Proof of Theorem fsum2dsub
| Step | Hyp | Ref
| Expression |
| 1 | | simpr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁)) |
| 2 | 1 | elfzelzd 13565 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℤ) |
| 3 | | 0zd 12625 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 0 ∈ ℤ) |
| 4 | | fzsum2sub.m |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
| 5 | 4 | nn0zd 12639 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 6 | 5 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℤ) |
| 7 | | simpll 767 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝜑) |
| 8 | | fz1ssnn 13595 |
. . . . . . . . . . . 12
⊢
(1...𝑁) ⊆
ℕ |
| 9 | | nnssnn0 12529 |
. . . . . . . . . . . 12
⊢ ℕ
⊆ ℕ0 |
| 10 | 8, 9 | sstri 3993 |
. . . . . . . . . . 11
⊢
(1...𝑁) ⊆
ℕ0 |
| 11 | 10, 1 | sselid 3981 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ0) |
| 12 | | nn0uz 12920 |
. . . . . . . . . 10
⊢
ℕ0 = (ℤ≥‘0) |
| 13 | 11, 12 | eleqtrdi 2851 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈
(ℤ≥‘0)) |
| 14 | | neg0 11555 |
. . . . . . . . . 10
⊢ -0 =
0 |
| 15 | | uzneg 12898 |
. . . . . . . . . 10
⊢ (𝑗 ∈
(ℤ≥‘0) → -0 ∈
(ℤ≥‘-𝑗)) |
| 16 | 14, 15 | eqeltrrid 2846 |
. . . . . . . . 9
⊢ (𝑗 ∈
(ℤ≥‘0) → 0 ∈
(ℤ≥‘-𝑗)) |
| 17 | | fzss1 13603 |
. . . . . . . . 9
⊢ (0 ∈
(ℤ≥‘-𝑗) → (0...𝑀) ⊆ (-𝑗...𝑀)) |
| 18 | 13, 16, 17 | 3syl 18 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0...𝑀) ⊆ (-𝑗...𝑀)) |
| 19 | | fzssuz 13605 |
. . . . . . . 8
⊢ (-𝑗...𝑀) ⊆
(ℤ≥‘-𝑗) |
| 20 | 18, 19 | sstrdi 3996 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0...𝑀) ⊆
(ℤ≥‘-𝑗)) |
| 21 | 20 | sselda 3983 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (ℤ≥‘-𝑗)) |
| 22 | 1 | adantr 480 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝑗 ∈ (1...𝑁)) |
| 23 | | fzsum2sub.2 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘-𝑗) ∧ 𝑗 ∈ (1...𝑁)) → 𝐴 ∈ ℂ) |
| 24 | 7, 21, 22, 23 | syl3anc 1373 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 ∈ ℂ) |
| 25 | | fzsum2sub.1 |
. . . . 5
⊢ (𝑖 = (𝑘 − 𝑗) → 𝐴 = 𝐵) |
| 26 | 2, 3, 6, 24, 25 | fsumshft 15816 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑖 ∈ (0...𝑀)𝐴 = Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵) |
| 27 | 4 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑀 ∈
ℕ0) |
| 28 | 8, 1 | sselid 3981 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ) |
| 29 | 28 | nnnn0d 12587 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ0) |
| 30 | 27, 29 | nn0addcld 12591 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈
ℕ0) |
| 31 | 30 | nn0red 12588 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ ℝ) |
| 32 | 31 | ltp1d 12198 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) < ((𝑀 + 𝑗) + 1)) |
| 33 | | fzdisj 13591 |
. . . . . . . 8
⊢ ((𝑀 + 𝑗) < ((𝑀 + 𝑗) + 1) → ((𝑗...(𝑀 + 𝑗)) ∩ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) = ∅) |
| 34 | 32, 33 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((𝑗...(𝑀 + 𝑗)) ∩ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) = ∅) |
| 35 | | fzsum2sub.n |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 36 | 35 | nn0zd 12639 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 37 | 5, 36 | zaddcld 12726 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℤ) |
| 38 | 37 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈ ℤ) |
| 39 | 30 | nn0zd 12639 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ ℤ) |
| 40 | 28 | nnred 12281 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℝ) |
| 41 | | nn0addge2 12573 |
. . . . . . . . . 10
⊢ ((𝑗 ∈ ℝ ∧ 𝑀 ∈ ℕ0)
