Proof of Theorem metakunt28
Step | Hyp | Ref
| Expression |
1 | | metakunt28.5 |
. . . . 5
⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) |
2 | 1 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))) |
3 | | metakunt28.7 |
. . . . . . . 8
⊢ (𝜑 → ¬ 𝑋 = 𝐼) |
4 | 3 | adantr 485 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ¬ 𝑋 = 𝐼) |
5 | | simpr 489 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) |
6 | 5 | eqeq1d 2761 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥 = 𝐼 ↔ 𝑋 = 𝐼)) |
7 | 6 | notbid 322 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (¬ 𝑥 = 𝐼 ↔ ¬ 𝑋 = 𝐼)) |
8 | 4, 7 | mpbird 260 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ¬ 𝑥 = 𝐼) |
9 | 8 | iffalsed 4424 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) |
10 | | metakunt28.8 |
. . . . . . . . 9
⊢ (𝜑 → ¬ 𝑋 < 𝐼) |
11 | 10 | adantr 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ¬ 𝑋 < 𝐼) |
12 | 5 | breq1d 5035 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥 < 𝐼 ↔ 𝑋 < 𝐼)) |
13 | 12 | notbid 322 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (¬ 𝑥 < 𝐼 ↔ ¬ 𝑋 < 𝐼)) |
14 | 11, 13 | mpbird 260 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ¬ 𝑥 < 𝐼) |
15 | 14 | iffalsed 4424 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) = (𝑥 − 1)) |
16 | 5 | oveq1d 7158 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥 − 1) = (𝑋 − 1)) |
17 | 15, 16 | eqtrd 2794 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) = (𝑋 − 1)) |
18 | 9, 17 | eqtrd 2794 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = (𝑋 − 1)) |
19 | | metakunt28.4 |
. . . 4
⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) |
20 | 19 | elfzelzd 12942 |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ ℤ) |
21 | | 1zzd 12037 |
. . . . 5
⊢ (𝜑 → 1 ∈
ℤ) |
22 | 20, 21 | zsubcld 12116 |
. . . 4
⊢ (𝜑 → (𝑋 − 1) ∈ ℤ) |
23 | 2, 18, 19, 22 | fvmptd 6759 |
. . 3
⊢ (𝜑 → (𝐴‘𝑋) = (𝑋 − 1)) |
24 | 23 | fveq2d 6655 |
. 2
⊢ (𝜑 → (𝐵‘(𝐴‘𝑋)) = (𝐵‘(𝑋 − 1))) |
25 | | metakunt28.6 |
. . . 4
⊢ 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) |
26 | 25 | a1i 11 |
. . 3
⊢ (𝜑 → 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))))) |
27 | 22 | zred 12111 |
. . . . . . . . 9
⊢ (𝜑 → (𝑋 − 1) ∈ ℝ) |
28 | 20 | zred 12111 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ∈ ℝ) |
29 | | metakunt28.1 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℕ) |
30 | 29 | nnred 11674 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ ℝ) |
31 | | 1rp 12419 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℝ+ |
32 | 31 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → 1 ∈
ℝ+) |
33 | 28, 32 | ltsubrpd 12489 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑋 − 1) < 𝑋) |
34 | | elfzle2 12945 |
. . . . . . . . . . 11
⊢ (𝑋 ∈ (1...𝑀) → 𝑋 ≤ 𝑀) |
35 | 19, 34 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ≤ 𝑀) |
36 | 27, 28, 30, 33, 35 | ltletrd 10823 |
. . . . . . . . 9
⊢ (𝜑 → (𝑋 − 1) < 𝑀) |
37 | 27, 36 | ltned 10799 |
. . . . . . . 8
⊢ (𝜑 → (𝑋 − 1) ≠ 𝑀) |
38 | 37 | adantr 485 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → (𝑋 − 1) ≠ 𝑀) |
39 | 38 | neneqd 2954 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → ¬ (𝑋 − 1) = 𝑀) |
40 | | simpr 489 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → 𝑧 = (𝑋 − 1)) |
41 | 40 | eqeq1d 2761 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → (𝑧 = 𝑀 ↔ (𝑋 − 1) = 𝑀)) |
42 | 41 | notbid 322 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → (¬ 𝑧 = 𝑀 ↔ ¬ (𝑋 − 1) = 𝑀)) |
