Step | Hyp | Ref
| Expression |
1 | | znle2.y |
. . . . 5
⊢ 𝑌 =
(ℤ/nℤ‘𝑁) |
2 | 1 | fvexi 6559 |
. . . 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 20384 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ0
→ 𝐹:𝑊–1-1-onto→𝑋) |
11 | | f1ocnv 6502 |
. . . . . . . . 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 6490 |
. . . . . . . 8
⊢ (◡𝐹:𝑋–1-1-onto→𝑊 → ◡𝐹:𝑋⟶𝑊) |
14 | 12, 13 | syl 17 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ0
→ ◡𝐹:𝑋⟶𝑊) |
15 | | sseq1 3919 |
. . . . . . . . . 10
⊢ (ℤ
= if(𝑁 = 0, ℤ,
(0..^𝑁)) → (ℤ
⊆ ℤ ↔ if(𝑁
= 0, ℤ, (0..^𝑁))
⊆ ℤ)) |
16 | | sseq1 3919 |
. . . . . . . . . 10
⊢
((0..^𝑁) = if(𝑁 = 0, ℤ, (0..^𝑁)) → ((0..^𝑁) ⊆ ℤ ↔
if(𝑁 = 0, ℤ,
(0..^𝑁)) ⊆
ℤ)) |
17 | | ssid 3916 |
. . . . . . . . . 10
⊢ ℤ
⊆ ℤ |
18 | | elfzoelz 12892 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (0..^𝑁) → 𝑥 ∈ ℤ) |
19 | 18 | ssriv 3899 |
. . . . . . . . . 10
⊢
(0..^𝑁) ⊆
ℤ |
20 | 15, 16, 17, 19 | keephyp 4456 |
. . . . . . . . 9
⊢ if(𝑁 = 0, ℤ, (0..^𝑁)) ⊆
ℤ |
21 | 9, 20 | eqsstri 3928 |
. . . . . . . 8
⊢ 𝑊 ⊆
ℤ |
22 | | zssre 11842 |
. . . . . . . 8
⊢ ℤ
⊆ ℝ |
23 | 21, 22 | sstri 3904 |
. . . . . . 7
⊢ 𝑊 ⊆
ℝ |
24 | | fss 6402 |
. . . . . . 7
⊢ ((◡𝐹:𝑋⟶𝑊 ∧ 𝑊 ⊆ ℝ) → ◡𝐹:𝑋⟶ℝ) |
25 | 14, 23, 24 | sylancl 586 |
. . . . . 6
⊢ (𝑁 ∈ ℕ0
→ ◡𝐹:𝑋⟶ℝ) |
26 | 25 | ffvelrnda 6723 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋) → (◡𝐹‘𝑥) ∈ ℝ) |
27 | 26 | leidd 11060 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋) → (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑥)) |
28 | 1, 8, 9, 6, 4 | znleval2 20388 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥 ≤ 𝑥 ↔ (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑥))) |
29 | 28 | 3anidm23 1414 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋) → (𝑥 ≤ 𝑥 ↔ (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑥))) |
30 | 27, 29 | mpbird 258 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋) → 𝑥 ≤ 𝑥) |
31 | 1, 8, 9, 6, 4 | znleval2 20388 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥 ≤ 𝑦 ↔ (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦))) |
32 | 1, 8, 9, 6, 4 | znleval2 20388 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑦 ≤ 𝑥 ↔ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑥))) |
33 | 32 | 3com23 1119 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑦 ≤ 𝑥 ↔ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑥))) |
34 | 31, 33 | anbi12d 630 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ ((◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦) ∧ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑥)))) |
35 | 26 | 3adant3 1125 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (◡𝐹‘𝑥) ∈ ℝ) |
36 | 25 | ffvelrnda 6723 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝑦 ∈ 𝑋) → (◡𝐹‘𝑦) ∈ ℝ) |
37 | 36 | 3adant2 1124 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (◡𝐹‘𝑦) ∈ ℝ) |
38 | 35, 37 | letri3d 10635 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((◡𝐹‘𝑥) = (◡𝐹‘𝑦) ↔ ((◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦) ∧ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑥)))) |
39 | | f1of1 6489 |
. . . . . . . 8
⊢ (◡𝐹:𝑋–1-1-onto→𝑊 → ◡𝐹:𝑋–1-1→𝑊) |