→ 𝑗 ≤ (𝑀 + 𝑗)) |
| 42 | 40, 27, 41 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ≤ (𝑀 + 𝑗)) |
| 43 | 35 | nn0red 12588 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 44 | 43 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑁 ∈ ℝ) |
| 45 | 27 | nn0red 12588 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℝ) |
| 46 | | elfzle2 13568 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ (1...𝑁) → 𝑗 ≤ 𝑁) |
| 47 | 46 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ≤ 𝑁) |
| 48 | 40, 44, 45, 47 | leadd2dd 11878 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ≤ (𝑀 + 𝑁)) |
| 49 | 2, 38, 39, 42, 48 | elfzd 13555 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ (𝑗...(𝑀 + 𝑁))) |
| 50 | | fzsplit 13590 |
. . . . . . . 8
⊢ ((𝑀 + 𝑗) ∈ (𝑗...(𝑀 + 𝑁)) → (𝑗...(𝑀 + 𝑁)) = ((𝑗...(𝑀 + 𝑗)) ∪ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)))) |
| 51 | 49, 50 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑁)) = ((𝑗...(𝑀 + 𝑗)) ∪ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)))) |
| 52 | | fzfid 14014 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑁)) ∈ Fin) |
| 53 | | simpll 767 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝜑) |
| 54 | 1 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑗 ∈ (1...𝑁)) |
| 55 | 10, 54 | sselid 3981 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑗 ∈ ℕ0) |
| 56 | | fz2ssnn0 32787 |
. . . . . . . . . 10
⊢ (𝑗 ∈ ℕ0
→ (𝑗...(𝑀 + 𝑁)) ⊆
ℕ0) |
| 57 | 55, 56 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → (𝑗...(𝑀 + 𝑁)) ⊆
ℕ0) |
| 58 | | simpr 484 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) |
| 59 | 57, 58 | sseldd 3984 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑘 ∈ ℕ0) |
| 60 | 25 | eleq1d 2826 |
. . . . . . . . 9
⊢ (𝑖 = (𝑘 − 𝑗) → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ)) |
| 61 | | simpll 767 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (ℤ≥‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝜑) |
| 62 | | simplr 769 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (ℤ≥‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (ℤ≥‘-𝑗)) |
| 63 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (ℤ≥‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁)) |
| 64 | 61, 62, 63, 23 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (ℤ≥‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝐴 ∈ ℂ) |
| 65 | 64 | an32s 652 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (ℤ≥‘-𝑗)) → 𝐴 ∈ ℂ) |
| 66 | 65 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ∀𝑖 ∈ (ℤ≥‘-𝑗)𝐴 ∈ ℂ) |
| 67 | 66 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) →
∀𝑖 ∈
(ℤ≥‘-𝑗)𝐴 ∈ ℂ) |
| 68 | | nnsscn 12271 |
. . . . . . . . . . . . 13
⊢ ℕ
⊆ ℂ |
| 69 | 8, 68 | sstri 3993 |
. . . . . . . . . . . 12
⊢
(1...𝑁) ⊆
ℂ |
| 70 | | simplr 769 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ (1...𝑁)) |
| 71 | 69, 70 | sselid 3981 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈
ℂ) |
| 72 | | simpr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℕ0) |
| 73 | 72 | nn0cnd 12589 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℂ) |
| 74 | 71, 73 | negsubdi2d 11636 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → -(𝑗 − 𝑘) = (𝑘 − 𝑗)) |
| 75 | 70 | elfzelzd 13565 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈
ℤ) |
| 76 | | eluzmn 12885 |
. . . . . . . . . . . 12
⊢ ((𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ0)
→ 𝑗 ∈
(ℤ≥‘(𝑗 − 𝑘))) |
| 77 | 75, 72, 76 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈
(ℤ≥‘(𝑗 − 𝑘))) |
