Step |
Hyp |
Ref |
Expression |
1 |
|
ig1pval.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
2 |
|
ig1pval.g |
⊢ 𝐺 = ( idlGen1p ‘ 𝑅 ) |
3 |
|
ig1pcl.u |
⊢ 𝑈 = ( LIdeal ‘ 𝑃 ) |
4 |
|
ig1pdvds.d |
⊢ ∥ = ( ∥r ‘ 𝑃 ) |
5 |
|
drngring |
⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring ) |
6 |
1
|
ply1ring |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring ) |
7 |
5 6
|
syl |
⊢ ( 𝑅 ∈ DivRing → 𝑃 ∈ Ring ) |
8 |
7
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) → 𝑃 ∈ Ring ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
10 |
9 3
|
lidlss |
⊢ ( 𝐼 ∈ 𝑈 → 𝐼 ⊆ ( Base ‘ 𝑃 ) ) |
11 |
10
|
3ad2ant2 |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) → 𝐼 ⊆ ( Base ‘ 𝑃 ) ) |
12 |
1 2 3
|
ig1pcl |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ) → ( 𝐺 ‘ 𝐼 ) ∈ 𝐼 ) |
13 |
12
|
3adant3 |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) → ( 𝐺 ‘ 𝐼 ) ∈ 𝐼 ) |
14 |
11 13
|
sseldd |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) → ( 𝐺 ‘ 𝐼 ) ∈ ( Base ‘ 𝑃 ) ) |
15 |
|
eqid |
⊢ ( 0g ‘ 𝑃 ) = ( 0g ‘ 𝑃 ) |
16 |
9 4 15
|
dvdsr01 |
⊢ ( ( 𝑃 ∈ Ring ∧ ( 𝐺 ‘ 𝐼 ) ∈ ( Base ‘ 𝑃 ) ) → ( 𝐺 ‘ 𝐼 ) ∥ ( 0g ‘ 𝑃 ) ) |
17 |
8 14 16
|
syl2anc |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) → ( 𝐺 ‘ 𝐼 ) ∥ ( 0g ‘ 𝑃 ) ) |
18 |
17
|
adantr |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 = { ( 0g ‘ 𝑃 ) } ) → ( 𝐺 ‘ 𝐼 ) ∥ ( 0g ‘ 𝑃 ) ) |
19 |
|
eleq2 |
⊢ ( 𝐼 = { ( 0g ‘ 𝑃 ) } → ( 𝑋 ∈ 𝐼 ↔ 𝑋 ∈ { ( 0g ‘ 𝑃 ) } ) ) |
20 |
19
|
biimpac |
⊢ ( ( 𝑋 ∈ 𝐼 ∧ 𝐼 = { ( 0g ‘ 𝑃 ) } ) → 𝑋 ∈ { ( 0g ‘ 𝑃 ) } ) |
21 |
20
|
3ad2antl3 |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 = { ( 0g ‘ 𝑃 ) } ) → 𝑋 ∈ { ( 0g ‘ 𝑃 ) } ) |
22 |
|
elsni |
⊢ ( 𝑋 ∈ { ( 0g ‘ 𝑃 ) } → 𝑋 = ( 0g ‘ 𝑃 ) ) |
23 |
21 22
|
syl |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 = { ( 0g ‘ 𝑃 ) } ) → 𝑋 = ( 0g ‘ 𝑃 ) ) |
24 |
18 23
|
breqtrrd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 = { ( 0g ‘ 𝑃 ) } ) → ( 𝐺 ‘ 𝐼 ) ∥ 𝑋 ) |
25 |
|
simpl1 |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → 𝑅 ∈ DivRing ) |
26 |
25 5
|
syl |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → 𝑅 ∈ Ring ) |
27 |
|
simpl2 |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → 𝐼 ∈ 𝑈 ) |
28 |
27 10
|
syl |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → 𝐼 ⊆ ( Base ‘ 𝑃 ) ) |
29 |
|
simpl3 |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → 𝑋 ∈ 𝐼 ) |
30 |
28 29
|
sseldd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → 𝑋 ∈ ( Base ‘ 𝑃 ) ) |
31 |
|
simpr |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) |
32 |
|
eqid |
⊢ ( deg1 ‘ 𝑅 ) = ( deg1 ‘ 𝑅 ) |
33 |
|
eqid |
⊢ ( Monic1p ‘ 𝑅 ) = ( Monic1p ‘ 𝑅 ) |
34 |
1 2 15 3 32 33
|
ig1pval3 |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( 𝐺 ‘ 𝐼 ) ∈ 𝐼 ∧ ( 𝐺 ‘ 𝐼 ) ∈ ( Monic1p ‘ 𝑅 ) ∧ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝐼 ) ) = inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ) ) |
35 |
25 27 31 34
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( 𝐺 ‘ 𝐼 ) ∈ 𝐼 ∧ ( 𝐺 ‘ 𝐼 ) ∈ ( Monic1p ‘ 𝑅 ) ∧ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝐼 ) ) = inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ) ) |
36 |
35
|
simp2d |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 𝐺 ‘ 𝐼 ) ∈ ( Monic1p ‘ 𝑅 ) ) |
37 |
|
eqid |
⊢ ( Unic1p ‘ 𝑅 ) = ( Unic1p ‘ 𝑅 ) |
38 |
37 33
|
mon1puc1p |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝐺 ‘ 𝐼 ) ∈ ( Monic1p ‘ 𝑅 ) ) → ( 𝐺 ‘ 𝐼 ) ∈ ( Unic1p ‘ 𝑅 ) ) |
39 |
26 36 38
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 𝐺 ‘ 𝐼 ) ∈ ( Unic1p ‘ 𝑅 ) ) |
40 |
|
eqid |
⊢ ( rem1p ‘ 𝑅 ) = ( rem1p ‘ 𝑅 ) |
41 |
40 1 9 37 32
|
