Step | Hyp | Ref
| Expression |
1 | | metakunt16.4 |
. 2
⊢ 𝐹 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) |
2 | | metakunt16.2 |
. . . . 5
⊢ (𝜑 → 𝐼 ∈ ℕ) |
3 | 2 | nnzd 12110 |
. . . 4
⊢ (𝜑 → 𝐼 ∈ ℤ) |
4 | 3 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝐼 ∈ ℤ) |
5 | | metakunt16.1 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℕ) |
6 | 5 | nnzd 12110 |
. . . . 5
⊢ (𝜑 → 𝑀 ∈ ℤ) |
7 | 6 | adantr 485 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝑀 ∈ ℤ) |
8 | | 1zzd 12037 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 1 ∈
ℤ) |
9 | 7, 8 | zsubcld 12116 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (𝑀 − 1) ∈ ℤ) |
10 | 8, 4 | zsubcld 12116 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (1 − 𝐼) ∈
ℤ) |
11 | | simpr 489 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝑥 ∈ (𝐼...(𝑀 − 1))) |
12 | | elfz3 12951 |
. . . 4
⊢ ((1
− 𝐼) ∈ ℤ
→ (1 − 𝐼) ∈
((1 − 𝐼)...(1 −
𝐼))) |
13 | 10, 12 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (1 − 𝐼) ∈ ((1 − 𝐼)...(1 − 𝐼))) |
14 | 4 | zcnd 12112 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝐼 ∈ ℂ) |
15 | | 1cnd 10659 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 1 ∈
ℂ) |
16 | 14, 15 | pncan3d 11023 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (𝐼 + (1 − 𝐼)) = 1) |
17 | 16 | eqcomd 2765 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 1 = (𝐼 + (1 − 𝐼))) |
18 | 5 | nncnd 11675 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℂ) |
19 | 18 | adantr 485 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝑀 ∈ ℂ) |
20 | 19, 15, 14 | npncand 11044 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → ((𝑀 − 1) + (1 − 𝐼)) = (𝑀 − 𝐼)) |
21 | 20 | eqcomd 2765 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (𝑀 − 𝐼) = ((𝑀 − 1) + (1 − 𝐼))) |
22 | 4, 9, 10, 10, 11, 13, 17, 21 | fzadd2d 39530 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (𝑥 + (1 − 𝐼)) ∈ (1...(𝑀 − 𝐼))) |
23 | 3 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝐼 ∈ ℤ) |
24 | 6 | adantr 485 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝑀 ∈ ℤ) |
25 | | 1zzd 12037 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 1 ∈
ℤ) |
26 | 24, 25 | zsubcld 12116 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝑀 − 1) ∈ ℤ) |
27 | | elfznn 12970 |
. . . . . 6
⊢ (𝑦 ∈ (1...(𝑀 − 𝐼)) → 𝑦 ∈ ℕ) |
28 | 27 | adantl 486 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝑦 ∈ ℕ) |
29 | | nnz 12028 |
. . . . 5
⊢ (𝑦 ∈ ℕ → 𝑦 ∈
ℤ) |
30 | 28, 29 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝑦 ∈ ℤ) |
31 | 25, 23 | zsubcld 12116 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (1 − 𝐼) ∈ ℤ) |
32 | 30, 31 | zsubcld 12116 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝑦 − (1 − 𝐼)) ∈ ℤ) |
33 | 23 | zred 12111 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝐼 ∈ ℝ) |
34 | 33 | recnd 10692 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝐼 ∈ ℂ) |
35 | | 1cnd 10659 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 1 ∈
ℂ) |
36 | 34, 35 | pncan3d 11023 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝐼 + (1 − 𝐼)) = 1) |
37 | 27 | nnge1d 11707 |
. . . . . 6
⊢ (𝑦 ∈ (1...(𝑀 − 𝐼)) → 1 ≤ 𝑦) |
38 | 37 | adantl 486 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 1 ≤ 𝑦) |
39 | 36, 38 | eqbrtrd 5047 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝐼 + (1 − 𝐼)) ≤ 𝑦) |
40 | | 1red 10665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 1 ∈
ℝ) |