| 78 | | uzneg 12898 |
. . . . . . . . . . 11
⊢ (𝑗 ∈
(ℤ≥‘(𝑗 − 𝑘)) → -(𝑗 − 𝑘) ∈ (ℤ≥‘-𝑗)) |
| 79 | 77, 78 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → -(𝑗 − 𝑘) ∈ (ℤ≥‘-𝑗)) |
| 80 | 74, 79 | eqeltrrd 2842 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → (𝑘 − 𝑗) ∈ (ℤ≥‘-𝑗)) |
| 81 | 60, 67, 80 | rspcdva 3623 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈
ℂ) |
| 82 | 53, 54, 59, 81 | syl21anc 838 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝐵 ∈ ℂ) |
| 83 | 34, 51, 52, 82 | fsumsplit 15777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 = (Σ𝑘 ∈ (𝑗...(𝑀 + 𝑗))𝐵 + Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵)) |
| 84 | 2 | zcnd 12723 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℂ) |
| 85 | 84 | addlidd 11462 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0 + 𝑗) = 𝑗) |
| 86 | 85 | oveq1d 7446 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((0 + 𝑗)...(𝑀 + 𝑗)) = (𝑗...(𝑀 + 𝑗))) |
| 87 | 86 | eqcomd 2743 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑗)) = ((0 + 𝑗)...(𝑀 + 𝑗))) |
| 88 | 87 | sumeq1d 15736 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑗))𝐵 = Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵) |
| 89 | | fzsum2sub.3 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) → 𝐵 = 0) |
| 90 | 89 | sumeq2dv 15738 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵 = Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0) |
| 91 | | fzfi 14013 |
. . . . . . . . 9
⊢ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ∈ Fin |
| 92 | | sumz 15758 |
. . . . . . . . . 10
⊢
(((((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ⊆ (ℤ≥‘0)
∨ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ∈ Fin) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0 = 0) |
| 93 | 92 | olcs 877 |
. . . . . . . . 9
⊢ ((((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ∈ Fin → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0 = 0) |
| 94 | 91, 93 | ax-mp 5 |
. . . . . . . 8
⊢
Σ𝑘 ∈
(((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0 = 0 |
| 95 | 90, 94 | eqtrdi 2793 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵 = 0) |
| 96 | 88, 95 | oveq12d 7449 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (Σ𝑘 ∈ (𝑗...(𝑀 + 𝑗))𝐵 + Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵) = (Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 + 0)) |
| 97 | | fzfid 14014 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((0 + 𝑗)...(𝑀 + 𝑗)) ∈ Fin) |
| 98 | | simpll 767 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝜑) |
| 99 | 1 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑗 ∈ (1...𝑁)) |
| 100 | | elfzuz3 13561 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘𝑗)) |
| 101 | 100 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑁 ∈ (ℤ≥‘𝑗)) |
| 102 | | eluzadd 12907 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈
(ℤ≥‘𝑗) ∧ 𝑀 ∈ ℤ) → (𝑁 + 𝑀) ∈
(ℤ≥‘(𝑗 + 𝑀))) |
| 103 | 101, 6, 102 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑁 + 𝑀) ∈
(ℤ≥‘(𝑗 + 𝑀))) |
| 104 | 35 | nn0cnd 12589 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 105 | 104 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑁 ∈ ℂ) |
| 106 | | zsscn 12621 |
. . . . . . . . . . . . . . . 16
⊢ ℤ
⊆ ℂ |
| 107 | 106, 6 | sselid 3981 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℂ) |
| 108 | 105, 107 | addcomd 11463 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑁 + 𝑀) = (𝑀 + 𝑁)) |
| 109 | 84, 107 | addcomd 11463 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑗 + 𝑀) = (𝑀 + 𝑗)) |
