Step |
Hyp |
Ref |
Expression |
1 |
|
hbtlem.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
2 |
|
hbtlem.u |
⊢ 𝑈 = ( LIdeal ‘ 𝑃 ) |
3 |
|
hbtlem.s |
⊢ 𝑆 = ( ldgIdlSeq ‘ 𝑅 ) |
4 |
|
hbtlem3.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
5 |
|
hbtlem3.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑈 ) |
6 |
|
hbtlem3.j |
⊢ ( 𝜑 → 𝐽 ∈ 𝑈 ) |
7 |
|
hbtlem3.ij |
⊢ ( 𝜑 → 𝐼 ⊆ 𝐽 ) |
8 |
|
hbtlem5.e |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℕ0 ( ( 𝑆 ‘ 𝐽 ) ‘ 𝑥 ) ⊆ ( ( 𝑆 ‘ 𝐼 ) ‘ 𝑥 ) ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
10 |
9 2
|
lidlss |
⊢ ( 𝐽 ∈ 𝑈 → 𝐽 ⊆ ( Base ‘ 𝑃 ) ) |
11 |
6 10
|
syl |
⊢ ( 𝜑 → 𝐽 ⊆ ( Base ‘ 𝑃 ) ) |
12 |
11
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐽 ) → 𝑎 ∈ ( Base ‘ 𝑃 ) ) |
13 |
|
eqid |
⊢ ( deg1 ‘ 𝑅 ) = ( deg1 ‘ 𝑅 ) |
14 |
13 1 9
|
deg1cl |
⊢ ( 𝑎 ∈ ( Base ‘ 𝑃 ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ ( ℕ0 ∪ { -∞ } ) ) |
15 |
12 14
|
syl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐽 ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ ( ℕ0 ∪ { -∞ } ) ) |
16 |
|
elun |
⊢ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ ( ℕ0 ∪ { -∞ } ) ↔ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ ℕ0 ∨ ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ { -∞ } ) ) |
17 |
|
nnssnn0 |
⊢ ℕ ⊆ ℕ0 |
18 |
|
nn0re |
⊢ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ ℕ0 → ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ ℝ ) |
19 |
|
arch |
⊢ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ ℝ → ∃ 𝑏 ∈ ℕ ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 ) |
20 |
18 19
|
syl |
⊢ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ ℕ0 → ∃ 𝑏 ∈ ℕ ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 ) |
21 |
|
ssrexv |
⊢ ( ℕ ⊆ ℕ0 → ( ∃ 𝑏 ∈ ℕ ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → ∃ 𝑏 ∈ ℕ0 ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 ) ) |
22 |
17 20 21
|
mpsyl |
⊢ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ ℕ0 → ∃ 𝑏 ∈ ℕ0 ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 ) |
23 |
|
elsni |
⊢ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ { -∞ } → ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) = -∞ ) |
24 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
25 |
|
mnflt0 |
⊢ -∞ < 0 |
26 |
|
breq2 |
⊢ ( 𝑏 = 0 → ( -∞ < 𝑏 ↔ -∞ < 0 ) ) |
27 |
26
|
rspcev |
⊢ ( ( 0 ∈ ℕ0 ∧ -∞ < 0 ) → ∃ 𝑏 ∈ ℕ0 -∞ < 𝑏 ) |
28 |
24 25 27
|
mp2an |
⊢ ∃ 𝑏 ∈ ℕ0 -∞ < 𝑏 |
29 |
|
breq1 |
⊢ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) = -∞ → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 ↔ -∞ < 𝑏 ) ) |
30 |
29
|
rexbidv |
⊢ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) = -∞ → ( ∃ 𝑏 ∈ ℕ0 ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 ↔ ∃ 𝑏 ∈ ℕ0 -∞ < 𝑏 ) ) |
31 |
28 30
|
mpbiri |
⊢ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) = -∞ → ∃ 𝑏 ∈ ℕ0 ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 ) |
32 |
23 31
|
syl |
⊢ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ { -∞ } → ∃ 𝑏 ∈ ℕ0 ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 ) |
33 |
22 32
|
jaoi |
