Step | Hyp | Ref
| Expression |
1 | | nn0uz 11722 |
. 2
⊢
ℕ0 = (ℤ≥‘0) |
2 | | nnuz 11723 |
. 2
⊢ ℕ =
(ℤ≥‘1) |
3 | | 0zd 11389 |
. 2
⊢ (𝜑 → 0 ∈
ℤ) |
4 | | 1zzd 11408 |
. 2
⊢ (𝜑 → 1 ∈
ℤ) |
5 | | 2nn0 11309 |
. . . . . 6
⊢ 2 ∈
ℕ0 |
6 | 5 | a1i 11 |
. . . . 5
⊢ (𝜑 → 2 ∈
ℕ0) |
7 | | nn0mulcl 11329 |
. . . . 5
⊢ ((2
∈ ℕ0 ∧ 𝑚 ∈ ℕ0) → (2
· 𝑚) ∈
ℕ0) |
8 | 6, 7 | sylan 488 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (2
· 𝑚) ∈
ℕ0) |
9 | | nn0p1nn 11332 |
. . . 4
⊢ ((2
· 𝑚) ∈
ℕ0 → ((2 · 𝑚) + 1) ∈ ℕ) |
10 | 8, 9 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((2
· 𝑚) + 1) ∈
ℕ) |
11 | | eqid 2622 |
. . 3
⊢ (𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1)) = (𝑚 ∈
ℕ0 ↦ ((2 · 𝑚) + 1)) |
12 | 10, 11 | fmptd 6385 |
. 2
⊢ (𝜑 → (𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1)):ℕ0⟶ℕ) |
13 | | nn0mulcl 11329 |
. . . . . 6
⊢ ((2
∈ ℕ0 ∧ 𝑖 ∈ ℕ0) → (2
· 𝑖) ∈
ℕ0) |
14 | 6, 13 | sylan 488 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (2
· 𝑖) ∈
ℕ0) |
15 | 14 | nn0red 11352 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (2
· 𝑖) ∈
ℝ) |
16 | | peano2nn0 11333 |
. . . . . 6
⊢ (𝑖 ∈ ℕ0
→ (𝑖 + 1) ∈
ℕ0) |
17 | | nn0mulcl 11329 |
. . . . . 6
⊢ ((2
∈ ℕ0 ∧ (𝑖 + 1) ∈ ℕ0) → (2
· (𝑖 + 1)) ∈
ℕ0) |
18 | 6, 16, 17 | syl2an 494 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (2
· (𝑖 + 1)) ∈
ℕ0) |
19 | 18 | nn0red 11352 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (2
· (𝑖 + 1)) ∈
ℝ) |
20 | | 1red 10055 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 1 ∈
ℝ) |
21 | | nn0re 11301 |
. . . . . . 7
⊢ (𝑖 ∈ ℕ0
→ 𝑖 ∈
ℝ) |
22 | 21 | adantl 482 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈
ℝ) |
23 | 22 | ltp1d 10954 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 < (𝑖 + 1)) |
24 | 16 | adantl 482 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑖 + 1) ∈
ℕ0) |
25 | 24 | nn0red 11352 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑖 + 1) ∈
ℝ) |
26 | | 2re 11090 |
. . . . . . 7
⊢ 2 ∈
ℝ |
27 | 26 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 2 ∈
ℝ) |
28 | | 2pos 11112 |
. . . . . . 7
⊢ 0 <
2 |
29 | 28 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 0 <
2) |
30 | | ltmul2 10874 |
. . . . . 6
⊢ ((𝑖 ∈ ℝ ∧ (𝑖 + 1) ∈ ℝ ∧ (2
∈ ℝ ∧ 0 < 2)) → (𝑖 < (𝑖 + 1) ↔ (2 · 𝑖) < (2 · (𝑖 + 1)))) |
31 | 22, 25, 27, 29, 30 | syl112anc 1330 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑖 < (𝑖 + 1) ↔ (2 · 𝑖) < (2 · (𝑖 + 1)))) |
32 | 23, 31 | mpbid 222 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (2
· 𝑖) < (2
· (𝑖 +
1))) |
33 | 15, 19, 20, 32 | ltadd1dd 10638 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((2
· 𝑖) + 1) < ((2
· (𝑖 + 1)) +
1)) |
34 | | oveq2 6658 |
. . . . . 6
⊢ (𝑚 = 𝑖 → (2 · 𝑚) = (2 · 𝑖)) |
35 | 34 | oveq1d 6665 |
. . . . 5
⊢ (𝑚 = 𝑖 → ((2 · 𝑚) + 1) = ((2 · 𝑖) + 1)) |
36 | | ovex 6678 |
. . . . 5
⊢ ((2
· 𝑖) + 1) ∈
V |
37 | 35, 11, 36 | fvmpt 6282 |
. . . 4
⊢ (𝑖 ∈ ℕ0
→ ((𝑚 ∈
ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑖) = ((2 · 𝑖) + 1)) |
38 | 37 | adantl 482 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1))‘𝑖) = ((2 ·