43 | 39, 42 | mpbird 260 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → ¬ 𝑧 = 𝑀) |
44 | 43 | iffalsed 4424 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))) = if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))) |
45 | 3 | neqned 2956 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ≠ 𝐼) |
46 | | metakunt28.2 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐼 ∈ ℕ) |
47 | 46 | nnred 11674 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐼 ∈ ℝ) |
48 | 47, 28, 10 | nltled 10813 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐼 ≤ 𝑋) |
49 | 47, 28, 48 | leltned 10816 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐼 < 𝑋 ↔ 𝑋 ≠ 𝐼)) |
50 | 45, 49 | mpbird 260 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 < 𝑋) |
51 | 46 | nnzd 12110 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐼 ∈ ℤ) |
52 | 51, 20 | zltlem1d 39531 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐼 < 𝑋 ↔ 𝐼 ≤ (𝑋 − 1))) |
53 | 50, 52 | mpbid 235 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ≤ (𝑋 − 1)) |
54 | 47, 27 | lenltd 10809 |
. . . . . . . . 9
⊢ (𝜑 → (𝐼 ≤ (𝑋 − 1) ↔ ¬ (𝑋 − 1) < 𝐼)) |
55 | 53, 54 | mpbid 235 |
. . . . . . . 8
⊢ (𝜑 → ¬ (𝑋 − 1) < 𝐼) |
56 | 55 | adantr 485 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → ¬ (𝑋 − 1) < 𝐼) |
57 | 40 | breq1d 5035 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → (𝑧 < 𝐼 ↔ (𝑋 − 1) < 𝐼)) |
58 | 57 | notbid 322 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → (¬ 𝑧 < 𝐼 ↔ ¬ (𝑋 − 1) < 𝐼)) |
59 | 56, 58 | mpbird 260 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → ¬ 𝑧 < 𝐼) |
60 | 59 | iffalsed 4424 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))) = (𝑧 + (1 − 𝐼))) |
61 | 40 | oveq1d 7158 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → (𝑧 + (1 − 𝐼)) = ((𝑋 − 1) + (1 − 𝐼))) |
62 | 20 | zcnd 12112 |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ ℂ) |
63 | | 1cnd 10659 |
. . . . . . . 8
⊢ (𝜑 → 1 ∈
ℂ) |
64 | 46 | nncnd 11675 |
. . . . . . . 8
⊢ (𝜑 → 𝐼 ∈ ℂ) |
65 | 62, 63, 64 | npncand 11044 |
. . . . . . 7
⊢ (𝜑 → ((𝑋 − 1) + (1 − 𝐼)) = (𝑋 − 𝐼)) |
66 | 65 | adantr 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → ((𝑋 − 1) + (1 − 𝐼)) = (𝑋 − 𝐼)) |
67 | 61, 66 | eqtrd 2794 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → (𝑧 + (1 − 𝐼)) = (𝑋 − 𝐼)) |
68 | 60, 67 | eqtrd 2794 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))) = (𝑋 − 𝐼)) |
69 | 44, 68 | eqtrd 2794 |
. . 3
⊢ ((𝜑 ∧ 𝑧 = (𝑋 − 1)) → if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))) = (𝑋 − 𝐼)) |
70 | 29 | nnzd 12110 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ℤ) |
71 | | 1red 10665 |
. . . . . 6
⊢ (𝜑 → 1 ∈
ℝ) |
72 | 46 | nnge1d 11707 |
. . . . . 6
⊢ (𝜑 → 1 ≤ 𝐼) |
73 | 71, 47, 28, 72, 50 | lelttrd 10821 |
. . . . 5
⊢ (𝜑 → 1 < 𝑋) |
74 | 21, 20 | zltlem1d 39531 |
. . . . 5
⊢ (𝜑 → (1 < 𝑋 ↔ 1 ≤ (𝑋 − 1))) |
75 | 73, 74 | mpbid 235 |
. . . 4
⊢ (𝜑 → 1 ≤ (𝑋 − 1)) |
76 | 28, 71 | resubcld 11091 |
. . . . 5
⊢ (𝜑 → (𝑋 − 1) ∈ ℝ) |
77 | | 0le1 11186 |
. . . . . . 7
⊢ 0 ≤
1 |
78 | 77 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 0 ≤ 1) |
79 | 28, 71 | subge02d 11255 |
. . . . . 6
⊢ (𝜑 → (0 ≤ 1 ↔ (𝑋 − 1) ≤ 𝑋)) |
80 | 78, 79 | mpbid 235 |
. . . . 5
⊢ (𝜑 → (𝑋 − 1) ≤ 𝑋) |
81 | 76, 28, 30, 80, 35 | letrd 10820 |
. . . 4
⊢ (𝜑 → (𝑋 − 1) ≤ 𝑀) |
82 | 21, 70, 22, 75, 81 | elfzd 12932 |
. . 3
⊢ (𝜑 → (𝑋 − 1) ∈ (1...𝑀)) |
83 | 20, 51 | zsubcld 12116 |
. . 3
⊢ (𝜑 → (𝑋 − 𝐼) ∈ ℤ) |
84 | 26, 69, 82, 83 | fvmptd 6759 |
. 2
⊢ (𝜑 → (𝐵‘(𝑋 − 1)) = (𝑋 − 𝐼)) |
85 | 24, 84 | eqtrd 2794 |
1
⊢ (𝜑 → (𝐵‘(𝐴‘𝑋)) = (𝑋 − 𝐼)) |