⊢ ( ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ ℕ0 ∨ ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ { -∞ } ) → ∃ 𝑏 ∈ ℕ0 ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 ) |
34 |
16 33
|
sylbi |
⊢ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) ∈ ( ℕ0 ∪ { -∞ } ) → ∃ 𝑏 ∈ ℕ0 ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 ) |
35 |
15 34
|
syl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐽 ) → ∃ 𝑏 ∈ ℕ0 ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 ) |
36 |
|
breq2 |
⊢ ( 𝑐 = 0 → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑐 ↔ ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 0 ) ) |
37 |
36
|
imbi1d |
⊢ ( 𝑐 = 0 → ( ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑐 → 𝑎 ∈ 𝐼 ) ↔ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 0 → 𝑎 ∈ 𝐼 ) ) ) |
38 |
37
|
ralbidv |
⊢ ( 𝑐 = 0 → ( ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑐 → 𝑎 ∈ 𝐼 ) ↔ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 0 → 𝑎 ∈ 𝐼 ) ) ) |
39 |
38
|
imbi2d |
⊢ ( 𝑐 = 0 → ( ( 𝜑 → ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑐 → 𝑎 ∈ 𝐼 ) ) ↔ ( 𝜑 → ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 0 → 𝑎 ∈ 𝐼 ) ) ) ) |
40 |
|
breq2 |
⊢ ( 𝑐 = 𝑏 → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑐 ↔ ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 ) ) |
41 |
40
|
imbi1d |
⊢ ( 𝑐 = 𝑏 → ( ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑐 → 𝑎 ∈ 𝐼 ) ↔ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ) |
42 |
41
|
ralbidv |
⊢ ( 𝑐 = 𝑏 → ( ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑐 → 𝑎 ∈ 𝐼 ) ↔ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ) |
43 |
42
|
imbi2d |
⊢ ( 𝑐 = 𝑏 → ( ( 𝜑 → ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑐 → 𝑎 ∈ 𝐼 ) ) ↔ ( 𝜑 → ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ) ) |
44 |
|
breq2 |
⊢ ( 𝑐 = ( 𝑏 + 1 ) → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑐 ↔ ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < ( 𝑏 + 1 ) ) ) |
45 |
44
|
imbi1d |
⊢ ( 𝑐 = ( 𝑏 + 1 ) → ( ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑐 → 𝑎 ∈ 𝐼 ) ↔ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < ( 𝑏 + 1 ) → 𝑎 ∈ 𝐼 ) ) ) |
46 |
45
|
ralbidv |
⊢ ( 𝑐 = ( 𝑏 + 1 ) → ( ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑐 → 𝑎 ∈ 𝐼 ) ↔ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < ( 𝑏 + 1 ) → 𝑎 ∈ 𝐼 ) ) ) |
47 |
|
fveq2 |
⊢ ( 𝑎 = 𝑑 → ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) = ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ) |
48 |
47
|
breq1d |
⊢ ( 𝑎 = 𝑑 → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < ( 𝑏 + 1 ) ↔ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) < ( 𝑏 + 1 ) ) ) |
49 |
|
eleq1 |
⊢ ( 𝑎 = 𝑑 → ( 𝑎 ∈ 𝐼 ↔ 𝑑 ∈ 𝐼 ) ) |
50 |
48 49
|
imbi12d |
⊢ ( 𝑎 = 𝑑 → ( ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < ( 𝑏 + 1 ) → 𝑎 ∈ 𝐼 ) ↔ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) < ( 𝑏 + 1 ) → 𝑑 ∈ 𝐼 ) ) ) |
51 |
50
|
cbvralvw |
⊢ ( ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < ( 𝑏 + 1 ) → 𝑎 ∈ 𝐼 ) ↔ ∀ 𝑑 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) < ( 𝑏 + 1 ) → 𝑑 ∈ 𝐼 ) ) |
52 |
46 51
|
syl6bb |
⊢ ( 𝑐 = ( 𝑏 + 1 ) → ( ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑐 → 𝑎 ∈ 𝐼 ) ↔ ∀ 𝑑 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) < ( 𝑏 + 1 ) → 𝑑 ∈ 𝐼 ) ) ) |
