Step | Hyp | Ref
| Expression |
1 | | nnnn0 9237 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℕ0) |
2 | | znchr.y |
. . . . . . . 8
⊢ 𝑌 =
(ℤ/nℤ‘𝑁) |
3 | | eqid 2193 |
. . . . . . . 8
⊢
(Base‘𝑌) =
(Base‘𝑌) |
4 | | eqid 2193 |
. . . . . . . 8
⊢
(ℤRHom‘𝑌) = (ℤRHom‘𝑌) |
5 | 2, 3, 4 | znzrhfo 14113 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ0
→ (ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌)) |
6 | 1, 5 | syl 14 |
. . . . . 6
⊢ (𝑁 ∈ ℕ →
(ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌)) |
7 | | znrrg.e |
. . . . . . . 8
⊢ 𝐸 = (RLReg‘𝑌) |
8 | 7, 3 | rrgss 13746 |
. . . . . . 7
⊢ 𝐸 ⊆ (Base‘𝑌) |
9 | 8 | sseli 3175 |
. . . . . 6
⊢ (𝑥 ∈ 𝐸 → 𝑥 ∈ (Base‘𝑌)) |
10 | | foelrn 5787 |
. . . . . 6
⊢
(((ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌) ∧ 𝑥 ∈ (Base‘𝑌)) → ∃𝑛 ∈ ℤ 𝑥 = ((ℤRHom‘𝑌)‘𝑛)) |
11 | 6, 9, 10 | syl2an 289 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝐸) → ∃𝑛 ∈ ℤ 𝑥 = ((ℤRHom‘𝑌)‘𝑛)) |
12 | 11 | ex 115 |
. . . 4
⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝐸 → ∃𝑛 ∈ ℤ 𝑥 = ((ℤRHom‘𝑌)‘𝑛))) |
13 | | nncn 8980 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℂ) |
14 | 13 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → 𝑁 ∈ ℂ) |
15 | | simplr 528 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → 𝑛 ∈ ℤ) |
16 | | nnz 9326 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℤ) |
17 | 16 | ad2antrr 488 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → 𝑁 ∈ ℤ) |
18 | | nnne0 9000 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) |
19 | 18 | ad2antrr 488 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → 𝑁 ≠ 0) |
20 | | simpr 110 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 = 0 ∧ 𝑁 = 0) → 𝑁 = 0) |
21 | 20 | necon3ai 2413 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ≠ 0 → ¬ (𝑛 = 0 ∧ 𝑁 = 0)) |
22 | 19, 21 | syl 14 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ¬ (𝑛 = 0 ∧ 𝑁 = 0)) |
23 | | gcdn0cl 12089 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑛 = 0 ∧ 𝑁 = 0)) → (𝑛 gcd 𝑁) ∈ ℕ) |
24 | 15, 17, 22, 23 | syl21anc 1248 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑛 gcd 𝑁) ∈ ℕ) |
25 | 24 | nncnd 8986 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑛 gcd 𝑁) ∈ ℂ) |
26 | 24 | nnap0d 9018 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑛 gcd 𝑁) # 0) |
27 | 14, 25, 26 | divcanap2d 8801 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((𝑛 gcd 𝑁) · (𝑁 / (𝑛 gcd 𝑁))) = 𝑁) |
28 | | gcddvds 12090 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑛 gcd 𝑁) ∥ 𝑛 ∧ (𝑛 gcd 𝑁) ∥ 𝑁)) |
29 | 15, 17, 28 | syl2anc 411 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((𝑛 gcd 𝑁) ∥ 𝑛 ∧ (𝑛 gcd 𝑁) ∥ 𝑁)) |
30 | 29 | simpld 112 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑛 gcd 𝑁) ∥ 𝑛) |
