Step |
Hyp |
Ref |
Expression |
1 |
|
mon1psubm.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
2 |
|
mon1psubm.m |
⊢ 𝑀 = ( Monic1p ‘ 𝑅 ) |
3 |
|
mon1psubm.u |
⊢ 𝑈 = ( mulGrp ‘ 𝑃 ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
5 |
1 4 2
|
mon1pcl |
⊢ ( 𝑥 ∈ 𝑀 → 𝑥 ∈ ( Base ‘ 𝑃 ) ) |
6 |
5
|
ssriv |
⊢ 𝑀 ⊆ ( Base ‘ 𝑃 ) |
7 |
6
|
a1i |
⊢ ( 𝑅 ∈ NzRing → 𝑀 ⊆ ( Base ‘ 𝑃 ) ) |
8 |
|
eqid |
⊢ ( 1r ‘ 𝑃 ) = ( 1r ‘ 𝑃 ) |
9 |
|
eqid |
⊢ ( deg1 ‘ 𝑅 ) = ( deg1 ‘ 𝑅 ) |
10 |
1 8 2 9
|
mon1pid |
⊢ ( 𝑅 ∈ NzRing → ( ( 1r ‘ 𝑃 ) ∈ 𝑀 ∧ ( ( deg1 ‘ 𝑅 ) ‘ ( 1r ‘ 𝑃 ) ) = 0 ) ) |
11 |
10
|
simpld |
⊢ ( 𝑅 ∈ NzRing → ( 1r ‘ 𝑃 ) ∈ 𝑀 ) |
12 |
1
|
ply1nz |
⊢ ( 𝑅 ∈ NzRing → 𝑃 ∈ NzRing ) |
13 |
|
nzrring |
⊢ ( 𝑃 ∈ NzRing → 𝑃 ∈ Ring ) |
14 |
12 13
|
syl |
⊢ ( 𝑅 ∈ NzRing → 𝑃 ∈ Ring ) |
15 |
14
|
adantr |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → 𝑃 ∈ Ring ) |
16 |
5
|
ad2antrl |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → 𝑥 ∈ ( Base ‘ 𝑃 ) ) |
17 |
|
simprr |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → 𝑦 ∈ 𝑀 ) |
18 |
6 17
|
sselid |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → 𝑦 ∈ ( Base ‘ 𝑃 ) ) |
19 |
|
eqid |
⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 ) |
20 |
4 19
|
ringcl |
⊢ ( ( 𝑃 ∈ Ring ∧ 𝑥 ∈ ( Base ‘ 𝑃 ) ∧ 𝑦 ∈ ( Base ‘ 𝑃 ) ) → ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ∈ ( Base ‘ 𝑃 ) ) |
21 |
15 16 18 20
|
syl3anc |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ∈ ( Base ‘ 𝑃 ) ) |
22 |
|
eqid |
⊢ ( RLReg ‘ 𝑅 ) = ( RLReg ‘ 𝑅 ) |
23 |
|
eqid |
⊢ ( 0g ‘ 𝑃 ) = ( 0g ‘ 𝑃 ) |
24 |
|
nzrring |
⊢ ( 𝑅 ∈ NzRing → 𝑅 ∈ Ring ) |
25 |
24
|
adantr |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → 𝑅 ∈ Ring ) |
26 |
1 23 2
|
mon1pn0 |
⊢ ( 𝑥 ∈ 𝑀 → 𝑥 ≠ ( 0g ‘ 𝑃 ) ) |
27 |
26
|
ad2antrl |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → 𝑥 ≠ ( 0g ‘ 𝑃 ) ) |
28 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
29 |
9 28 2
|
mon1pldg |
⊢ ( 𝑥 ∈ 𝑀 → ( ( coe1 ‘ 𝑥 ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ 𝑥 ) ) = ( 1r ‘ 𝑅 ) ) |
30 |
29
|
ad2antrl |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( coe1 ‘ 𝑥 ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ 𝑥 ) ) = ( 1r ‘ 𝑅 ) ) |
31 |
|
eqid |
⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 ) |
32 |
22 31
|
unitrrg |
⊢ ( 𝑅 ∈ Ring → ( Unit ‘ 𝑅 ) ⊆ ( RLReg ‘ 𝑅 ) ) |
33 |
24 32
|
syl |
⊢ ( 𝑅 ∈ NzRing → ( Unit ‘ 𝑅 ) ⊆ ( RLReg ‘ 𝑅 ) ) |
34 |
31 28
|
1unit |
⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ ( Unit ‘ 𝑅 ) ) |
35 |
24 34
|
syl |
⊢ ( 𝑅 ∈ NzRing → ( 1r ‘ 𝑅 ) ∈ ( Unit ‘ 𝑅 ) ) |
36 |
33 35
|
sseldd |
⊢ ( 𝑅 ∈ NzRing → ( 1r ‘ 𝑅 ) ∈ ( RLReg ‘ 𝑅 ) ) |
37 |
36
|
adantr |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( 1r ‘ 𝑅 ) ∈ ( RLReg ‘ 𝑅 ) ) |
