Step | Hyp | Ref
| Expression |
1 | | fzfid 12634 |
. . . . . 6
⊢ (𝜑 → (𝑀...𝑚) ∈ Fin) |
2 | | simpl 472 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑚)) → 𝜑) |
3 | | elfzuz 12209 |
. . . . . . . . 9
⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ (ℤ≥‘𝑀)) |
4 | | fsumsermpt.z |
. . . . . . . . 9
⊢ 𝑍 =
(ℤ≥‘𝑀) |
5 | 3, 4 | syl6eleqr 2699 |
. . . . . . . 8
⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ 𝑍) |
6 | 5 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑚)) → 𝑘 ∈ 𝑍) |
7 | | fsumsermpt.a |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
8 | 2, 6, 7 | syl2anc 691 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑚)) → 𝐴 ∈ ℂ) |
9 | 1, 8 | fsumcl 14311 |
. . . . 5
⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ) |
10 | 9 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ) |
11 | 10 | ralrimiva 2949 |
. . 3
⊢ (𝜑 → ∀𝑚 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ) |
12 | | fsumsermpt.f |
. . . . 5
⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐴) |
13 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑛 = 𝑚 → (𝑀...𝑛) = (𝑀...𝑚)) |
14 | 13 | sumeq1d 14279 |
. . . . . 6
⊢ (𝑛 = 𝑚 → Σ𝑘 ∈ (𝑀...𝑛)𝐴 = Σ𝑘 ∈ (𝑀...𝑚)𝐴) |
15 | 14 | cbvmptv 4678 |
. . . . 5
⊢ (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐴) = (𝑚 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑚)𝐴) |
16 | 12, 15 | eqtri 2632 |
. . . 4
⊢ 𝐹 = (𝑚 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑚)𝐴) |
17 | 16 | fnmpt 5933 |
. . 3
⊢
(∀𝑚 ∈
𝑍 Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ → 𝐹 Fn 𝑍) |
18 | 11, 17 | syl 17 |
. 2
⊢ (𝜑 → 𝐹 Fn 𝑍) |
19 | | fsumsermpt.m |
. . . . 5
⊢ (𝜑 → 𝑀 ∈ ℤ) |
20 | | simpr 476 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) |
21 | | nfv 1830 |
. . . . . . . . 9
⊢
Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
22 | | nfcv 2751 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘𝑗 |
23 | 22 | nfcsb1 3514 |
. . . . . . . . . 10
⊢
Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 |
24 | 23 | nfel1 2765 |
. . . . . . . . 9
⊢
Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ |
25 | 21, 24 | nfim 1813 |
. . . . . . . 8
⊢
Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) |
26 | | eleq1 2676 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) |
27 | 26 | anbi2d 736 |
. . . . . . . . 9
⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
28 | | csbeq1a 3508 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴) |
29 | 28 | eleq1d 2672 |
. . . . . . . . 9
⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℂ ↔ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ)) |
30 | 27, 29 | imbi12d 333 |
. . . . . . . 8
⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ))) |
31 | 25, 30, 7 | chvar 2250 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) |
32 | | eqid 2610 |
. . . . . . . 8
⊢ (𝑘 ∈ 𝑍 ↦ 𝐴) = (𝑘 ∈ 𝑍 ↦ 𝐴) |
33 | 22, 23, 28, 32 | fvmptf 6209 |
. . . . . . 7
⊢ ((𝑗 ∈ 𝑍 ∧ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
34 | 20, 31, 33 | syl2anc 691 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
35 | 34, 31 | eqeltrd 2688 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) ∈ ℂ) |
36 | | addcl 9897 |
. . . . . 6
⊢ ((𝑗 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑗 + 𝑥) ∈ ℂ) |