r1pdeglt |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ ( Base ‘ 𝑃 ) ∧ ( 𝐺 ‘ 𝐼 ) ∈ ( Unic1p ‘ 𝑅 ) ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) < ( ( deg1 ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝐼 ) ) ) |
42 |
26 30 39 41
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) < ( ( deg1 ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝐼 ) ) ) |
43 |
35
|
simp3d |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝐼 ) ) = inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ) |
44 |
42 43
|
breqtrd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) < inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ) |
45 |
32 1 9
|
deg1xrf |
⊢ ( deg1 ‘ 𝑅 ) : ( Base ‘ 𝑃 ) ⟶ ℝ* |
46 |
35
|
simp1d |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 𝐺 ‘ 𝐼 ) ∈ 𝐼 ) |
47 |
28 46
|
sseldd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 𝐺 ‘ 𝐼 ) ∈ ( Base ‘ 𝑃 ) ) |
48 |
|
eqid |
⊢ ( quot1p ‘ 𝑅 ) = ( quot1p ‘ 𝑅 ) |
49 |
|
eqid |
⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 ) |
50 |
|
eqid |
⊢ ( -g ‘ 𝑃 ) = ( -g ‘ 𝑃 ) |
51 |
40 1 9 48 49 50
|
r1pval |
⊢ ( ( 𝑋 ∈ ( Base ‘ 𝑃 ) ∧ ( 𝐺 ‘ 𝐼 ) ∈ ( Base ‘ 𝑃 ) ) → ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) = ( 𝑋 ( -g ‘ 𝑃 ) ( ( 𝑋 ( quot1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ( .r ‘ 𝑃 ) ( 𝐺 ‘ 𝐼 ) ) ) ) |
52 |
30 47 51
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) = ( 𝑋 ( -g ‘ 𝑃 ) ( ( 𝑋 ( quot1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ( .r ‘ 𝑃 ) ( 𝐺 ‘ 𝐼 ) ) ) ) |
53 |
26 6
|
syl |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → 𝑃 ∈ Ring ) |
54 |
48 1 9 37
|
q1pcl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ ( Base ‘ 𝑃 ) ∧ ( 𝐺 ‘ 𝐼 ) ∈ ( Unic1p ‘ 𝑅 ) ) → ( 𝑋 ( quot1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ∈ ( Base ‘ 𝑃 ) ) |
55 |
26 30 39 54
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 𝑋 ( quot1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ∈ ( Base ‘ 𝑃 ) ) |
56 |
3 9 49
|
lidlmcl |
⊢ ( ( ( 𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) ∧ ( ( 𝑋 ( quot1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ∈ ( Base ‘ 𝑃 ) ∧ ( 𝐺 ‘ 𝐼 ) ∈ 𝐼 ) ) → ( ( 𝑋 ( quot1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ( .r ‘ 𝑃 ) ( 𝐺 ‘ 𝐼 ) ) ∈ 𝐼 ) |
57 |
53 27 55 46 56
|
syl22anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( 𝑋 ( quot1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ( .r ‘ 𝑃 ) ( 𝐺 ‘ 𝐼 ) ) ∈ 𝐼 ) |
58 |
3 50
|
lidlsubcl |
⊢ ( ( ( 𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) ∧ ( 𝑋 ∈ 𝐼 ∧ ( ( 𝑋 ( quot1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ( .r ‘ 𝑃 ) ( 𝐺 ‘ 𝐼 ) ) ∈ 𝐼 ) ) → ( 𝑋 ( -g ‘ 𝑃 ) ( ( 𝑋 ( quot1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ( .r ‘ 𝑃 ) ( 𝐺 ‘ 𝐼 ) ) ) ∈ 𝐼 ) |
59 |
53 27 29 57 58
|
syl22anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 𝑋 ( -g ‘ 𝑃 ) ( ( 𝑋 ( quot1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ( .r ‘ 𝑃 ) ( 𝐺 ‘ 𝐼 ) ) ) ∈ 𝐼 ) |
60 |
52 59
|
eqeltrd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ∈ 𝐼 ) |
61 |
28 60
|
sseldd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ∈ ( Base ‘ 𝑃 ) ) |
62 |
|
ffvelrn |
⊢ ( ( ( deg1 ‘ 𝑅 ) : ( Base ‘ 𝑃 ) ⟶ ℝ* ∧ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ∈ ( Base ‘ 𝑃 ) ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) ∈ ℝ* ) |
63 |
45 61 62
|
sylancr |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) ∈ ℝ* ) |
64 |
28
|
ssdifd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ⊆ ( ( Base ‘ 𝑃 ) ∖ { ( 0g ‘ 𝑃 ) } ) ) |
65 |
|
imass2 |
⊢ ( ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ⊆ ( ( Base ‘ 𝑃 ) ∖ { ( 0g ‘ 𝑃 ) } ) → ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ⊆ ( ( deg1 ‘ 𝑅 ) “ ( ( Base ‘ 𝑃 ) ∖ { ( 0g ‘ 𝑃 ) } ) ) ) |