53 |
52
|
imbi2d |
⊢ ( 𝑐 = ( 𝑏 + 1 ) → ( ( 𝜑 → ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑐 → 𝑎 ∈ 𝐼 ) ) ↔ ( 𝜑 → ∀ 𝑑 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) < ( 𝑏 + 1 ) → 𝑑 ∈ 𝐼 ) ) ) ) |
54 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐽 ) → 𝑅 ∈ Ring ) |
55 |
|
eqid |
⊢ ( 0g ‘ 𝑃 ) = ( 0g ‘ 𝑃 ) |
56 |
13 1 55 9
|
deg1lt0 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑎 ∈ ( Base ‘ 𝑃 ) ) → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 0 ↔ 𝑎 = ( 0g ‘ 𝑃 ) ) ) |
57 |
54 12 56
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐽 ) → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 0 ↔ 𝑎 = ( 0g ‘ 𝑃 ) ) ) |
58 |
1
|
ply1ring |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring ) |
59 |
4 58
|
syl |
⊢ ( 𝜑 → 𝑃 ∈ Ring ) |
60 |
2 55
|
lidl0cl |
⊢ ( ( 𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) → ( 0g ‘ 𝑃 ) ∈ 𝐼 ) |
61 |
59 5 60
|
syl2anc |
⊢ ( 𝜑 → ( 0g ‘ 𝑃 ) ∈ 𝐼 ) |
62 |
|
eleq1a |
⊢ ( ( 0g ‘ 𝑃 ) ∈ 𝐼 → ( 𝑎 = ( 0g ‘ 𝑃 ) → 𝑎 ∈ 𝐼 ) ) |
63 |
61 62
|
syl |
⊢ ( 𝜑 → ( 𝑎 = ( 0g ‘ 𝑃 ) → 𝑎 ∈ 𝐼 ) ) |
64 |
63
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐽 ) → ( 𝑎 = ( 0g ‘ 𝑃 ) → 𝑎 ∈ 𝐼 ) ) |
65 |
57 64
|
sylbid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐽 ) → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 0 → 𝑎 ∈ 𝐼 ) ) |
66 |
65
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 0 → 𝑎 ∈ 𝐼 ) ) |
67 |
11
|
3ad2ant2 |
⊢ ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) → 𝐽 ⊆ ( Base ‘ 𝑃 ) ) |
68 |
67
|
sselda |
⊢ ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ 𝑑 ∈ 𝐽 ) → 𝑑 ∈ ( Base ‘ 𝑃 ) ) |
69 |
13 1 9
|
deg1cl |
⊢ ( 𝑑 ∈ ( Base ‘ 𝑃 ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ∈ ( ℕ0 ∪ { -∞ } ) ) |
70 |
68 69
|
syl |
⊢ ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ 𝑑 ∈ 𝐽 ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ∈ ( ℕ0 ∪ { -∞ } ) ) |
71 |
|
simpl1 |
⊢ ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ 𝑑 ∈ 𝐽 ) → 𝑏 ∈ ℕ0 ) |
72 |
71
|
nn0zd |
⊢ ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ 𝑑 ∈ 𝐽 ) → 𝑏 ∈ ℤ ) |
73 |
|
degltp1le |
⊢ ( ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ∈ ( ℕ0 ∪ { -∞ } ) ∧ 𝑏 ∈ ℤ ) → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) < ( 𝑏 + 1 ) ↔ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) |
74 |
70 72 73
|
syl2anc |
⊢ ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ 𝑑 ∈ 𝐽 ) → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) < ( 𝑏 + 1 ) ↔ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) |
75 |
|
fveq2 |
⊢ ( 𝑥 = 𝑏 → ( ( 𝑆 ‘ 𝐽 ) ‘ 𝑥 ) = ( ( 𝑆 ‘ 𝐽 ) ‘ 𝑏 ) ) |
76 |
|
fveq2 |
⊢ ( 𝑥 = 𝑏 → ( ( 𝑆 ‘ 𝐼 ) ‘ 𝑥 ) = ( ( 𝑆 ‘ 𝐼 ) ‘ 𝑏 ) ) |
77 |
75 76
|
sseq12d |
⊢ ( 𝑥 = 𝑏 → ( ( ( 𝑆 ‘ 𝐽 ) ‘ 𝑥 ) ⊆ ( ( 𝑆 ‘ 𝐼 ) ‘ 𝑥 ) ↔ ( ( 𝑆 ‘ 𝐽 ) ‘ 𝑏 ) ⊆ ( ( 𝑆 ‘ 𝐼 ) ‘ 𝑏 ) ) ) |
78 |
77
|
rspcva |
⊢ ( ( 𝑏 ∈ ℕ0 ∧ ∀ 𝑥 ∈ ℕ0 ( ( 𝑆 ‘ 𝐽 ) ‘ 𝑥 ) ⊆ ( ( 𝑆 ‘ 𝐼 ) ‘ 𝑥 ) ) → ( ( 𝑆 ‘ 𝐽 ) ‘ 𝑏 ) ⊆ ( ( 𝑆 ‘ 𝐼 ) ‘ 𝑏 ) ) |
79 |
8 78
|
sylan2 |
⊢ ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ) → ( ( 𝑆 ‘ 𝐽 ) ‘ 𝑏 ) ⊆ ( ( 𝑆 ‘ 𝐼 ) ‘ 𝑏 ) ) |
80 |
4
|
adantl |