𝑖) + 1)) |
39 | | oveq2 6658 |
. . . . . 6
⊢ (𝑚 = (𝑖 + 1) → (2 · 𝑚) = (2 · (𝑖 + 1))) |
40 | 39 | oveq1d 6665 |
. . . . 5
⊢ (𝑚 = (𝑖 + 1) → ((2 · 𝑚) + 1) = ((2 · (𝑖 + 1)) + 1)) |
41 | | ovex 6678 |
. . . . 5
⊢ ((2
· (𝑖 + 1)) + 1)
∈ V |
42 | 40, 11, 41 | fvmpt 6282 |
. . . 4
⊢ ((𝑖 + 1) ∈ ℕ0
→ ((𝑚 ∈
ℕ0 ↦ ((2 · 𝑚) + 1))‘(𝑖 + 1)) = ((2 · (𝑖 + 1)) + 1)) |
43 | 24, 42 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1))‘(𝑖 + 1)) = ((2
· (𝑖 + 1)) +
1)) |
44 | 33, 38, 43 | 3brtr4d 4685 |
. 2
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1))‘𝑖) < ((𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1))‘(𝑖 +
1))) |
45 | | eldifi 3732 |
. . . . . . 7
⊢ (𝑛 ∈ (ℕ ∖ ran
(𝑚 ∈
ℕ0 ↦ ((2 · 𝑚) + 1))) → 𝑛 ∈ ℕ) |
46 | | simpr 477 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) |
47 | | 0cnd 10033 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 0 ∈
ℂ) |
48 | | nnz 11399 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℤ) |
49 | 48 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ) |
50 | | odd2np1 15065 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℤ → (¬ 2
∥ 𝑛 ↔
∃𝑘 ∈ ℤ ((2
· 𝑘) + 1) = 𝑛)) |
51 | 49, 50 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (¬ 2 ∥
𝑛 ↔ ∃𝑘 ∈ ℤ ((2 ·
𝑘) + 1) = 𝑛)) |
52 | | simprl 794 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 𝑘 ∈ ℤ) |
53 | | nnm1nn0 11334 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈
ℕ0) |
54 | 53 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → (𝑛 − 1) ∈
ℕ0) |
55 | 54 | nn0red 11352 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → (𝑛 − 1) ∈ ℝ) |
56 | 54 | nn0ge0d 11354 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 0 ≤ (𝑛 − 1)) |
57 | 26 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 2 ∈ ℝ) |
58 | 28 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 0 < 2) |
59 | | divge0 10892 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑛 − 1) ∈ ℝ ∧
0 ≤ (𝑛 − 1)) ∧
(2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ ((𝑛 − 1) / 2)) |
60 | 55, 56, 57, 58, 59 | syl22anc 1327 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 0 ≤ ((𝑛 − 1) / 2)) |
61 | | simprr 796 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → ((2 · 𝑘) + 1) = 𝑛) |
62 | 61 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → (((2 · 𝑘) + 1) − 1) = (𝑛 − 1)) |
63 | | 2cn 11091 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ∈
ℂ |
64 | | zcn 11382 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ ℤ → 𝑘 ∈
ℂ) |
65 | 64 | ad2antrl 764 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 𝑘 ∈ ℂ) |
66 | | mulcl 10020 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((2
∈ ℂ ∧ 𝑘
∈ ℂ) → (2 · 𝑘) ∈ ℂ) |
67 | 63, 65, 66 | sylancr 695 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → (2 · 𝑘) ∈ ℂ) |
68 | | ax-1cn 9994 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 1 ∈
ℂ |
69 | | pncan 10287 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((2
· 𝑘) ∈ ℂ
∧ 1 ∈ ℂ) → (((2 · 𝑘) + 1) − 1) = (2 · 𝑘)) |
