Proof of Theorem metakunt1
Step | Hyp | Ref
| Expression |
1 | | eleq1 2838 |
. . 3
⊢ (𝑀 = if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) → (𝑀 ∈ (1...𝑀) ↔ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) ∈ (1...𝑀))) |
2 | | eleq1 2838 |
. . 3
⊢ (if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) = if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) → (if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) ∈ (1...𝑀) ↔ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) ∈ (1...𝑀))) |
3 | | 1zzd 12037 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑀)) ∧ 𝑥 = 𝐼) → 1 ∈ ℤ) |
4 | | metakunt1.1 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℕ) |
5 | 4 | nnzd 12110 |
. . . . 5
⊢ (𝜑 → 𝑀 ∈ ℤ) |
6 | 5 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑀)) ∧ 𝑥 = 𝐼) → 𝑀 ∈ ℤ) |
7 | 4 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑀)) ∧ 𝑥 = 𝐼) → 𝑀 ∈ ℕ) |
8 | 7 | nnge1d 11707 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑀)) ∧ 𝑥 = 𝐼) → 1 ≤ 𝑀) |
9 | 4 | nnred 11674 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℝ) |
10 | 9 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑀)) ∧ 𝑥 = 𝐼) → 𝑀 ∈ ℝ) |
11 | 10 | leidd 11229 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑀)) ∧ 𝑥 = 𝐼) → 𝑀 ≤ 𝑀) |
12 | 3, 6, 6, 8, 11 | elfzd 12932 |
. . 3
⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑀)) ∧ 𝑥 = 𝐼) → 𝑀 ∈ (1...𝑀)) |
13 | | eleq1 2838 |
. . . 4
⊢ (𝑥 = if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) → (𝑥 ∈ (1...𝑀) ↔ if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) ∈ (1...𝑀))) |
14 | | eleq1 2838 |
. . . 4
⊢ ((𝑥 − 1) = if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) → ((𝑥 − 1) ∈ (1...𝑀) ↔ if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) ∈ (1...𝑀))) |
15 | | simpllr 776 |
. . . 4
⊢ ((((𝜑 ∧ 𝑥 ∈ (1...𝑀)) ∧ ¬ 𝑥 = 𝐼) ∧ 𝑥 < 𝐼) → 𝑥 ∈ (1...𝑀)) |
16 | | pm4.56 987 |
. . . . . . 7
⊢ ((¬
𝑥 = 𝐼 ∧ ¬ 𝑥 < 𝐼) ↔ ¬ (𝑥 = 𝐼 ∨ 𝑥 < 𝐼)) |
17 | 16 | anbi2i 626 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑀)) ∧ (¬ 𝑥 = 𝐼 ∧ ¬ 𝑥 < 𝐼)) ↔ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) ∧ ¬ (𝑥 = 𝐼 ∨ 𝑥 < 𝐼))) |
18 | | metakunt1.2 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐼 ∈ ℕ) |
19 | 18 | nnred 11674 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐼 ∈ ℝ) |
20 | 19 | adantr 485 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) → 𝐼 ∈ ℝ) |
21 | | elfznn 12970 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℕ) |
22 | 21 | nnred 11674 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℝ) |
23 | 22 | adantl 486 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) → 𝑥 ∈ ℝ) |
24 | 20, 23 | jca 516 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) → (𝐼 ∈ ℝ ∧ 𝑥 ∈ ℝ)) |
25 | | axlttri 10735 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐼 < 𝑥 ↔ ¬ (𝐼 = 𝑥 ∨ 𝑥 < 𝐼))) |
26 | 24, 25 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) → (𝐼 < 𝑥 ↔ ¬ (𝐼 = 𝑥 ∨ 𝑥 < 𝐼))) |
27 | | eqcom 2766 |
. . . . . . . . . . 11
⊢ (𝐼 = 𝑥 ↔ 𝑥 = 𝐼) |
28 | 27 | orbi1i 912 |
. . . . . . . . . 10
⊢ ((𝐼 = 𝑥 ∨ 𝑥 < 𝐼) ↔ (𝑥 = 𝐼 ∨ 𝑥 < 𝐼)) |
29 | 28 | notbii 324 |
. . . . . . . . 9
⊢ (¬
(𝐼 = 𝑥 ∨ 𝑥 < 𝐼) ↔ ¬ (𝑥 = 𝐼 ∨ 𝑥 < 𝐼)) |
30 | 26, 29 | syl6bb 291 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) → (𝐼 < 𝑥 ↔ ¬ (𝑥 = 𝐼 ∨ 𝑥 < 𝐼))) |
31 | | 1zzd 12037 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → 1 ∈ ℤ) |
32 | 5 | 3ad2ant1 1131 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → 𝑀 ∈ ℤ) |