| 110 | 109 | fveq2d 6910 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) →
(ℤ≥‘(𝑗 + 𝑀)) = (ℤ≥‘(𝑀 + 𝑗))) |
| 111 | 103, 108,
110 | 3eltr3d 2855 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈
(ℤ≥‘(𝑀 + 𝑗))) |
| 112 | 111 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → (𝑀 + 𝑁) ∈
(ℤ≥‘(𝑀 + 𝑗))) |
| 113 | | fzss2 13604 |
. . . . . . . . . . . 12
⊢ ((𝑀 + 𝑁) ∈
(ℤ≥‘(𝑀 + 𝑗)) → (𝑗...(𝑀 + 𝑗)) ⊆ (𝑗...(𝑀 + 𝑁))) |
| 114 | 112, 113 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → (𝑗...(𝑀 + 𝑗)) ⊆ (𝑗...(𝑀 + 𝑁))) |
| 115 | | simpr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) |
| 116 | 86 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → ((0 + 𝑗)...(𝑀 + 𝑗)) = (𝑗...(𝑀 + 𝑗))) |
| 117 | 115, 116 | eleqtrd 2843 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑗))) |
| 118 | 114, 117 | sseldd 3984 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) |
| 119 | 98, 99, 118, 59 | syl21anc 838 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ ℕ0) |
| 120 | 98, 99, 119, 81 | syl21anc 838 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝐵 ∈ ℂ) |
| 121 | 97, 120 | fsumcl 15769 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 ∈ ℂ) |
| 122 | 121 | addridd 11461 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 + 0) = Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵) |
| 123 | 83, 96, 122 | 3eqtrrd 2782 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 = Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵) |
| 124 | | fzval3 13773 |
. . . . . . . . . 10
⊢ ((𝑀 + 𝑁) ∈ ℤ → (𝑗...(𝑀 + 𝑁)) = (𝑗..^((𝑀 + 𝑁) + 1))) |
| 125 | 38, 124 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑁)) = (𝑗..^((𝑀 + 𝑁) + 1))) |
| 126 | 125 | ineq2d 4220 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∩ (𝑗...(𝑀 + 𝑁))) = ((0..^𝑗) ∩ (𝑗..^((𝑀 + 𝑁) + 1)))) |
| 127 | | fzodisj 13733 |
. . . . . . . 8
⊢
((0..^𝑗) ∩
(𝑗..^((𝑀 + 𝑁) + 1))) = ∅ |
| 128 | 126, 127 | eqtrdi 2793 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∩ (𝑗...(𝑀 + 𝑁))) = ∅) |
| 129 | 38 | peano2zd 12725 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((𝑀 + 𝑁) + 1) ∈ ℤ) |
| 130 | 29 | nn0ge0d 12590 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 0 ≤ 𝑗) |
| 131 | 129 | zred 12722 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((𝑀 + 𝑁) + 1) ∈ ℝ) |
| 132 | 38 | zred 12722 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈ ℝ) |
| 133 | | nn0addge2 12573 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℕ0)
→ 𝑁 ≤ (𝑀 + 𝑁)) |
| 134 | 43, 4, 133 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ≤ (𝑀 + 𝑁)) |
| 135 | 134 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑁 ≤ (𝑀 + 𝑁)) |
| 136 | 132 | lep1d 12199 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ≤ ((𝑀 + 𝑁) + 1)) |
| 137 | 44, 132, 131, 135, 136 | letrd 11418 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑁 ≤ ((𝑀 + 𝑁) + 1)) |
| 138 | 40, 44, 131, 47, 137 | letrd 11418 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ≤ ((𝑀 + 𝑁) + 1)) |
| 139 | 3, 129, 2, 130, 138 | elfzd 13555 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (0...((𝑀 + 𝑁) + 1))) |
| 140 | | fzosplit 13732 |
. . . . . . . . 9
⊢ (𝑗 ∈ (0...((𝑀 + 𝑁) + 1)) → (0..^((𝑀 + 𝑁) + 1)) = ((0..^𝑗) ∪ (𝑗..^((𝑀 + 𝑁) + 1)))) |
| 141 | 139, 140 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0..^((𝑀 + 𝑁) + 1)) = ((0..^𝑗) ∪ (𝑗..^((𝑀 + 𝑁) + 1)))) |
| 142 | | fzval3 13773 |
. . . . . . . . 9
⊢ ((𝑀 + 𝑁) ∈ ℤ → (0...(𝑀 + 𝑁)) = (0..^((𝑀 + 𝑁) + 1))) |
| 143 | 38, 142 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0...(𝑀 + 𝑁)) = (0..^((𝑀 + 𝑁) + 1))) |