70 | 67, 68, 69 | sylancl 694 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → (((2 · 𝑘) + 1) − 1) = (2 · 𝑘)) |
71 | 62, 70 | eqtr3d 2658 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → (𝑛 − 1) = (2 · 𝑘)) |
72 | 71 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → ((𝑛 − 1) / 2) = ((2 · 𝑘) / 2)) |
73 | | 2cnd 11093 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 2 ∈ ℂ) |
74 | | 2ne0 11113 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 2 ≠
0 |
75 | 74 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 2 ≠ 0) |
76 | 65, 73, 75 | divcan3d 10806 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → ((2 · 𝑘) / 2) = 𝑘) |
77 | 72, 76 | eqtrd 2656 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → ((𝑛 − 1) / 2) = 𝑘) |
78 | 60, 77 | breqtrd 4679 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 0 ≤ 𝑘) |
79 | | elnn0z 11390 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℕ0
↔ (𝑘 ∈ ℤ
∧ 0 ≤ 𝑘)) |
80 | 52, 78, 79 | sylanbrc 698 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛)) → 𝑘 ∈ ℕ0) |
81 | 80 | ex 450 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛) → 𝑘 ∈
ℕ0)) |
82 | | simpr 477 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ ℤ ∧ ((2
· 𝑘) + 1) = 𝑛) → ((2 · 𝑘) + 1) = 𝑛) |
83 | 82 | eqcomd 2628 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ ℤ ∧ ((2
· 𝑘) + 1) = 𝑛) → 𝑛 = ((2 · 𝑘) + 1)) |
84 | 83 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛) → 𝑛 = ((2 · 𝑘) + 1))) |
85 | 81, 84 | jcad 555 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℤ ∧ ((2 · 𝑘) + 1) = 𝑛) → (𝑘 ∈ ℕ0 ∧ 𝑛 = ((2 · 𝑘) + 1)))) |
86 | 85 | reximdv2 3014 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∃𝑘 ∈ ℤ ((2 ·
𝑘) + 1) = 𝑛 → ∃𝑘 ∈ ℕ0
𝑛 = ((2 · 𝑘) + 1))) |
87 | 51, 86 | sylbid 230 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (¬ 2 ∥
𝑛 → ∃𝑘 ∈ ℕ0
𝑛 = ((2 · 𝑘) + 1))) |
88 | | iserodd.f |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈
ℂ) |
89 | | iserodd.h |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = ((2 · 𝑘) + 1) → 𝐵 = 𝐶) |
90 | 89 | eleq1d 2686 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = ((2 · 𝑘) + 1) → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
91 | 88, 90 | syl5ibrcom 237 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑛 = ((2 · 𝑘) + 1) → 𝐵 ∈ ℂ)) |
92 | 91 | rexlimdva 3031 |
. . . . . . . . . . . 12
⊢ (𝜑 → (∃𝑘 ∈ ℕ0 𝑛 = ((2 · 𝑘) + 1) → 𝐵 ∈ ℂ)) |
93 | 92 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∃𝑘 ∈ ℕ0
𝑛 = ((2 · 𝑘) + 1) → 𝐵 ∈ ℂ)) |
94 | 87, 93 | syld 47 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (¬ 2 ∥
𝑛 → 𝐵 ∈ ℂ)) |
95 | 94 | imp 445 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 𝐵 ∈ ℂ) |
96 | 47, 95 | ifclda 4120 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(2 ∥ 𝑛, 0, 𝐵) ∈ ℂ) |
97 | | eqid 2622 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ ↦ if(2
∥ 𝑛, 0, 𝐵)) = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵)) |
98 | 97 | fvmpt2 6291 |
. . . . . . . 8
⊢ ((𝑛 ∈ ℕ ∧ if(2
∥ 𝑛, 0, 𝐵) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(2
