Proof of Theorem metakunt10
Step | Hyp | Ref
| Expression |
1 | | metakunt10.4 |
. . 3
⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) |
2 | 1 | a1i 11 |
. 2
⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))) |
3 | | eqeq1 2763 |
. . . . 5
⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 = 𝐼 ↔ (𝐶‘𝑋) = 𝐼)) |
4 | | breq1 5028 |
. . . . . 6
⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 < 𝐼 ↔ (𝐶‘𝑋) < 𝐼)) |
5 | | id 22 |
. . . . . 6
⊢ (𝑥 = (𝐶‘𝑋) → 𝑥 = (𝐶‘𝑋)) |
6 | | oveq1 7150 |
. . . . . 6
⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 − 1) = ((𝐶‘𝑋) − 1)) |
7 | 4, 5, 6 | ifbieq12d 4441 |
. . . . 5
⊢ (𝑥 = (𝐶‘𝑋) → if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) = if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) |
8 | 3, 7 | ifbieq2d 4439 |
. . . 4
⊢ (𝑥 = (𝐶‘𝑋) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)))) |
9 | 8 | adantl 486 |
. . 3
⊢ (((𝜑 ∧ 𝑋 = 𝑀) ∧ 𝑥 = (𝐶‘𝑋)) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)))) |
10 | | fveq2 6651 |
. . . . . . . 8
⊢ (𝑋 = 𝑀 → (𝐶‘𝑋) = (𝐶‘𝑀)) |
11 | 10 | adantl 486 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 = 𝑀) → (𝐶‘𝑋) = (𝐶‘𝑀)) |
12 | | metakunt10.5 |
. . . . . . . . . 10
⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) |
13 | 12 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))) |
14 | | iftrue 4419 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑀 → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝐼) |
15 | 14 | adantl 486 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝐼) |
16 | | 1zzd 12037 |
. . . . . . . . . 10
⊢ (𝜑 → 1 ∈
ℤ) |
17 | | metakunt10.1 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℕ) |
18 | 17 | nnzd 12110 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ ℤ) |
19 | 17 | nnge1d 11707 |
. . . . . . . . . 10
⊢ (𝜑 → 1 ≤ 𝑀) |
20 | 17 | nnred 11674 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℝ) |
21 | 20 | leidd 11229 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ≤ 𝑀) |
22 | 16, 18, 18, 19, 21 | elfzd 12932 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ (1...𝑀)) |
23 | | metakunt10.2 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈ ℕ) |
24 | 13, 15, 22, 23 | fvmptd 6759 |
. . . . . . . 8
⊢ (𝜑 → (𝐶‘𝑀) = 𝐼) |
25 | 24 | adantr 485 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 = 𝑀) → (𝐶‘𝑀) = 𝐼) |
26 | 11, 25 | eqtrd 2794 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 = 𝑀) → (𝐶‘𝑋) = 𝐼) |
27 | | iftrue 4419 |
. . . . . 6
⊢ ((𝐶‘𝑋) = 𝐼 → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑀) |
28 | 26, 27 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 = 𝑀) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑀) |
29 | | simpr 489 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝑋 = 𝑀) |
30 | 29 | eqcomd 2765 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝑀 = 𝑋) |
31 | 28, 30 | eqtrd 2794 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 = 𝑀) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋) |
32 | 31 | adantr 485 |
. . 3
⊢ (((𝜑 ∧ 𝑋 = 𝑀) ∧ 𝑥 = (𝐶‘𝑋)) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋) |
33 | 9, 32 | eqtrd 2794 |
. 2
⊢ (((𝜑 ∧ 𝑋 = 𝑀) ∧ 𝑥 = (𝐶‘𝑋)) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑋) |
34 | | metakunt10.3 |
. . . . 5
⊢ (𝜑 → 𝐼 ≤ 𝑀) |
35 | 17, 23, 34, 12 | metakunt2 39633 |
. . . 4
⊢ (𝜑 → 𝐶:(1...𝑀)⟶(1...𝑀)) |
36 | 35 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝐶:(1...𝑀)⟶(1...𝑀)) |
37 | | metakunt10.6 |
. . . 4
⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) |
38 | 37 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝑋 ∈ (1...𝑀)) |
39 | 36, 38 | ffvelrnd 6836 |
. 2
⊢ ((𝜑 ∧ 𝑋 = 𝑀) → (𝐶‘𝑋) ∈ (1...𝑀)) |
40 | 2, 33, 39, 38 | fvmptd 6759 |
1
⊢ ((𝜑 ∧ 𝑋 = 𝑀) → (𝐴‘(𝐶‘𝑋)) = 𝑋) |