38 |
30 37
|
eqeltrd |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( coe1 ‘ 𝑥 ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ 𝑥 ) ) ∈ ( RLReg ‘ 𝑅 ) ) |
39 |
1 23 2
|
mon1pn0 |
⊢ ( 𝑦 ∈ 𝑀 → 𝑦 ≠ ( 0g ‘ 𝑃 ) ) |
40 |
39
|
ad2antll |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → 𝑦 ≠ ( 0g ‘ 𝑃 ) ) |
41 |
9 1 22 4 19 23 25 16 27 38 18 40
|
deg1mul2 |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) = ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑥 ) + ( ( deg1 ‘ 𝑅 ) ‘ 𝑦 ) ) ) |
42 |
9 1 23 4
|
deg1nn0cl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ ( Base ‘ 𝑃 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑃 ) ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝑥 ) ∈ ℕ0 ) |
43 |
25 16 27 42
|
syl3anc |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝑥 ) ∈ ℕ0 ) |
44 |
9 1 23 4
|
deg1nn0cl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑦 ∈ ( Base ‘ 𝑃 ) ∧ 𝑦 ≠ ( 0g ‘ 𝑃 ) ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝑦 ) ∈ ℕ0 ) |
45 |
25 18 40 44
|
syl3anc |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝑦 ) ∈ ℕ0 ) |
46 |
43 45
|
nn0addcld |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑥 ) + ( ( deg1 ‘ 𝑅 ) ‘ 𝑦 ) ) ∈ ℕ0 ) |
47 |
41 46
|
eqeltrd |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) ∈ ℕ0 ) |
48 |
9 1 23 4
|
deg1nn0clb |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ∈ ( Base ‘ 𝑃 ) ) → ( ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ≠ ( 0g ‘ 𝑃 ) ↔ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) ∈ ℕ0 ) ) |
49 |
25 21 48
|
syl2anc |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ≠ ( 0g ‘ 𝑃 ) ↔ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) ∈ ℕ0 ) ) |
50 |
47 49
|
mpbird |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ≠ ( 0g ‘ 𝑃 ) ) |
51 |
41
|
fveq2d |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( coe1 ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) ) = ( ( coe1 ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) ‘ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑥 ) + ( ( deg1 ‘ 𝑅 ) ‘ 𝑦 ) ) ) ) |
52 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
53 |
1 19 52 4 9 23 25 16 27 18 40
|
coe1mul4 |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( coe1 ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) ‘ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑥 ) + ( ( deg1 ‘ 𝑅 ) ‘ 𝑦 ) ) ) = ( ( ( coe1 ‘ 𝑥 ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ 𝑥 ) ) ( .r ‘ 𝑅 ) ( ( coe1 ‘ 𝑦 ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ 𝑦 ) ) ) ) |
54 |
9 28 2
|
mon1pldg |
⊢ ( 𝑦 ∈ 𝑀 → ( ( coe1 ‘ 𝑦 ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ 𝑦 ) ) = ( 1r ‘ 𝑅 ) ) |
55 |
54
|