66 |
64 65
|
syl |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ⊆ ( ( deg1 ‘ 𝑅 ) “ ( ( Base ‘ 𝑃 ) ∖ { ( 0g ‘ 𝑃 ) } ) ) ) |
67 |
32 1 15 9
|
deg1n0ima |
⊢ ( 𝑅 ∈ Ring → ( ( deg1 ‘ 𝑅 ) “ ( ( Base ‘ 𝑃 ) ∖ { ( 0g ‘ 𝑃 ) } ) ) ⊆ ℕ0 ) |
68 |
26 67
|
syl |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( deg1 ‘ 𝑅 ) “ ( ( Base ‘ 𝑃 ) ∖ { ( 0g ‘ 𝑃 ) } ) ) ⊆ ℕ0 ) |
69 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
70 |
68 69
|
sseqtrdi |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( deg1 ‘ 𝑅 ) “ ( ( Base ‘ 𝑃 ) ∖ { ( 0g ‘ 𝑃 ) } ) ) ⊆ ( ℤ≥ ‘ 0 ) ) |
71 |
66 70
|
sstrd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ⊆ ( ℤ≥ ‘ 0 ) ) |
72 |
|
uzssz |
⊢ ( ℤ≥ ‘ 0 ) ⊆ ℤ |
73 |
|
zssre |
⊢ ℤ ⊆ ℝ |
74 |
|
ressxr |
⊢ ℝ ⊆ ℝ* |
75 |
73 74
|
sstri |
⊢ ℤ ⊆ ℝ* |
76 |
72 75
|
sstri |
⊢ ( ℤ≥ ‘ 0 ) ⊆ ℝ* |
77 |
71 76
|
sstrdi |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ⊆ ℝ* ) |
78 |
3 15
|
lidl0cl |
⊢ ( ( 𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) → ( 0g ‘ 𝑃 ) ∈ 𝐼 ) |
79 |
53 27 78
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 0g ‘ 𝑃 ) ∈ 𝐼 ) |
80 |
79
|
snssd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → { ( 0g ‘ 𝑃 ) } ⊆ 𝐼 ) |
81 |
31
|
necomd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → { ( 0g ‘ 𝑃 ) } ≠ 𝐼 ) |
82 |
|
pssdifn0 |
⊢ ( ( { ( 0g ‘ 𝑃 ) } ⊆ 𝐼 ∧ { ( 0g ‘ 𝑃 ) } ≠ 𝐼 ) → ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ≠ ∅ ) |
83 |
80 81 82
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ≠ ∅ ) |
84 |
|
ffn |
⊢ ( ( deg1 ‘ 𝑅 ) : ( Base ‘ 𝑃 ) ⟶ ℝ* → ( deg1 ‘ 𝑅 ) Fn ( Base ‘ 𝑃 ) ) |
85 |
45 84
|
ax-mp |
⊢ ( deg1 ‘ 𝑅 ) Fn ( Base ‘ 𝑃 ) |
86 |
28
|
ssdifssd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ⊆ ( Base ‘ 𝑃 ) ) |
87 |
|
fnimaeq0 |
⊢ ( ( ( deg1 ‘ 𝑅 ) Fn ( Base ‘ 𝑃 ) ∧ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ⊆ ( Base ‘ 𝑃 ) ) → ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) = ∅ ↔ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) = ∅ ) ) |
88 |
85 86 87
|
sylancr |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) = ∅ ↔ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) = ∅ ) ) |
89 |
88
|
necon3bid |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ≠ ∅ ↔ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ≠ ∅ ) ) |
90 |
83 89
|
mpbird |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ≠ ∅ ) |
91 |
|
infssuzcl |
⊢ ( ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ⊆ ( ℤ≥ ‘ 0 ) ∧ ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ≠ ∅ ) → inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ∈ ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ) |
92 |
71 90 91
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ∈ ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ) |
93 |
77 92
|
sseldd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ∈ ℝ* ) |
94 |
|
xrltnle |
⊢ ( ( ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) ∈ ℝ* ∧ inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ∈ ℝ* ) → ( ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) < inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ↔ ¬ inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ≤ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) ) ) |
95 |
63 93 94
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) < inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ↔ ¬ inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ≤ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) ) ) |
96 |
44 95
|