33 | | simp2 1135 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → 𝑥 ∈ (1...𝑀)) |
34 | 33 | elfzelzd 12942 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → 𝑥 ∈ ℤ) |
35 | 34, 31 | zsubcld 12116 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → (𝑥 − 1) ∈ ℤ) |
36 | | 1red 10665 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → 1 ∈ ℝ) |
37 | 20 | 3adant3 1130 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → 𝐼 ∈ ℝ) |
38 | 33, 22 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → 𝑥 ∈ ℝ) |
39 | 18 | nnge1d 11707 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 1 ≤ 𝐼) |
40 | 39 | 3ad2ant1 1131 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → 1 ≤ 𝐼) |
41 | | simp3 1136 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → 𝐼 < 𝑥) |
42 | 36, 37, 38, 40, 41 | lelttrd 10821 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → 1 < 𝑥) |
43 | 31, 34 | zltlem1d 39531 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → (1 < 𝑥 ↔ 1 ≤ (𝑥 − 1))) |
44 | 42, 43 | mpbid 235 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → 1 ≤ (𝑥 − 1)) |
45 | | 1red 10665 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) → 1 ∈ ℝ) |
46 | 23, 45 | resubcld 11091 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) → (𝑥 − 1) ∈ ℝ) |
47 | 9 | adantr 485 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) → 𝑀 ∈ ℝ) |
48 | | 0le1 11186 |
. . . . . . . . . . . . . . 15
⊢ 0 ≤
1 |
49 | 48 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (1...𝑀) → 0 ≤ 1) |
50 | | 1red 10665 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (1...𝑀) → 1 ∈ ℝ) |
51 | 22, 50 | subge02d 11255 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (1...𝑀) → (0 ≤ 1 ↔ (𝑥 − 1) ≤ 𝑥)) |
52 | 49, 51 | mpbid 235 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (1...𝑀) → (𝑥 − 1) ≤ 𝑥) |
53 | 52 | adantl 486 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) → (𝑥 − 1) ≤ 𝑥) |
54 | | elfzle2 12945 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ≤ 𝑀) |
55 | 54 | adantl 486 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) → 𝑥 ≤ 𝑀) |
56 | 46, 23, 47, 53, 55 | letrd 10820 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) → (𝑥 − 1) ≤ 𝑀) |
57 | 56 | 3adant3 1130 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → (𝑥 − 1) ≤ 𝑀) |
58 | 31, 32, 35, 44, 57 | elfzd 12932 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀) ∧ 𝐼 < 𝑥) → (𝑥 − 1) ∈ (1...𝑀)) |
59 | 58 | 3expia 1119 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) → (𝐼 < 𝑥 → (𝑥 − 1) ∈ (1...𝑀))) |
60 | 30, 59 | sylbird 263 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) → (¬ (𝑥 = 𝐼 ∨ 𝑥 < 𝐼) → (𝑥 − 1) ∈ (1...𝑀))) |
61 | 60 | imp 411 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑀)) ∧ ¬ (𝑥 = 𝐼 ∨ 𝑥 < 𝐼)) → (𝑥 − 1) ∈ (1...𝑀)) |
62 | 17, 61 | sylbi 220 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑀)) ∧ (¬ 𝑥 = 𝐼 ∧ ¬ 𝑥 < 𝐼)) → (𝑥 − 1) ∈ (1...𝑀)) |
63 | 62 | anassrs 472 |
. . . 4
⊢ ((((𝜑 ∧ 𝑥 ∈ (1...𝑀)) ∧ ¬ 𝑥 = 𝐼) ∧ ¬ 𝑥 < 𝐼) → (𝑥 − 1) ∈ (1...𝑀)) |
64 | 13, 14, 15, 63 | ifbothda 4451 |
. . 3
⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑀)) ∧ ¬ 𝑥 = 𝐼) → if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) ∈ (1...𝑀)) |
65 | 1, 2, 12, 64 | ifbothda 4451 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑀)) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) ∈ (1...𝑀)) |
66 | | metakunt1.4 |
. 2
⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) |
67 | 65, 66 | fmptd 6862 |
1
⊢ (𝜑 → 𝐴:(1...𝑀)⟶(1...𝑀)) |