40 | 12, 39 | syl 17 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ0
→ ◡𝐹:𝑋–1-1→𝑊) |
41 | | f1fveq 6892 |
. . . . . . 7
⊢ ((◡𝐹:𝑋–1-1→𝑊 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((◡𝐹‘𝑥) = (◡𝐹‘𝑦) ↔ 𝑥 = 𝑦)) |
42 | 40, 41 | sylan 580 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((◡𝐹‘𝑥) = (◡𝐹‘𝑦) ↔ 𝑥 = 𝑦)) |
43 | 42 | 3impb 1108 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((◡𝐹‘𝑥) = (◡𝐹‘𝑦) ↔ 𝑥 = 𝑦)) |
44 | 34, 38, 43 | 3bitr2d 308 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦)) |
45 | 44 | biimpd 230 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦)) |
46 | 26 | 3ad2antr1 1181 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (◡𝐹‘𝑥) ∈ ℝ) |
47 | 36 | 3ad2antr2 1182 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (◡𝐹‘𝑦) ∈ ℝ) |
48 | 25 | ffvelrnda 6723 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑧 ∈ 𝑋) → (◡𝐹‘𝑧) ∈ ℝ) |
49 | 48 | 3ad2antr3 1183 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (◡𝐹‘𝑧) ∈ ℝ) |
50 | | letr 10587 |
. . . . 5
⊢ (((◡𝐹‘𝑥) ∈ ℝ ∧ (◡𝐹‘𝑦) ∈ ℝ ∧ (◡𝐹‘𝑧) ∈ ℝ) → (((◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦) ∧ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑧)) → (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑧))) |
51 | 46, 47, 49, 50 | syl3anc 1364 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦) ∧ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑧)) → (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑧))) |
52 | 31 | 3adant3r3 1177 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 ≤ 𝑦 ↔ (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦))) |
53 | 1, 8, 9, 6, 4 | znleval2 20388 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦 ≤ 𝑧 ↔ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑧))) |
54 | 53 | 3adant3r1 1175 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦 ≤ 𝑧 ↔ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑧))) |
55 | 52, 54 | anbi12d 630 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) ↔ ((◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦) ∧ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑧)))) |
56 | 1, 8, 9, 6, 4 | znleval2 20388 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥 ≤ 𝑧 ↔ (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑧))) |
57 | 56 | 3adant3r2 1176 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 ≤ 𝑧 ↔ (◡𝐹‘𝑥) ≤ (◡𝐹‘𝑧))) |
58 | 51, 55, 57 | 3imtr4d 295 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) |
59 | 3, 5, 7, 30, 45, 58 | isposd 17398 |
. 2
⊢ (𝑁 ∈ ℕ0
→ 𝑌 ∈
Poset) |
60 | 35, 37 | letrid 10645 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦) ∨ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑥))) |
61 | 31, 33 | orbi12d 913 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ((◡𝐹‘𝑥) ≤ (◡𝐹‘𝑦) ∨ (◡𝐹‘𝑦) ≤ (◡𝐹‘𝑥)))) |
62 | 60, 61 | mpbird 258 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) |
63 | 62 | 3expb 1113 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) |
64 | 63 | ralrimivva 3160 |
. 2
⊢ (𝑁 ∈ ℕ0
→ ∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) |
65 | 4, 6 | istos 17478 |
. 2
⊢ (𝑌 ∈ Toset ↔ (𝑌 ∈ Poset ∧
∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))) |
66 | 59, 64, 65 | sylanbrc 583 |
1
⊢ (𝑁 ∈ ℕ0
→ 𝑌 ∈
Toset) |