∥ 𝑛, 0, 𝐵))‘𝑛) = if(2 ∥ 𝑛, 0, 𝐵)) |
99 | 46, 96, 98 | syl2anc 693 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑛) = if(2 ∥ 𝑛, 0, 𝐵)) |
100 | 45, 99 | sylan2 491 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1)))) → ((𝑛 ∈
ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑛) = if(2 ∥ 𝑛, 0, 𝐵)) |
101 | | eldif 3584 |
. . . . . . . 8
⊢ (𝑛 ∈ (ℕ ∖ ran
(𝑚 ∈
ℕ0 ↦ ((2 · 𝑚) + 1))) ↔ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ ran (𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1)))) |
102 | | vex 3203 |
. . . . . . . . . . . 12
⊢ 𝑛 ∈ V |
103 | | oveq2 6658 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑘 → (2 · 𝑚) = (2 · 𝑘)) |
104 | 103 | oveq1d 6665 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑘 → ((2 · 𝑚) + 1) = ((2 · 𝑘) + 1)) |
105 | 104 | cbvmptv 4750 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1)) = (𝑘 ∈
ℕ0 ↦ ((2 · 𝑘) + 1)) |
106 | 105 | elrnmpt 5372 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ V → (𝑛 ∈ ran (𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) + 1)) ↔
∃𝑘 ∈
ℕ0 𝑛 = ((2
· 𝑘) +
1))) |
107 | 102, 106 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ran (𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) + 1)) ↔
∃𝑘 ∈
ℕ0 𝑛 = ((2
· 𝑘) +
1)) |
108 | 87, 107 | syl6ibr 242 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (¬ 2 ∥
𝑛 → 𝑛 ∈ ran (𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1)))) |
109 | 108 | con1d 139 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (¬ 𝑛 ∈ ran (𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) + 1)) → 2
∥ 𝑛)) |
110 | 109 | impr 649 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ ran (𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) + 1)))) →
2 ∥ 𝑛) |
111 | 101, 110 | sylan2b 492 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1)))) → 2 ∥ 𝑛) |
112 | 111 | iftrued 4094 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1)))) → if(2 ∥ 𝑛, 0, 𝐵) = 0) |
113 | 100, 112 | eqtrd 2656 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1)))) → ((𝑛 ∈
ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑛) = 0) |
114 | 113 | ralrimiva 2966 |
. . . 4
⊢ (𝜑 → ∀𝑛 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1)))((𝑛 ∈ ℕ
↦ if(2 ∥ 𝑛, 0,
𝐵))‘𝑛) = 0) |
115 | | nfv 1843 |
. . . . 5
⊢
Ⅎ𝑗((𝑛 ∈ ℕ ↦ if(2
∥ 𝑛, 0, 𝐵))‘𝑛) = 0 |
116 | | nffvmpt1 6199 |
. . . . . 6
⊢
Ⅎ𝑛((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑗) |
117 | 116 | nfeq1 2778 |
. . . . 5
⊢
Ⅎ𝑛((𝑛 ∈ ℕ ↦ if(2
∥ 𝑛, 0, 𝐵))‘𝑗) = 0 |
118 | | fveq2 6191 |
. . . . . 6
⊢ (𝑛 = 𝑗 → ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑛) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑗)) |
119 | 118 | eqeq1d 2624 |
. . . . 5
⊢ (𝑛 = 𝑗 → (((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑛) = 0 ↔ ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑗) = 0)) |
120 | 115, 117,
119 | cbvral 3167 |
. . . 4
⊢
(∀𝑛 ∈
(ℕ ∖ ran (𝑚
∈ ℕ0 ↦ ((2 · 𝑚) + 1)))((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑛) = 0 ↔ ∀𝑗 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1)))((𝑛 ∈ ℕ