ad2antll |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( coe1 ‘ 𝑦 ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ 𝑦 ) ) = ( 1r ‘ 𝑅 ) ) |
56 |
30 55
|
oveq12d |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( ( coe1 ‘ 𝑥 ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ 𝑥 ) ) ( .r ‘ 𝑅 ) ( ( coe1 ‘ 𝑦 ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ 𝑦 ) ) ) = ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) ) |
57 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
58 |
57 28
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
59 |
57 52 28
|
ringlidm |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
60 |
24 58 59
|
syl2anc2 |
⊢ ( 𝑅 ∈ NzRing → ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
61 |
60
|
adantr |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
62 |
56 61
|
eqtrd |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( ( coe1 ‘ 𝑥 ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ 𝑥 ) ) ( .r ‘ 𝑅 ) ( ( coe1 ‘ 𝑦 ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ 𝑦 ) ) ) = ( 1r ‘ 𝑅 ) ) |
63 |
53 62
|
eqtrd |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( coe1 ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) ‘ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑥 ) + ( ( deg1 ‘ 𝑅 ) ‘ 𝑦 ) ) ) = ( 1r ‘ 𝑅 ) ) |
64 |
51 63
|
eqtrd |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( ( coe1 ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) ) = ( 1r ‘ 𝑅 ) ) |
65 |
1 4 23 9 2 28
|
ismon1p |
⊢ ( ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ∈ 𝑀 ↔ ( ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ∈ ( Base ‘ 𝑃 ) ∧ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ≠ ( 0g ‘ 𝑃 ) ∧ ( ( coe1 ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) ) = ( 1r ‘ 𝑅 ) ) ) |
66 |
21 50 64 65
|
syl3anbrc |
⊢ ( ( 𝑅 ∈ NzRing ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ∈ 𝑀 ) |
67 |
66
|
ralrimivva |
⊢ ( 𝑅 ∈ NzRing → ∀ 𝑥 ∈ 𝑀 ∀ 𝑦 ∈ 𝑀 ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ∈ 𝑀 ) |
68 |
3
|
ringmgp |
⊢ ( 𝑃 ∈ Ring → 𝑈 ∈ Mnd ) |
69 |
14 68
|
syl |
⊢ ( 𝑅 ∈ NzRing → 𝑈 ∈ Mnd ) |
70 |
3 4
|
mgpbas |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑈 ) |
71 |
3 8
|
ringidval |
⊢ ( 1r ‘ 𝑃 ) = ( 0g ‘ 𝑈 ) |
72 |
3 19
|
mgpplusg |
⊢ ( .r ‘ 𝑃 ) = ( +g ‘ 𝑈 ) |
73 |
70 71 72
|
issubm |
⊢ ( 𝑈 ∈ Mnd → ( 𝑀 ∈ ( SubMnd ‘ 𝑈 ) ↔ ( 𝑀 ⊆ ( Base ‘ 𝑃 ) ∧ ( 1r ‘ 𝑃 ) ∈ 𝑀 ∧ ∀ 𝑥 ∈ 𝑀 ∀ 𝑦 ∈ 𝑀 ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ∈ 𝑀 ) ) ) |
74 |
69 73
|
syl |
⊢ ( 𝑅 ∈ NzRing → ( 𝑀 ∈ ( SubMnd ‘ 𝑈 ) ↔ ( 𝑀 ⊆ ( Base ‘ 𝑃 ) ∧ ( 1r ‘ 𝑃 ) ∈ 𝑀 ∧ ∀ 𝑥 ∈ 𝑀 ∀ 𝑦 ∈ 𝑀 ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ∈ 𝑀 ) ) ) |
75 |
7 11 67 74
|
mpbir3and |
⊢ ( 𝑅 ∈ NzRing → 𝑀 ∈ ( SubMnd ‘ 𝑈 ) ) |