31 | 24 | nnzd 9428 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑛 gcd 𝑁) ∈ ℤ) |
32 | 29 | simprd 114 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑛 gcd 𝑁) ∥ 𝑁) |
33 | | simpll 527 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → 𝑁 ∈ ℕ) |
34 | | nndivdvds 11929 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ ℕ ∧ (𝑛 gcd 𝑁) ∈ ℕ) → ((𝑛 gcd 𝑁) ∥ 𝑁 ↔ (𝑁 / (𝑛 gcd 𝑁)) ∈ ℕ)) |
35 | 33, 24, 34 | syl2anc 411 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((𝑛 gcd 𝑁) ∥ 𝑁 ↔ (𝑁 / (𝑛 gcd 𝑁)) ∈ ℕ)) |
36 | 32, 35 | mpbid 147 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑁 / (𝑛 gcd 𝑁)) ∈ ℕ) |
37 | 36 | nnzd 9428 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑁 / (𝑛 gcd 𝑁)) ∈ ℤ) |
38 | | dvdsmulc 11952 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑛 gcd 𝑁) ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ (𝑁 / (𝑛 gcd 𝑁)) ∈ ℤ) → ((𝑛 gcd 𝑁) ∥ 𝑛 → ((𝑛 gcd 𝑁) · (𝑁 / (𝑛 gcd 𝑁))) ∥ (𝑛 · (𝑁 / (𝑛 gcd 𝑁))))) |
39 | 31, 15, 37, 38 | syl3anc 1249 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((𝑛 gcd 𝑁) ∥ 𝑛 → ((𝑛 gcd 𝑁) · (𝑁 / (𝑛 gcd 𝑁))) ∥ (𝑛 · (𝑁 / (𝑛 gcd 𝑁))))) |
40 | 30, 39 | mpd 13 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((𝑛 gcd 𝑁) · (𝑁 / (𝑛 gcd 𝑁))) ∥ (𝑛 · (𝑁 / (𝑛 gcd 𝑁)))) |
41 | 27, 40 | eqbrtrrd 4053 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → 𝑁 ∥ (𝑛 · (𝑁 / (𝑛 gcd 𝑁)))) |
42 | | simpr 110 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) |
43 | 1 | ad2antrr 488 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → 𝑁 ∈
ℕ0) |
44 | 43, 5 | syl 14 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌)) |
45 | | fof 5468 |
. . . . . . . . . . . . . . . . 17
⊢
((ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌) → (ℤRHom‘𝑌):ℤ⟶(Base‘𝑌)) |
46 | 44, 45 | syl 14 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (ℤRHom‘𝑌):ℤ⟶(Base‘𝑌)) |
47 | 46, 37 | ffvelcdmd 5686 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((ℤRHom‘𝑌)‘(𝑁 / (𝑛 gcd 𝑁))) ∈ (Base‘𝑌)) |
48 | | eqid 2193 |
. . . . . . . . . . . . . . . 16
⊢
(.r‘𝑌) = (.r‘𝑌) |
49 | | eqid 2193 |
. . . . . . . . . . . . . . . 16
⊢
(0g‘𝑌) = (0g‘𝑌) |
50 | 7, 3, 48, 49 | rrgeq0i 13744 |
. . . . . . . . . . . . . . 15
⊢
((((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸 ∧ ((ℤRHom‘𝑌)‘(𝑁 / (𝑛 gcd 𝑁))) ∈ (Base‘𝑌)) → ((((ℤRHom‘𝑌)‘𝑛)(.r‘𝑌)((ℤRHom‘𝑌)‘(𝑁 / (𝑛 gcd 𝑁)))) = (0g‘𝑌) →
((ℤRHom‘𝑌)‘(𝑁 / (𝑛 gcd 𝑁))) = (0g‘𝑌))) |
51 | 42, 47, 50 | syl2anc 411 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((((ℤRHom‘𝑌)‘𝑛)(.r‘𝑌)((ℤRHom‘𝑌)‘(𝑁 / (𝑛 gcd 𝑁)))) = (0g‘𝑌) →
((ℤRHom‘𝑌)‘(𝑁 / (𝑛 gcd 𝑁))) = (0g‘𝑌))) |
52 | 2 | zncrng 14110 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ ℕ0
→ 𝑌 ∈
CRing) |
53 | 1, 52 | syl 14 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ → 𝑌 ∈ CRing) |