↦ if(2 ∥ 𝑛, 0,
𝐵))‘𝑗) = 0) |
121 | 114, 120 | sylib 208 |
. . 3
⊢ (𝜑 → ∀𝑗 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1)))((𝑛 ∈ ℕ
↦ if(2 ∥ 𝑛, 0,
𝐵))‘𝑗) = 0) |
122 | 121 | r19.21bi 2932 |
. 2
⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ ran (𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1)))) → ((𝑛 ∈
ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑗) = 0) |
123 | 96, 97 | fmptd 6385 |
. . 3
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵)):ℕ⟶ℂ) |
124 | 123 | ffvelrnda 6359 |
. 2
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘𝑗) ∈ ℂ) |
125 | | simpr 477 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℕ0) |
126 | | eqid 2622 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ0
↦ 𝐶) = (𝑘 ∈ ℕ0
↦ 𝐶) |
127 | 126 | fvmpt2 6291 |
. . . . . . 7
⊢ ((𝑘 ∈ ℕ0
∧ 𝐶 ∈ ℂ)
→ ((𝑘 ∈
ℕ0 ↦ 𝐶)‘𝑘) = 𝐶) |
128 | 125, 88, 127 | syl2anc 693 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑘 ∈ ℕ0
↦ 𝐶)‘𝑘) = 𝐶) |
129 | | ovex 6678 |
. . . . . . . . . 10
⊢ ((2
· 𝑘) + 1) ∈
V |
130 | 104, 11, 129 | fvmpt 6282 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ0
→ ((𝑚 ∈
ℕ0 ↦ ((2 · 𝑚) + 1))‘𝑘) = ((2 · 𝑘) + 1)) |
131 | 130 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1))‘𝑘) = ((2 ·
𝑘) + 1)) |
132 | 131 | fveq2d 6195 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ ↦ if(2
∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑘)) = ((𝑛 ∈ ℕ ↦ if(2
∥ 𝑛, 0, 𝐵))‘((2 · 𝑘) + 1))) |
133 | | nn0mulcl 11329 |
. . . . . . . . . 10
⊢ ((2
∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (2
· 𝑘) ∈
ℕ0) |
134 | 6, 133 | sylan 488 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2
· 𝑘) ∈
ℕ0) |
135 | | nn0p1nn 11332 |
. . . . . . . . 9
⊢ ((2
· 𝑘) ∈
ℕ0 → ((2 · 𝑘) + 1) ∈ ℕ) |
136 | 134, 135 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((2
· 𝑘) + 1) ∈
ℕ) |
137 | | 2z 11409 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℤ |
138 | | nn0z 11400 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℤ) |
139 | 138 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℤ) |
140 | | dvdsmul1 15003 |
. . . . . . . . . . . 12
⊢ ((2
∈ ℤ ∧ 𝑘
∈ ℤ) → 2 ∥ (2 · 𝑘)) |
141 | 137, 139,
140 | sylancr 695 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 2 ∥
(2 · 𝑘)) |
142 | 134 | nn0zd 11480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2
· 𝑘) ∈
ℤ) |
143 | | 2nn 11185 |
. . . . . . . . . . . . 13
⊢ 2 ∈
ℕ |
144 | 143 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 2 ∈
ℕ) |
145 | | 1lt2 11194 |
. . . . . . . . . . . . 13
⊢ 1 <
2 |
146 | 145 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 1 <
2) |
147 | | ndvdsp1 15135 |
. . . . . . . . . . . 12
⊢ (((2
· 𝑘) ∈ ℤ
∧ 2 ∈ ℕ ∧ 1 < 2) → (2 ∥ (2 · 𝑘) → ¬ 2 ∥ ((2
· 𝑘) +
1))) |
148 | 142, 144,
146, 147 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2
∥ (2 · 𝑘)
→ ¬ 2 ∥ ((2 · 𝑘) + 1))) |
149 | 141, 148 | mpd 15 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ¬ 2
∥ ((2 · 𝑘) +
1)) |
150 | 149 | iffalsed 4097 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if(2