⊢ ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ) → 𝑅 ∈ Ring ) |
81 |
6
|
adantl |
⊢ ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ) → 𝐽 ∈ 𝑈 ) |
82 |
|
simpl |
⊢ ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ) → 𝑏 ∈ ℕ0 ) |
83 |
1 2 3 13
|
hbtlem1 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐽 ∈ 𝑈 ∧ 𝑏 ∈ ℕ0 ) → ( ( 𝑆 ‘ 𝐽 ) ‘ 𝑏 ) = { 𝑐 ∣ ∃ 𝑒 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ) |
84 |
80 81 82 83
|
syl3anc |
⊢ ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ) → ( ( 𝑆 ‘ 𝐽 ) ‘ 𝑏 ) = { 𝑐 ∣ ∃ 𝑒 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ) |
85 |
5
|
adantl |
⊢ ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ) → 𝐼 ∈ 𝑈 ) |
86 |
1 2 3 13
|
hbtlem1 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑏 ∈ ℕ0 ) → ( ( 𝑆 ‘ 𝐼 ) ‘ 𝑏 ) = { 𝑐 ∣ ∃ 𝑒 ∈ 𝐼 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ) |
87 |
80 85 82 86
|
syl3anc |
⊢ ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ) → ( ( 𝑆 ‘ 𝐼 ) ‘ 𝑏 ) = { 𝑐 ∣ ∃ 𝑒 ∈ 𝐼 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ) |
88 |
79 84 87
|
3sstr3d |
⊢ ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ) → { 𝑐 ∣ ∃ 𝑒 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ⊆ { 𝑐 ∣ ∃ 𝑒 ∈ 𝐼 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ) |
89 |
88
|
3adant3 |
⊢ ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) → { 𝑐 ∣ ∃ 𝑒 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ⊆ { 𝑐 ∣ ∃ 𝑒 ∈ 𝐼 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ) |
90 |
89
|
adantr |
⊢ ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) → { 𝑐 ∣ ∃ 𝑒 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ⊆ { 𝑐 ∣ ∃ 𝑒 ∈ 𝐼 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ) |
91 |
|
simpl |
⊢ ( ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) → 𝑑 ∈ 𝐽 ) |
92 |
|
simpr |
⊢ ( ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) |
93 |
|
eqidd |
⊢ ( ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) → ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) ) |
94 |
|
fveq2 |
⊢ ( 𝑒 = 𝑑 → ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) = ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ) |
95 |
94
|
breq1d |
⊢ ( 𝑒 = 𝑑 → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ↔ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) |
96 |
|
fveq2 |
⊢ ( 𝑒 = 𝑑 → ( coe1 ‘ 𝑒 ) = ( coe1 ‘ 𝑑 ) ) |
97 |
96
|
fveq1d |
⊢ ( 𝑒 = 𝑑 → ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) ) |
98 |
97
|
eqeq2d |
⊢ ( 𝑒 = 𝑑 → ( ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ↔ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) ) ) |
99 |
95 98
|
anbi12d |
⊢ ( 𝑒 = 𝑑 → ( ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ↔ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) ) ) ) |
100 |
99
|
rspcev |
⊢ ( ( 𝑑 ∈ 𝐽 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) ) ) → ∃ 𝑒 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) |
101 |
91 92 93 100
|
syl12anc |
⊢ ( ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) → ∃ 𝑒 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) |
102 |
|
fvex |
⊢ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) ∈ V |
103 |
|
eqeq1 |