mpbid |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ¬ inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ≤ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) ) |
97 |
71
|
adantr |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) ∧ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ≠ ( 0g ‘ 𝑃 ) ) → ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ⊆ ( ℤ≥ ‘ 0 ) ) |
98 |
60
|
adantr |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) ∧ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ≠ ( 0g ‘ 𝑃 ) ) → ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ∈ 𝐼 ) |
99 |
|
simpr |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) ∧ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ≠ ( 0g ‘ 𝑃 ) ) → ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ≠ ( 0g ‘ 𝑃 ) ) |
100 |
|
eldifsn |
⊢ ( ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ∈ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ↔ ( ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ∈ 𝐼 ∧ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ≠ ( 0g ‘ 𝑃 ) ) ) |
101 |
98 99 100
|
sylanbrc |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) ∧ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ≠ ( 0g ‘ 𝑃 ) ) → ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ∈ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) |
102 |
|
fnfvima |
⊢ ( ( ( deg1 ‘ 𝑅 ) Fn ( Base ‘ 𝑃 ) ∧ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ⊆ ( Base ‘ 𝑃 ) ∧ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ∈ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) ∈ ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ) |
103 |
85 86 101 102
|
mp3an2ani |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) ∧ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ≠ ( 0g ‘ 𝑃 ) ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) ∈ ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ) |
104 |
|
infssuzle |
⊢ ( ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ⊆ ( ℤ≥ ‘ 0 ) ∧ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) ∈ ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) ) → inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ≤ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) ) |
105 |
97 103 104
|
syl2anc |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) ∧ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ≠ ( 0g ‘ 𝑃 ) ) → inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ≤ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) ) |
106 |
105
|
ex |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ≠ ( 0g ‘ 𝑃 ) → inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ≤ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) ) ) |
107 |
106
|
necon1bd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ¬ inf ( ( ( deg1 ‘ 𝑅 ) “ ( 𝐼 ∖ { ( 0g ‘ 𝑃 ) } ) ) , ℝ , < ) ≤ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) ) → ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) = ( 0g ‘ 𝑃 ) ) ) |
108 |
96 107
|
mpd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) = ( 0g ‘ 𝑃 ) ) |
109 |
1 4 9 37 15 40
|
dvdsr1p |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ ( Base ‘ 𝑃 ) ∧ ( 𝐺 ‘ 𝐼 ) ∈ ( Unic1p ‘ 𝑅 ) ) → ( ( 𝐺 ‘ 𝐼 ) ∥ 𝑋 ↔ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) = ( 0g ‘ 𝑃 ) ) ) |
110 |
26 30 39 109
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( ( 𝐺 ‘ 𝐼 ) ∥ 𝑋 ↔ ( 𝑋 ( rem1p ‘ 𝑅 ) ( 𝐺 ‘ 𝐼 ) ) = ( 0g ‘ 𝑃 ) ) ) |
111 |
108 110
|
mpbird |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) ∧ 𝐼 ≠ { ( 0g ‘ 𝑃 ) } ) → ( 𝐺 ‘ 𝐼 ) ∥ 𝑋 ) |
112 |
24 111
|
pm2.61dane |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼 ) → ( 𝐺 ‘ 𝐼 ) ∥ 𝑋 ) |