54 | 53 | crngringd 13489 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℕ → 𝑌 ∈ Ring) |
55 | 54 | ad2antrr 488 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → 𝑌 ∈ Ring) |
56 | 4 | zrhrhm 14088 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑌 ∈ Ring →
(ℤRHom‘𝑌)
∈ (ℤring RingHom 𝑌)) |
57 | 55, 56 | syl 14 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (ℤRHom‘𝑌) ∈ (ℤring
RingHom 𝑌)) |
58 | | zringbas 14062 |
. . . . . . . . . . . . . . . . . 18
⊢ ℤ =
(Base‘ℤring) |
59 | | zringmulr 14065 |
. . . . . . . . . . . . . . . . . 18
⊢ ·
= (.r‘ℤring) |
60 | 58, 59, 48 | rhmmul 13644 |
. . . . . . . . . . . . . . . . 17
⊢
(((ℤRHom‘𝑌) ∈ (ℤring RingHom
𝑌) ∧ 𝑛 ∈ ℤ ∧ (𝑁 / (𝑛 gcd 𝑁)) ∈ ℤ) →
((ℤRHom‘𝑌)‘(𝑛 · (𝑁 / (𝑛 gcd 𝑁)))) = (((ℤRHom‘𝑌)‘𝑛)(.r‘𝑌)((ℤRHom‘𝑌)‘(𝑁 / (𝑛 gcd 𝑁))))) |
61 | 57, 15, 37, 60 | syl3anc 1249 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((ℤRHom‘𝑌)‘(𝑛 · (𝑁 / (𝑛 gcd 𝑁)))) = (((ℤRHom‘𝑌)‘𝑛)(.r‘𝑌)((ℤRHom‘𝑌)‘(𝑁 / (𝑛 gcd 𝑁))))) |
62 | 61 | eqeq1d 2202 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (((ℤRHom‘𝑌)‘(𝑛 · (𝑁 / (𝑛 gcd 𝑁)))) = (0g‘𝑌) ↔
(((ℤRHom‘𝑌)‘𝑛)(.r‘𝑌)((ℤRHom‘𝑌)‘(𝑁 / (𝑛 gcd 𝑁)))) = (0g‘𝑌))) |
63 | 15, 37 | zmulcld 9435 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑛 · (𝑁 / (𝑛 gcd 𝑁))) ∈ ℤ) |
64 | 2, 4, 49 | zndvds0 14115 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ0
∧ (𝑛 · (𝑁 / (𝑛 gcd 𝑁))) ∈ ℤ) →
(((ℤRHom‘𝑌)‘(𝑛 · (𝑁 / (𝑛 gcd 𝑁)))) = (0g‘𝑌) ↔ 𝑁 ∥ (𝑛 · (𝑁 / (𝑛 gcd 𝑁))))) |
65 | 43, 63, 64 | syl2anc 411 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (((ℤRHom‘𝑌)‘(𝑛 · (𝑁 / (𝑛 gcd 𝑁)))) = (0g‘𝑌) ↔ 𝑁 ∥ (𝑛 · (𝑁 / (𝑛 gcd 𝑁))))) |
66 | 62, 65 | bitr3d 190 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((((ℤRHom‘𝑌)‘𝑛)(.r‘𝑌)((ℤRHom‘𝑌)‘(𝑁 / (𝑛 gcd 𝑁)))) = (0g‘𝑌) ↔ 𝑁 ∥ (𝑛 · (𝑁 / (𝑛 gcd 𝑁))))) |
67 | 2, 4, 49 | zndvds0 14115 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ0
∧ (𝑁 / (𝑛 gcd 𝑁)) ∈ ℤ) →
(((ℤRHom‘𝑌)‘(𝑁 / (𝑛 gcd 𝑁))) = (0g‘𝑌) ↔ 𝑁 ∥ (𝑁 / (𝑛 gcd 𝑁)))) |
68 | 43, 37, 67 | syl2anc 411 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (((ℤRHom‘𝑌)‘(𝑁 / (𝑛 gcd 𝑁))) = (0g‘𝑌) ↔ 𝑁 ∥ (𝑁 / (𝑛 gcd 𝑁)))) |
69 | 51, 66, 68 | 3imtr3d 202 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑁 ∥ (𝑛 · (𝑁 / (𝑛 gcd 𝑁))) → 𝑁 ∥ (𝑁 / (𝑛 gcd 𝑁)))) |
70 | 41, 69 | mpd 13 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → 𝑁 ∥ (𝑁 / (𝑛 gcd 𝑁))) |
71 | 14, 25, 26 | divcanap1d 8800 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((𝑁 / (𝑛 gcd 𝑁)) · (𝑛 gcd 𝑁)) = 𝑁) |
72 | 36 | nncnd 8986 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑁 / (𝑛 gcd 𝑁)) ∈ ℂ) |
73 | 72 | mulridd 8026 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((𝑁 / (𝑛 gcd 𝑁)) · 1) = (𝑁 / (𝑛 gcd 𝑁))) |
