Proof of Theorem indsumin
Step | Hyp | Ref
| Expression |
1 | | inindif 29353 |
. . . 4
⊢ ((𝐴 ∩ 𝐵) ∩ (𝐴 ∖ 𝐵)) = ∅ |
2 | 1 | a1i 11 |
. . 3
⊢ (𝜑 → ((𝐴 ∩ 𝐵) ∩ (𝐴 ∖ 𝐵)) = ∅) |
3 | | inundif 4046 |
. . . . 5
⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
4 | 3 | eqcomi 2631 |
. . . 4
⊢ 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) |
5 | 4 | a1i 11 |
. . 3
⊢ (𝜑 → 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) |
6 | | indsumin.2 |
. . 3
⊢ (𝜑 → 𝐴 ∈ Fin) |
7 | | pr01ssre 29570 |
. . . . . 6
⊢ {0, 1}
⊆ ℝ |
8 | | ax-resscn 9993 |
. . . . . 6
⊢ ℝ
⊆ ℂ |
9 | 7, 8 | sstri 3612 |
. . . . 5
⊢ {0, 1}
⊆ ℂ |
10 | | indsumin.1 |
. . . . . . . 8
⊢ (𝜑 → 𝑂 ∈ 𝑉) |
11 | | indsumin.4 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ⊆ 𝑂) |
12 | | indf 30077 |
. . . . . . . 8
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐵 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐵):𝑂⟶{0, 1}) |
13 | 10, 11, 12 | syl2anc 693 |
. . . . . . 7
⊢ (𝜑 → ((𝟭‘𝑂)‘𝐵):𝑂⟶{0, 1}) |
14 | 13 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝟭‘𝑂)‘𝐵):𝑂⟶{0, 1}) |
15 | | indsumin.3 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ⊆ 𝑂) |
16 | 15 | sselda 3603 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑂) |
17 | 14, 16 | ffvelrnd 6360 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (((𝟭‘𝑂)‘𝐵)‘𝑘) ∈ {0, 1}) |
18 | 9, 17 | sseldi 3601 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (((𝟭‘𝑂)‘𝐵)‘𝑘) ∈ ℂ) |
19 | | indsumin.5 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
20 | 18, 19 | mulcld 10060 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) ∈ ℂ) |
21 | 2, 5, 6, 20 | fsumsplit 14471 |
. 2
⊢ (𝜑 → Σ𝑘 ∈ 𝐴 ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = (Σ𝑘 ∈ (𝐴 ∩ 𝐵)((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) + Σ𝑘 ∈ (𝐴 ∖ 𝐵)((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶))) |
22 | 10 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → 𝑂 ∈ 𝑉) |
23 | 11 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → 𝐵 ⊆ 𝑂) |
24 | | inss2 3834 |
. . . . . . . . 9
⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 |
25 | 24 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵) |
26 | 25 | sselda 3603 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → 𝑘 ∈ 𝐵) |
27 | | ind1 30079 |
. . . . . . 7
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐵 ⊆ 𝑂 ∧ 𝑘 ∈ 𝐵) → (((𝟭‘𝑂)‘𝐵)‘𝑘) = 1) |
28 | 22, 23, 26, 27 | syl3anc 1326 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → (((𝟭‘𝑂)‘𝐵)‘𝑘) = 1) |
29 | 28 | oveq1d 6665 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = (1 · 𝐶)) |
30 | | inss1 3833 |
. . . . . . . . 9
⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 |
31 | 30 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐴) |
32 | 31 | sselda 3603 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → 𝑘 ∈ 𝐴) |
33 | 32, 19 | syldan 487 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → 𝐶 ∈ ℂ) |
34 | 33 | mulid2d 10058 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → (1 · 𝐶) = 𝐶) |
35 | 29, 34 | eqtrd 2656 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = 𝐶) |
36 | 35 | sumeq2dv 14433 |
. . 3
⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∩ 𝐵)((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶) |
37 | 10 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝑂 ∈ 𝑉) |
38 | 11 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝐵 ⊆ 𝑂) |
39 | 15 | ssdifd 3746 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ (𝑂 ∖ 𝐵)) |
40 | 39 | sselda 3603 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝑘 ∈ (𝑂 ∖ 𝐵)) |
41 | | ind0 30080 |
. . . . . . . 8
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐵 ⊆ 𝑂 ∧ 𝑘 ∈ (𝑂 ∖ 𝐵)) → (((𝟭‘𝑂)‘𝐵)‘𝑘) = 0) |
42 | 37, 38, 40, 41 | syl3anc 1326 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → (((𝟭‘𝑂)‘𝐵)‘𝑘) = 0) |
43 | 42 | oveq1d 6665 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = (0 · 𝐶)) |
44 | | difssd 3738 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴) |
45 | 44 | sselda 3603 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝑘 ∈ 𝐴) |
46 | 45, 19 | syldan 487 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝐶 ∈ ℂ) |
47 | 46 | mul02d 10234 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → (0 · 𝐶) = 0) |
48 | 43, 47 | eqtrd 2656 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = 0) |
49 | 48 | sumeq2dv 14433 |
. . . 4
⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∖ 𝐵)((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = Σ𝑘 ∈ (𝐴 ∖ 𝐵)0) |
50 | | diffi 8192 |
. . . . . 6
⊢ (𝐴 ∈ Fin → (𝐴 ∖ 𝐵) ∈ Fin) |
51 | 6, 50 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ Fin) |
52 | | sumz 14453 |
. . . . . 6
⊢ (((𝐴 ∖ 𝐵) ⊆ (ℤ≥‘0)
∨ (𝐴 ∖ 𝐵) ∈ Fin) →
Σ𝑘 ∈ (𝐴 ∖ 𝐵)0 = 0) |
53 | 52 | olcs 410 |
. . . . 5
⊢ ((𝐴 ∖ 𝐵) ∈ Fin → Σ𝑘 ∈ (𝐴 ∖ 𝐵)0 = 0) |
54 | 51, 53 | syl 17 |
. . . 4
⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∖ 𝐵)0 = 0) |
55 | 49, 54 | eqtrd 2656 |
. . 3
⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∖ 𝐵)((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = 0) |
56 | 36, 55 | oveq12d 6668 |
. 2
⊢ (𝜑 → (Σ𝑘 ∈ (𝐴 ∩ 𝐵)((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) + Σ𝑘 ∈ (𝐴 ∖ 𝐵)((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶)) = (Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶 + 0)) |
57 | | infi 8184 |
. . . . 5
⊢ (𝐴 ∈ Fin → (𝐴 ∩ 𝐵) ∈ Fin) |
58 | 6, 57 | syl 17 |
. . . 4
⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ Fin) |
59 | 58, 33 | fsumcl 14464 |
. . 3
⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶 ∈ ℂ) |
60 | 59 | addid1d 10236 |
. 2
⊢ (𝜑 → (Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶 + 0) = Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶) |
61 | 21, 56, 60 | 3eqtrd 2660 |
1
⊢ (𝜑 → Σ𝑘 ∈ 𝐴 ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶) |