Step | Hyp | Ref
| Expression |
1 | | znle2.y |
. . . . 5
⊢ 𝑌 =
(ℤ/nℤ‘𝑁) |
2 | 1 | fvexi 6701 |
. . . 4
⊢ 𝑌 ∈ V |
3 | 2 | a1i 11 |
. . 3
⊢ (𝑁 ∈ ℕ0
→ 𝑌 ∈
V) |
4 | | znleval.x |
. . . 4
⊢ 𝑋 = (Base‘𝑌) |
5 | 4 | a1i 11 |
. . 3
⊢ (𝑁 ∈ ℕ0
→ 𝑋 =
(Base‘𝑌)) |
6 | | znle2.l |
. . . 4
⊢ ≤ =
(le‘𝑌) |
7 | 6 | a1i 11 |
. . 3
⊢ (𝑁 ∈ ℕ0
→ ≤ = (le‘𝑌)) |
8 | | znle2.f |
. . . . . . . . . 10
⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) |
9 | | znle2.w |
. . . . . . . . . 10
⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) |
10 | 1, 4, 8, 9 | znf1o 20383 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ0
→ 𝐹:𝑊–1-1-onto→𝑋) |
11 | | f1ocnv 6643 |
. . . . . . . . 9
⊢ (𝐹:𝑊–1-1-onto→𝑋 → ◡𝐹:𝑋–1-1-onto→𝑊) |
12 | 10, 11 | syl 17 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ0
→ ◡𝐹:𝑋–1-1-onto→𝑊) |
13 | | f1of 6631 |
. . . . . . . 8
⊢ (◡𝐹:𝑋–1-1-onto→𝑊 → ◡𝐹:𝑋⟶𝑊) |
14 | 12, 13 | syl 17 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ0
→ ◡𝐹:𝑋⟶𝑊) |
15 | | sseq1 3912 |
. . . . . . . . . 10
⊢ (ℤ
= if(𝑁 = 0, ℤ,
(0..^𝑁)) → (ℤ
⊆ ℤ ↔ if(𝑁
= 0, ℤ, (0..^𝑁))
⊆ ℤ)) |
16 | | sseq1 3912 |
. . . . . . . . . 10
⊢
((0..^𝑁) = if(𝑁 = 0, ℤ, (0..^𝑁)) → ((0..^𝑁) ⊆ ℤ ↔
if(𝑁 = 0, ℤ,
(0..^𝑁)) ⊆
ℤ)) |
17 | | ssid 3909 |
. . . . . . . . . 10
⊢ ℤ
⊆ ℤ |
18 | | fzossz 13161 |
. . . . . . . . . 10
⊢
(0..^𝑁) ⊆
ℤ |
19 | 15, 16, 17, 18 | keephyp 4495 |
. . . . . . . . 9
⊢ if(𝑁 = 0, ℤ, (0..^𝑁)) ⊆
ℤ |
20 | 9, 19 | eqsstri 3921 |
. . . . . . . 8
⊢ 𝑊 ⊆
ℤ |
21 | | zssre 12082 |
. . . . . . . 8
⊢ ℤ
⊆ ℝ |
22 | 20, 21 | sstri 3896 |
. . . . . . 7
⊢ 𝑊 ⊆
ℝ |
23 | | fss 6532 |
. . . . . . 7
⊢ ((◡𝐹:𝑋⟶𝑊 ∧ 𝑊 ⊆ ℝ) → ◡𝐹:𝑋⟶ℝ) |
24 | 14, 22, 23 | sylancl 589 |
. . . . . 6
⊢ (𝑁 ∈ ℕ0
→ ◡𝐹:𝑋⟶ℝ) |
25 | 24 | ffvelrnda 6874 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋) → (◡𝐹‘𝑥) ∈ ℝ) |
26 | 25 | leidd 11297 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋) → (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑥)) |
27 | 1, 8, 9, 6, 4 | znleval2 20387 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥 ≤ 𝑥 ↔ (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑥))) |
28 | 27 | 3anidm23 1422 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋) → (𝑥 ≤ 𝑥 ↔ (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑥))) |
29 | 26, 28 | mpbird 260 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋) → 𝑥 ≤ 𝑥) |
30 | 1, 8, 9, 6, 4 | znleval2 20387 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥 ≤ 𝑦 ↔ (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦))) |
31 | 1, 8, 9, 6, 4 | znleval2 20387 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑦 ≤ 𝑥 ↔ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑥))) |
32 | 31 | 3com23 1127 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑦 ≤ 𝑥 ↔ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑥))) |
33 | 30, 32 | anbi12d 634 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ ((◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦) ∧ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑥)))) |
34 | 25 | 3adant3 1133 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (◡𝐹‘𝑥) ∈ ℝ) |
35 | 24 | ffvelrnda 6874 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝑦 ∈ 𝑋) → (◡𝐹‘𝑦) ∈ ℝ) |
36 | 35 | 3adant2 1132 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (◡𝐹‘𝑦) ∈ ℝ) |
37 | 34, 36 | letri3d 10873 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((◡𝐹‘𝑥) = (◡𝐹‘𝑦) ↔ ((◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦) ∧ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑥)))) |
38 | | f1of1 6630 |
. . . . . . . 8
⊢ (◡𝐹:𝑋–1-1-onto→𝑊 → ◡𝐹:𝑋–1-1→𝑊) |
39 | 12, 38 | syl 17 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ0
→ ◡𝐹:𝑋–1-1→𝑊) |
40 | | f1fveq 7044 |
. . . . . . 7
⊢ ((◡𝐹:𝑋–1-1→𝑊 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((◡𝐹‘𝑥) = (◡𝐹‘𝑦) ↔ 𝑥 = 𝑦)) |
41 | 39, 40 | sylan 583 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((◡𝐹‘𝑥) = (◡𝐹‘𝑦) ↔ 𝑥 = 𝑦)) |
42 | 41 | 3impb 1116 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((◡𝐹‘𝑥) = (◡𝐹‘𝑦) ↔ 𝑥 = 𝑦)) |
43 | 33, 37, 42 | 3bitr2d 310 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦)) |
44 | 43 | biimpd 232 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦)) |
45 | 25 | 3ad2antr1 1189 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (◡𝐹‘𝑥) ∈ ℝ) |
46 | 35 | 3ad2antr2 1190 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (◡𝐹‘𝑦) ∈ ℝ) |
47 | 24 | ffvelrnda 6874 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑧 ∈ 𝑋) → (◡𝐹‘𝑧) ∈ ℝ) |
48 | 47 | 3ad2antr3 1191 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (◡𝐹‘𝑧) ∈ ℝ) |
49 | | letr 10825 |
. . . . 5
⊢ (((◡𝐹‘𝑥) ∈ ℝ ∧ (◡𝐹‘𝑦) ∈ ℝ ∧ (◡𝐹‘𝑧) ∈ ℝ) → (((◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦) ∧ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑧)) → (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑧))) |
50 | 45, 46, 48, 49 | syl3anc 1372 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦) ∧ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑧)) → (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑧))) |
51 | 30 | 3adant3r3 1185 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 ≤ 𝑦 ↔ (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦))) |
52 | 1, 8, 9, 6, 4 | znleval2 20387 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦 ≤ 𝑧 ↔ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑧))) |
53 | 52 | 3adant3r1 1183 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦 ≤ 𝑧 ↔ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑧))) |
54 | 51, 53 | anbi12d 634 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) ↔ ((◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦) ∧ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑧)))) |
55 | 1, 8, 9, 6, 4 | znleval2 20387 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥 ≤ 𝑧 ↔ (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑧))) |
56 | 55 | 3adant3r2 1184 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 ≤ 𝑧 ↔ (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑧))) |
57 | 50, 54, 56 | 3imtr4d 297 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) |
58 | 3, 5, 7, 29, 44, 57 | isposd 17694 |
. 2
⊢ (𝑁 ∈ ℕ0
→ 𝑌 ∈
Poset) |
59 | 34, 36 | letrid 10883 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦) ∨ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑥))) |
60 | 30, 32 | orbi12d 918 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ((◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦) ∨ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑥)))) |
61 | 59, 60 | mpbird 260 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) |
62 | 61 | 3expb 1121 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) |
63 | 62 | ralrimivva 3104 |
. 2
⊢ (𝑁 ∈ ℕ0
→ ∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) |
64 | 4, 6 | istos 17774 |
. 2
⊢ (𝑌 ∈ Toset ↔ (𝑌 ∈ Poset ∧
∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))) |
65 | 58, 63, 64 | sylanbrc 586 |
1
⊢ (𝑁 ∈ ℕ0
→ 𝑌 ∈
Toset) |