∥ ((2 · 𝑘) +
1), 0, 𝐶) = 𝐶) |
151 | 150, 88 | eqeltrd 2701 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if(2
∥ ((2 · 𝑘) +
1), 0, 𝐶) ∈
ℂ) |
152 | | breq2 4657 |
. . . . . . . . . 10
⊢ (𝑛 = ((2 · 𝑘) + 1) → (2 ∥ 𝑛 ↔ 2 ∥ ((2 ·
𝑘) + 1))) |
153 | 152, 89 | ifbieq2d 4111 |
. . . . . . . . 9
⊢ (𝑛 = ((2 · 𝑘) + 1) → if(2 ∥ 𝑛, 0, 𝐵) = if(2 ∥ ((2 · 𝑘) + 1), 0, 𝐶)) |
154 | 153, 97 | fvmptg 6280 |
. . . . . . . 8
⊢ ((((2
· 𝑘) + 1) ∈
ℕ ∧ if(2 ∥ ((2 · 𝑘) + 1), 0, 𝐶) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(2
∥ 𝑛, 0, 𝐵))‘((2 · 𝑘) + 1)) = if(2 ∥ ((2
· 𝑘) + 1), 0, 𝐶)) |
155 | 136, 151,
154 | syl2anc 693 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ ↦ if(2
∥ 𝑛, 0, 𝐵))‘((2 · 𝑘) + 1)) = if(2 ∥ ((2
· 𝑘) + 1), 0, 𝐶)) |
156 | 132, 155,
150 | 3eqtrd 2660 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ ↦ if(2
∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑘)) = 𝐶) |
157 | 128, 156 | eqtr4d 2659 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑘 ∈ ℕ0
↦ 𝐶)‘𝑘) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑘))) |
158 | 157 | ralrimiva 2966 |
. . . 4
⊢ (𝜑 → ∀𝑘 ∈ ℕ0 ((𝑘 ∈ ℕ0
↦ 𝐶)‘𝑘) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑘))) |
159 | | nfv 1843 |
. . . . 5
⊢
Ⅎ𝑖((𝑘 ∈ ℕ0
↦ 𝐶)‘𝑘) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑘)) |
160 | | nffvmpt1 6199 |
. . . . . 6
⊢
Ⅎ𝑘((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑖) |
161 | 160 | nfeq1 2778 |
. . . . 5
⊢
Ⅎ𝑘((𝑘 ∈ ℕ0
↦ 𝐶)‘𝑖) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑖)) |
162 | | fveq2 6191 |
. . . . . 6
⊢ (𝑘 = 𝑖 → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑘) = ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑖)) |
163 | | fveq2 6191 |
. . . . . . 7
⊢ (𝑘 = 𝑖 → ((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑘) = ((𝑚 ∈ ℕ0
↦ ((2 · 𝑚) +
1))‘𝑖)) |
164 | 163 | fveq2d 6195 |
. . . . . 6
⊢ (𝑘 = 𝑖 → ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑘)) = ((𝑛 ∈ ℕ ↦ if(2
∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑖))) |
165 | 162, 164 | eqeq12d 2637 |
. . . . 5
⊢ (𝑘 = 𝑖 → (((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑘) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑘)) ↔
((𝑘 ∈
ℕ0 ↦ 𝐶)‘𝑖) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑖)))) |
166 | 159, 161,
165 | cbvral 3167 |
. . . 4
⊢
(∀𝑘 ∈
ℕ0 ((𝑘
∈ ℕ0 ↦ 𝐶)‘𝑘) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑘)) ↔
∀𝑖 ∈
ℕ0 ((𝑘
∈ ℕ0 ↦ 𝐶)‘𝑖) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑖))) |
167 | 158, 166 | sylib 208 |
. . 3
⊢ (𝜑 → ∀𝑖 ∈ ℕ0 ((𝑘 ∈ ℕ0
↦ 𝐶)‘𝑖) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑖))) |
168 | 167 | r19.21bi 2932 |
. 2
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((𝑘 ∈ ℕ0
↦ 𝐶)‘𝑖) = ((𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 𝐵))‘((𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) +
1))‘𝑖))) |
169 | 1, 2, 3, 4, 12, 44, 122, 124, 168 | isercoll2 14399 |
1
⊢ (𝜑 → (seq0( + , (𝑘 ∈ ℕ0
↦ 𝐶)) ⇝ 𝐴 ↔ seq1( + , (𝑛 ∈ ℕ ↦ if(2
∥ 𝑛, 0, 𝐵))) ⇝ 𝐴)) |