41 | 40, 33 | resubcld 11091 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (1 − 𝐼) ∈ ℝ) |
42 | 28 | nnred 11674 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝑦 ∈ ℝ) |
43 | 33, 41, 42 | 3jca 1126 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝐼 ∈ ℝ ∧ (1 − 𝐼) ∈ ℝ ∧ 𝑦 ∈
ℝ)) |
44 | | leaddsub 11139 |
. . . . 5
⊢ ((𝐼 ∈ ℝ ∧ (1 −
𝐼) ∈ ℝ ∧
𝑦 ∈ ℝ) →
((𝐼 + (1 − 𝐼)) ≤ 𝑦 ↔ 𝐼 ≤ (𝑦 − (1 − 𝐼)))) |
45 | 43, 44 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → ((𝐼 + (1 − 𝐼)) ≤ 𝑦 ↔ 𝐼 ≤ (𝑦 − (1 − 𝐼)))) |
46 | 39, 45 | mpbid 235 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝐼 ≤ (𝑦 − (1 − 𝐼))) |
47 | | elfzle2 12945 |
. . . . . 6
⊢ (𝑦 ∈ (1...(𝑀 − 𝐼)) → 𝑦 ≤ (𝑀 − 𝐼)) |
48 | 47 | adantl 486 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝑦 ≤ (𝑀 − 𝐼)) |
49 | 18 | adantr 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝑀 ∈ ℂ) |
50 | 23 | zcnd 12112 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝐼 ∈ ℂ) |
51 | 49, 35, 50 | npncand 11044 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → ((𝑀 − 1) + (1 − 𝐼)) = (𝑀 − 𝐼)) |
52 | 48, 51 | breqtrrd 5053 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝑦 ≤ ((𝑀 − 1) + (1 − 𝐼))) |
53 | 31 | zred 12111 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (1 − 𝐼) ∈ ℝ) |
54 | 26 | zred 12111 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝑀 − 1) ∈ ℝ) |
55 | 42, 53, 54 | lesubaddd 11260 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → ((𝑦 − (1 − 𝐼)) ≤ (𝑀 − 1) ↔ 𝑦 ≤ ((𝑀 − 1) + (1 − 𝐼)))) |
56 | 52, 55 | mpbird 260 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝑦 − (1 − 𝐼)) ≤ (𝑀 − 1)) |
57 | 23, 26, 32, 46, 56 | elfzd 12932 |
. 2
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝑦 − (1 − 𝐼)) ∈ (𝐼...(𝑀 − 1))) |
58 | | 1cnd 10659 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → 1 ∈
ℂ) |
59 | 34 | adantrl 716 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → 𝐼 ∈ ℂ) |
60 | 58, 59 | subcld 11020 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → (1 − 𝐼) ∈ ℂ) |
61 | | elfzelz 12941 |
. . . . . . 7
⊢ (𝑥 ∈ (𝐼...(𝑀 − 1)) → 𝑥 ∈ ℤ) |
62 | 61 | ad2antrl 728 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → 𝑥 ∈ ℤ) |
63 | | zcn 12010 |
. . . . . 6
⊢ (𝑥 ∈ ℤ → 𝑥 ∈
ℂ) |
64 | 62, 63 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → 𝑥 ∈ ℂ) |
65 | 28 | adantrl 716 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → 𝑦 ∈ ℕ) |
66 | | nncn 11667 |
. . . . . 6
⊢ (𝑦 ∈ ℕ → 𝑦 ∈
ℂ) |
67 | 65, 66 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → 𝑦 ∈ ℂ) |
68 | 60, 64, 67 | addrsub 11080 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → (((1 − 𝐼) + 𝑥) = 𝑦 ↔ 𝑥 = (𝑦 − (1 − 𝐼)))) |
69 | 68 | bicomd 226 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → (𝑥 = (𝑦 − (1 − 𝐼)) ↔ ((1 − 𝐼) + 𝑥) = 𝑦)) |
70 | 60, 64 | addcomd 10865 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → ((1 − 𝐼) + 𝑥) = (𝑥 + (1 − 𝐼))) |
71 | 70 | eqeq1d 2761 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → (((1 − 𝐼) + 𝑥) = 𝑦 ↔ (𝑥 + (1 − 𝐼)) = 𝑦)) |
72 | | eqcom 2766 |
. . . . 5
⊢ ((𝑥 + (1 − 𝐼)) = 𝑦 ↔ 𝑦 = (𝑥 + (1 − 𝐼))) |
73 | 72 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → ((𝑥 + (1 − 𝐼)) = 𝑦 ↔ 𝑦 = (𝑥 + (1 − 𝐼)))) |
74 | 71, 73 | bitrd 282 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → (((1 − 𝐼) + 𝑥) = 𝑦 ↔ 𝑦 = (𝑥 + (1 − 𝐼)))) |
75 | 69, 74 | bitrd 282 |
. 2
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → (𝑥 = (𝑦 − (1 − 𝐼)) ↔ 𝑦 = (𝑥 + (1 − 𝐼)))) |
76 | 1, 22, 57, 75 | f1o2d 7388 |
1
⊢ (𝜑 → 𝐹:(𝐼...(𝑀 − 1))–1-1-onto→(1...(𝑀 − 𝐼))) |