⊢ ( 𝑐 = ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) → ( 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ↔ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) |
104 |
103
|
anbi2d |
⊢ ( 𝑐 = ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) → ( ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ↔ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) |
105 |
104
|
rexbidv |
⊢ ( 𝑐 = ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) → ( ∃ 𝑒 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ↔ ∃ 𝑒 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) |
106 |
102 105
|
elab |
⊢ ( ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) ∈ { 𝑐 ∣ ∃ 𝑒 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ↔ ∃ 𝑒 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) |
107 |
101 106
|
sylibr |
⊢ ( ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) → ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) ∈ { 𝑐 ∣ ∃ 𝑒 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ) |
108 |
107
|
adantl |
⊢ ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) → ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) ∈ { 𝑐 ∣ ∃ 𝑒 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ) |
109 |
90 108
|
sseldd |
⊢ ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) → ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) ∈ { 𝑐 ∣ ∃ 𝑒 ∈ 𝐼 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ) |
110 |
104
|
rexbidv |
⊢ ( 𝑐 = ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) → ( ∃ 𝑒 ∈ 𝐼 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ↔ ∃ 𝑒 ∈ 𝐼 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) |
111 |
102 110
|
elab |
⊢ ( ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) ∈ { 𝑐 ∣ ∃ 𝑒 ∈ 𝐼 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } ↔ ∃ 𝑒 ∈ 𝐼 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) |
112 |
|
simpll2 |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝜑 ) |
113 |
112 59
|
syl |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝑃 ∈ Ring ) |
114 |
|
ringgrp |
⊢ ( 𝑃 ∈ Ring → 𝑃 ∈ Grp ) |
115 |
113 114
|
syl |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝑃 ∈ Grp ) |
116 |
112 11
|
syl |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝐽 ⊆ ( Base ‘ 𝑃 ) ) |
117 |
|
simplrl |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝑑 ∈ 𝐽 ) |
118 |
116 117
|
sseldd |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝑑 ∈ ( Base ‘ 𝑃 ) ) |
119 |
9 2
|
lidlss |
⊢ ( 𝐼 ∈ 𝑈 → 𝐼 ⊆ ( Base ‘ 𝑃 ) ) |
120 |
5 119
|
syl |
⊢ ( 𝜑 → 𝐼 ⊆ ( Base ‘ 𝑃 ) ) |
121 |
112 120
|
syl |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝐼 ⊆ ( Base ‘ 𝑃 ) ) |
122 |
|
simprl |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝑒 ∈ 𝐼 ) |
123 |
121 122
|
sseldd |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝑒 ∈ ( Base ‘ 𝑃 ) ) |
124 |
|
eqid |
⊢ ( +g ‘ 𝑃 ) = ( +g ‘ 𝑃 ) |
125 |
|
eqid |
⊢ ( -g ‘ 𝑃 ) = ( -g ‘ 𝑃 ) |
126 |
9 124 125
|
grpnpcan |
⊢ ( ( 𝑃 ∈ Grp ∧ 𝑑 ∈ ( Base ‘ 𝑃 ) ∧ 𝑒 ∈ ( Base ‘ 𝑃 ) ) → ( ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ( +g ‘ 𝑃 ) 𝑒 ) = 𝑑 ) |
127 |
115 118 123 126
|
syl3anc |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → ( ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ( +g ‘ 𝑃 ) 𝑒 ) = 𝑑 ) |