| 144 | 125 | uneq2d 4168 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∪ (𝑗...(𝑀 + 𝑁))) = ((0..^𝑗) ∪ (𝑗..^((𝑀 + 𝑁) + 1)))) |
| 145 | 141, 143,
144 | 3eqtr4d 2787 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0...(𝑀 + 𝑁)) = ((0..^𝑗) ∪ (𝑗...(𝑀 + 𝑁)))) |
| 146 | | fzfid 14014 |
. . . . . . . 8
⊢ (𝜑 → (0...(𝑀 + 𝑁)) ∈ Fin) |
| 147 | 146 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0...(𝑀 + 𝑁)) ∈ Fin) |
| 148 | | simpl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝜑) |
| 149 | 1 | adantrl 716 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑗 ∈ (1...𝑁)) |
| 150 | | fz0ssnn0 13662 |
. . . . . . . . . 10
⊢
(0...(𝑀 + 𝑁)) ⊆
ℕ0 |
| 151 | | simprl 771 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑘 ∈ (0...(𝑀 + 𝑁))) |
| 152 | 150, 151 | sselid 3981 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑘 ∈ ℕ0) |
| 153 | 148, 149,
152, 81 | syl21anc 838 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝐵 ∈ ℂ) |
| 154 | 153 | anass1rs 655 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → 𝐵 ∈ ℂ) |
| 155 | 128, 145,
147, 154 | fsumsplit 15777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵 = (Σ𝑘 ∈ (0..^𝑗)𝐵 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵)) |
| 156 | | fzsum2sub.4 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0..^𝑗)) → 𝐵 = 0) |
| 157 | 156 | sumeq2dv 15738 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (0..^𝑗)𝐵 = Σ𝑘 ∈ (0..^𝑗)0) |
| 158 | | fzofi 14015 |
. . . . . . . . 9
⊢
(0..^𝑗) ∈
Fin |
| 159 | | sumz 15758 |
. . . . . . . . . 10
⊢
(((0..^𝑗) ⊆
(ℤ≥‘0) ∨ (0..^𝑗) ∈ Fin) → Σ𝑘 ∈ (0..^𝑗)0 = 0) |
| 160 | 159 | olcs 877 |
. . . . . . . . 9
⊢
((0..^𝑗) ∈ Fin
→ Σ𝑘 ∈
(0..^𝑗)0 =
0) |
| 161 | 158, 160 | ax-mp 5 |
. . . . . . . 8
⊢
Σ𝑘 ∈
(0..^𝑗)0 =
0 |
| 162 | 157, 161 | eqtrdi 2793 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (0..^𝑗)𝐵 = 0) |
| 163 | 162 | oveq1d 7446 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0..^𝑗)𝐵 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵) = (0 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵)) |
| 164 | 52, 82 | fsumcl 15769 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 ∈ ℂ) |
| 165 | 164 | addlidd 11462 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵) = Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵) |
| 166 | 155, 163,
165 | 3eqtrrd 2782 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵) |
| 167 | 123, 166 | eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵) |
| 168 | 26, 167 | eqtrd 2777 |
. . 3
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑖 ∈ (0...𝑀)𝐴 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵) |
| 169 | 168 | sumeq2dv 15738 |
. 2
⊢ (𝜑 → Σ𝑗 ∈ (1...𝑁)Σ𝑖 ∈ (0...𝑀)𝐴 = Σ𝑗 ∈ (1...𝑁)Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵) |
| 170 | | fzfid 14014 |
. . 3
⊢ (𝜑 → (0...𝑀) ∈ Fin) |
| 171 | | fzfid 14014 |
. . 3
⊢ (𝜑 → (1...𝑁) ∈ Fin) |
| 172 | 24 | anasss 466 |
. . . 4
⊢ ((𝜑 ∧ (𝑗 ∈ (1...𝑁) ∧ 𝑖 ∈ (0...𝑀))) → 𝐴 ∈ ℂ) |
| 173 | 172 | ancom2s 650 |
. . 3
⊢ ((𝜑 ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (1...𝑁))) → 𝐴 ∈ ℂ) |
| 174 | 170, 171,
173 | fsumcom 15811 |
. 2
⊢ (𝜑 → Σ𝑖 ∈ (0...𝑀)Σ𝑗 ∈ (1...𝑁)𝐴 = Σ𝑗 ∈ (1...𝑁)Σ𝑖 ∈ (0...𝑀)𝐴) |
| 175 | 146, 171,
153 | fsumcom 15811 |
. 2
⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑗 ∈ (1...𝑁)𝐵 = Σ𝑗 ∈ (1...𝑁)Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵) |
| 176 | 169, 174,
175 | 3eqtr4d 2787 |
1
⊢ (𝜑 → Σ𝑖 ∈ (0...𝑀)Σ𝑗 ∈ (1...𝑁)𝐴 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑗 ∈ (1...𝑁)𝐵) |