |
. . . . . . . 8
⊢ ∅
∈ dom vol |
37 | | mblvol 24282 |
. . . . . . . 8
⊢ (∅
∈ dom vol → (vol‘∅) =
(vol*‘∅)) |
38 | 36, 37 | ax-mp 5 |
. . . . . . 7
⊢
(vol‘∅) = (vol*‘∅) |
39 | | ovol0 24245 |
. . . . . . 7
⊢
(vol*‘∅) = 0 |
40 | 38, 39 | eqtri 2761 |
. . . . . 6
⊢
(vol‘∅) = 0 |
41 | 35, 40 | eqtrdi 2789 |
. . . . 5
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → (vol‘(◡𝐹 “ {𝑥})) = 0) |
42 | 41 | oveq2d 7186 |
. . . 4
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → (𝑥 · (vol‘(◡𝐹 “ {𝑥}))) = (𝑥 · 0)) |
43 | | eldifi 4017 |
. . . . . . 7
⊢ (𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0})) → 𝑥 ∈ 𝐴) |
44 | 43, 8 | sylan2 596 |
. . . . . 6
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → 𝑥 ∈ ℝ) |
45 | 44 | recnd 10747 |
. . . . 5
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → 𝑥 ∈ ℂ) |
46 | 45 | mul01d 10917 |
. . . 4
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → (𝑥 · 0) = 0) |
47 | 42, 46 | eqtrd 2773 |
. . 3
⊢ (((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) ∧ 𝑥 ∈ (𝐴 ∖ (ran 𝐹 ∖ {0}))) → (𝑥 · (vol‘(◡𝐹 “ {𝑥}))) = 0) |
48 | | simpr1 1195 |
. . 3
⊢ ((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) →
𝐴 ∈
Fin) |
49 | 3, 14, 47, 48 | fsumss 15175 |
. 2
⊢ ((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) →
Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥}))) = Σ𝑥 ∈ 𝐴 (𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
50 | 2, 49 | eqtrd 2773 |
1
⊢ ((𝐹 ∈ dom ∫1
∧ (𝐴 ∈ Fin ∧
(ran 𝐹 ∖ {0}) ⊆
𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) →
(∫1‘𝐹)
= Σ𝑥 ∈ 𝐴 (𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |