Step | Hyp | Ref
| Expression |
1 | | fzfid 13546 |
. . . . . 6
⊢ (𝜑 → (𝑀...𝑚) ∈ Fin) |
2 | | simpl 486 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑚)) → 𝜑) |
3 | | elfzuz 13108 |
. . . . . . . . 9
⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ (ℤ≥‘𝑀)) |
4 | | fsumsermpt.z |
. . . . . . . . 9
⊢ 𝑍 =
(ℤ≥‘𝑀) |
5 | 3, 4 | eleqtrrdi 2849 |
. . . . . . . 8
⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ 𝑍) |
6 | 5 | adantl 485 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑚)) → 𝑘 ∈ 𝑍) |
7 | | fsumsermpt.a |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
8 | 2, 6, 7 | syl2anc 587 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑚)) → 𝐴 ∈ ℂ) |
9 | 1, 8 | fsumcl 15297 |
. . . . 5
⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ) |
10 | 9 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ) |
11 | 10 | ralrimiva 3105 |
. . 3
⊢ (𝜑 → ∀𝑚 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ) |
12 | | fsumsermpt.f |
. . . . 5
⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐴) |
13 | | oveq2 7221 |
. . . . . . 7
⊢ (𝑛 = 𝑚 → (𝑀...𝑛) = (𝑀...𝑚)) |
14 | 13 | sumeq1d 15265 |
. . . . . 6
⊢ (𝑛 = 𝑚 → Σ𝑘 ∈ (𝑀...𝑛)𝐴 = Σ𝑘 ∈ (𝑀...𝑚)𝐴) |
15 | 14 | cbvmptv 5158 |
. . . . 5
⊢ (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐴) = (𝑚 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑚)𝐴) |
16 | 12, 15 | eqtri 2765 |
. . . 4
⊢ 𝐹 = (𝑚 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑚)𝐴) |
17 | 16 | fnmpt 6518 |
. . 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 488 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) |
21 | | nfv 1922 |
. . . . . . . . 9
⊢
Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
22 | | nfcv 2904 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘𝑗 |
23 | 22 | nfcsb1 3835 |
. . . . . . . . . 10
⊢
Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 |
24 | 23 | nfel1 2920 |
. . . . . . . . 9
⊢
Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ |
25 | 21, 24 | nfim 1904 |
. . . . . . . 8
⊢
Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) |
26 | | eleq1w 2820 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) |
27 | 26 | anbi2d 632 |
. . . . . . . . 9
⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
28 | | csbeq1a 3825 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴) |
29 | 28 | eleq1d 2822 |
. . . . . . . . 9
⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℂ ↔ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ)) |
30 | 27, 29 | imbi12d 348 |
. . . . . . . 8
⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ))) |
31 | 25, 30, 7 | chvarfv 2238 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) |
32 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑘 ∈ 𝑍 ↦ 𝐴) = (𝑘 ∈ 𝑍 ↦ 𝐴) |
33 | 22, 23, 28, 32 | fvmptf 6839 |
. . . . . . 7
⊢ ((𝑗 ∈ 𝑍 ∧ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
34 | 20, 31, 33 | syl2anc 587 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
35 | 34, 31 | eqeltrd 2838 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) ∈ ℂ) |
36 | | addcl 10811 |
. . . . . 6
⊢ ((𝑗 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑗 + 𝑥) ∈ ℂ) |
37 | 36 | adantl 485 |
. . . . 5
⊢ ((𝜑 ∧ (𝑗 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑗 + 𝑥) ∈ ℂ) |
38 | 4, 19, 35, 37 | seqf 13597 |
. . . 4
⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)):𝑍⟶ℂ) |
39 | 38 | ffnd 6546 |
. . 3
⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) Fn 𝑍) |
40 | | fsumsermpt.g |
. . . . 5
⊢ 𝐺 = seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) |
41 | 40 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝐺 = seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))) |
42 | 41 | fneq1d 6472 |
. . 3
⊢ (𝜑 → (𝐺 Fn 𝑍 ↔ seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) Fn 𝑍)) |
43 | 39, 42 | mpbird 260 |
. 2
⊢ (𝜑 → 𝐺 Fn 𝑍) |
44 | | simpr 488 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍) |
45 | 16 | fvmpt2 6829 |
. . . . 5
⊢ ((𝑚 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑚)𝐴 ∈ ℂ) → (𝐹‘𝑚) = Σ𝑘 ∈ (𝑀...𝑚)𝐴) |
46 | 44, 10, 45 | syl2anc 587 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = Σ𝑘 ∈ (𝑀...𝑚)𝐴) |
47 | | nfcv 2904 |
. . . . . 6
⊢
Ⅎ𝑗(𝑀...𝑚) |
48 | | nfcv 2904 |
. . . . . 6
⊢
Ⅎ𝑘(𝑀...𝑚) |
49 | | nfcv 2904 |
. . . . . 6
⊢
Ⅎ𝑗𝐴 |
50 | 28, 47, 48, 49, 23 | cbvsum 15259 |
. . . . 5
⊢
Σ𝑘 ∈
(𝑀...𝑚)𝐴 = Σ𝑗 ∈ (𝑀...𝑚)⦋𝑗 / 𝑘⦌𝐴 |
51 | 50 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑚)𝐴 = Σ𝑗 ∈ (𝑀...𝑚)⦋𝑗 / 𝑘⦌𝐴) |
52 | 46, 51 | eqtrd 2777 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = Σ𝑗 ∈ (𝑀...𝑚)⦋𝑗 / 𝑘⦌𝐴) |
53 | | simpl 486 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑚)) → 𝜑) |
54 | | elfzuz 13108 |
. . . . . . . 8
⊢ (𝑗 ∈ (𝑀...𝑚) → 𝑗 ∈ (ℤ≥‘𝑀)) |
55 | 54, 4 | eleqtrrdi 2849 |
. . . . . . 7
⊢ (𝑗 ∈ (𝑀...𝑚) → 𝑗 ∈ 𝑍) |
56 | 55 | adantl 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑚)) → 𝑗 ∈ 𝑍) |
57 | 53, 56, 34 | syl2anc 587 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑚)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
58 | 57 | adantlr 715 |
. . . 4
⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑗 ∈ (𝑀...𝑚)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
59 | | id 22 |
. . . . . 6
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ 𝑍) |
60 | 59, 4 | eleqtrdi 2848 |
. . . . 5
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (ℤ≥‘𝑀)) |
61 | 60 | adantl 485 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ (ℤ≥‘𝑀)) |
62 | 53, 56, 31 | syl2anc 587 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑚)) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) |
63 | 62 | adantlr 715 |
. . . 4
⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑗 ∈ (𝑀...𝑚)) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) |
64 | 58, 61, 63 | fsumser 15294 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → Σ𝑗 ∈ (𝑀...𝑚)⦋𝑗 / 𝑘⦌𝐴 = (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑚)) |
65 | 40 | fveq1i 6718 |
. . . . 5
⊢ (𝐺‘𝑚) = (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑚) |
66 | 65 | eqcomi 2746 |
. . . 4
⊢ (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑚) = (𝐺‘𝑚) |
67 | 66 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑚) = (𝐺‘𝑚)) |
68 | 52, 64, 67 | 3eqtrd 2781 |
. 2
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = (𝐺‘𝑚)) |
69 | 18, 43, 68 | eqfnfvd 6855 |
1
⊢ (𝜑 → 𝐹 = 𝐺) |