37 | 36 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ (𝑗 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑗 + 𝑥) ∈ ℂ) |
38 | 4, 19, 35, 37 | seqf 12684 |
. . . 4
⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)):𝑍⟶ℂ) |
39 | | ffn 5958 |
. . . 4
⊢ (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)):𝑍⟶ℂ → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) Fn 𝑍) |
40 | 38, 39 | syl 17 |
. . 3
⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) Fn 𝑍) |
41 | | fsumsermpt.g |
. . . . 5
⊢ 𝐺 = seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) |
42 | 41 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝐺 = seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))) |
43 | 42 | fneq1d 5895 |
. . 3
⊢ (𝜑 → (𝐺 Fn 𝑍 ↔ seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) Fn 𝑍)) |
44 | 40, 43 | mpbird 246 |
. 2
⊢ (𝜑 → 𝐺 Fn 𝑍) |
45 | | simpr 476 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍) |
46 | 16 | fvmpt2 6200 |
. . . . 5
⊢ ((𝑚 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ) → (𝐹‘𝑚) = Σ𝑘 ∈ (𝑀...𝑚)𝐴) |
47 | 45, 10, 46 | syl2anc 691 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = Σ𝑘 ∈ (𝑀...𝑚)𝐴) |
48 | | nfcv 2751 |
. . . . . 6
⊢
Ⅎ𝑗(𝑀...𝑚) |
49 | | nfcv 2751 |
. . . . . 6
⊢
Ⅎ𝑘(𝑀...𝑚) |
50 | | nfcv 2751 |
. . . . . 6
⊢
Ⅎ𝑗𝐴 |
51 | 28, 48, 49, 50, 23 | cbvsum 14273 |
. . . . 5
⊢
Σ𝑘 ∈
(𝑀...𝑚)𝐴 = Σ𝑗 ∈ (𝑀...𝑚)⦋𝑗 / 𝑘⦌𝐴 |
52 | 51 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑚)𝐴 = Σ𝑗 ∈ (𝑀...𝑚)⦋𝑗 / 𝑘⦌𝐴) |
53 | 47, 52 | eqtrd 2644 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = Σ𝑗 ∈ (𝑀...𝑚)⦋𝑗 / 𝑘⦌𝐴) |
54 | | simpl 472 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑚)) → 𝜑) |
55 | | elfzuz 12209 |
. . . . . . . 8
⊢ (𝑗 ∈ (𝑀...𝑚) → 𝑗 ∈ (ℤ≥‘𝑀)) |
56 | 55, 4 | syl6eleqr 2699 |
. . . . . . 7
⊢ (𝑗 ∈ (𝑀...𝑚) → 𝑗 ∈ 𝑍) |
57 | 56 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑚)) → 𝑗 ∈ 𝑍) |
58 | 54, 57, 34 | syl2anc 691 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑚)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
59 | 58 | adantlr 747 |
. . . 4
⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑗 ∈ (𝑀...𝑚)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
60 | | id 22 |
. . . . . 6
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ 𝑍) |
61 | 60, 4 | syl6eleq 2698 |
. . . . 5
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (ℤ≥‘𝑀)) |
62 | 61 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ (ℤ≥‘𝑀)) |
63 | 54, 57, 31 | syl2anc 691 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑚)) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) |
64 | 63 | adantlr 747 |
. . . 4
⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑗 ∈ (𝑀...𝑚)) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) |
65 | 59, 62, 64 | fsumser 14308 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → Σ𝑗 ∈ (𝑀...𝑚)⦋𝑗 / 𝑘⦌𝐴 = (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑚)) |
66 | 41 | fveq1i 6104 |
. . . . 5
⊢ (𝐺‘𝑚) = (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑚) |
67 | 66 | eqcomi 2619 |
. . . 4
⊢ (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑚) = (𝐺‘𝑚) |
68 | 67 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑚) = (𝐺‘𝑚)) |
69 | 53, 65, 68 | 3eqtrd 2648 |
. 2
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = (𝐺‘𝑚)) |
70 | 18, 44, 69 | eqfnfvd 6222 |
1
⊢ (𝜑 → 𝐹 = 𝐺) |