74 | 70, 71, 73 | 3brtr4d 4061 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((𝑁 / (𝑛 gcd 𝑁)) · (𝑛 gcd 𝑁)) ∥ ((𝑁 / (𝑛 gcd 𝑁)) · 1)) |
75 | | 1zzd 9334 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → 1 ∈ ℤ) |
76 | 36 | nnne0d 9017 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑁 / (𝑛 gcd 𝑁)) ≠ 0) |
77 | | dvdscmulr 11953 |
. . . . . . . . . . . 12
⊢ (((𝑛 gcd 𝑁) ∈ ℤ ∧ 1 ∈ ℤ
∧ ((𝑁 / (𝑛 gcd 𝑁)) ∈ ℤ ∧ (𝑁 / (𝑛 gcd 𝑁)) ≠ 0)) → (((𝑁 / (𝑛 gcd 𝑁)) · (𝑛 gcd 𝑁)) ∥ ((𝑁 / (𝑛 gcd 𝑁)) · 1) ↔ (𝑛 gcd 𝑁) ∥ 1)) |
78 | 31, 75, 37, 76, 77 | syl112anc 1253 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (((𝑁 / (𝑛 gcd 𝑁)) · (𝑛 gcd 𝑁)) ∥ ((𝑁 / (𝑛 gcd 𝑁)) · 1) ↔ (𝑛 gcd 𝑁) ∥ 1)) |
79 | 74, 78 | mpbid 147 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑛 gcd 𝑁) ∥ 1) |
80 | 15, 17 | gcdcld 12095 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑛 gcd 𝑁) ∈
ℕ0) |
81 | | dvds1 11985 |
. . . . . . . . . . 11
⊢ ((𝑛 gcd 𝑁) ∈ ℕ0 → ((𝑛 gcd 𝑁) ∥ 1 ↔ (𝑛 gcd 𝑁) = 1)) |
82 | 80, 81 | syl 14 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((𝑛 gcd 𝑁) ∥ 1 ↔ (𝑛 gcd 𝑁) = 1)) |
83 | 79, 82 | mpbid 147 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (𝑛 gcd 𝑁) = 1) |
84 | | znunit.u |
. . . . . . . . . . 11
⊢ 𝑈 = (Unit‘𝑌) |
85 | 2, 84, 4 | znunit 14124 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ 𝑛 ∈ ℤ)
→ (((ℤRHom‘𝑌)‘𝑛) ∈ 𝑈 ↔ (𝑛 gcd 𝑁) = 1)) |
86 | 43, 15, 85 | syl2anc 411 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → (((ℤRHom‘𝑌)‘𝑛) ∈ 𝑈 ↔ (𝑛 gcd 𝑁) = 1)) |
87 | 83, 86 | mpbird 167 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) ∧
((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸) → ((ℤRHom‘𝑌)‘𝑛) ∈ 𝑈) |
88 | 87 | ex 115 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) →
(((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸 → ((ℤRHom‘𝑌)‘𝑛) ∈ 𝑈)) |
89 | | eleq1 2256 |
. . . . . . . 8
⊢ (𝑥 = ((ℤRHom‘𝑌)‘𝑛) → (𝑥 ∈ 𝐸 ↔ ((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸)) |
90 | | eleq1 2256 |
. . . . . . . 8
⊢ (𝑥 = ((ℤRHom‘𝑌)‘𝑛) → (𝑥 ∈ 𝑈 ↔ ((ℤRHom‘𝑌)‘𝑛) ∈ 𝑈)) |
91 | 89, 90 | imbi12d 234 |
. . . . . . 7
⊢ (𝑥 = ((ℤRHom‘𝑌)‘𝑛) → ((𝑥 ∈ 𝐸 → 𝑥 ∈ 𝑈) ↔ (((ℤRHom‘𝑌)‘𝑛) ∈ 𝐸 → ((ℤRHom‘𝑌)‘𝑛) ∈ 𝑈))) |
92 | 88, 91 | syl5ibrcom 157 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) → (𝑥 = ((ℤRHom‘𝑌)‘𝑛) → (𝑥 ∈ 𝐸 → 𝑥 ∈ 𝑈))) |
93 | 92 | rexlimdva 2611 |
. . . . 5
⊢ (𝑁 ∈ ℕ →
(∃𝑛 ∈ ℤ
𝑥 =
((ℤRHom‘𝑌)‘𝑛) → (𝑥 ∈ 𝐸 → 𝑥 ∈ 𝑈))) |
94 | 93 | com23 78 |
. . . 4
⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝐸 → (∃𝑛 ∈ ℤ 𝑥 = ((ℤRHom‘𝑌)‘𝑛) → 𝑥 ∈ 𝑈))) |
95 | 12, 94 | mpdd 41 |
. . 3
⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝐸 → 𝑥 ∈ 𝑈)) |
96 | 95 | ssrdv 3185 |
. 2
⊢ (𝑁 ∈ ℕ → 𝐸 ⊆ 𝑈) |
97 | 7, 84 | unitrrg 13747 |
. . 3
⊢ (𝑌 ∈ Ring → 𝑈 ⊆ 𝐸) |
98 | 54, 97 | syl 14 |
. 2
⊢ (𝑁 ∈ ℕ → 𝑈 ⊆ 𝐸) |
99 | 96, 98 | eqssd 3196 |
1
⊢ (𝑁 ∈ ℕ → 𝐸 = 𝑈) |