128 |
5
|
3ad2ant2 |
⊢ ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) → 𝐼 ∈ 𝑈 ) |
129 |
128
|
ad2antrr |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝐼 ∈ 𝑈 ) |
130 |
|
simpll1 |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝑏 ∈ ℕ0 ) |
131 |
112 4
|
syl |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝑅 ∈ Ring ) |
132 |
|
simplrr |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) |
133 |
|
simprrl |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ) |
134 |
|
eqid |
⊢ ( coe1 ‘ 𝑑 ) = ( coe1 ‘ 𝑑 ) |
135 |
|
eqid |
⊢ ( coe1 ‘ 𝑒 ) = ( coe1 ‘ 𝑒 ) |
136 |
|
simprrr |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) |
137 |
13 1 9 125 130 131 118 132 123 133 134 135 136
|
deg1sublt |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ) < 𝑏 ) |
138 |
112 6
|
syl |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝐽 ∈ 𝑈 ) |
139 |
7
|
3ad2ant2 |
⊢ ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) → 𝐼 ⊆ 𝐽 ) |
140 |
139
|
ad2antrr |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝐼 ⊆ 𝐽 ) |
141 |
140 122
|
sseldd |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝑒 ∈ 𝐽 ) |
142 |
2 125
|
lidlsubcl |
⊢ ( ( ( 𝑃 ∈ Ring ∧ 𝐽 ∈ 𝑈 ) ∧ ( 𝑑 ∈ 𝐽 ∧ 𝑒 ∈ 𝐽 ) ) → ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ∈ 𝐽 ) |
143 |
113 138 117 141 142
|
syl22anc |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ∈ 𝐽 ) |
144 |
|
simpll3 |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) |
145 |
|
fveq2 |
⊢ ( 𝑎 = ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) = ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ) ) |
146 |
145
|
breq1d |
⊢ ( 𝑎 = ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 ↔ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ) < 𝑏 ) ) |
147 |
|
eleq1 |
⊢ ( 𝑎 = ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) → ( 𝑎 ∈ 𝐼 ↔ ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ∈ 𝐼 ) ) |
148 |
146 147
|
imbi12d |
⊢ ( 𝑎 = ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) → ( ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ↔ ( ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ) < 𝑏 → ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ∈ 𝐼 ) ) ) |
149 |
148
|
rspcva |
⊢ ( ( ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ∈ 𝐽 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) → ( ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ) < 𝑏 → ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ∈ 𝐼 ) ) |
150 |
143 144 149
|
syl2anc |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → ( ( ( deg1 ‘ 𝑅 ) ‘ ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ) < 𝑏 → ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ∈ 𝐼 ) ) |
151 |
137 150
|
mpd |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ∈ 𝐼 ) |
152 |
2 124
|
lidlacl |
⊢ ( ( ( 𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) ∧ ( ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ∈ 𝐼 ∧ 𝑒 ∈ 𝐼 ) ) → ( ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ( +g ‘ 𝑃 ) 𝑒 ) ∈ 𝐼 ) |
153 |
113 129 151 122 152
|
syl22anc |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → ( ( 𝑑 ( -g ‘ 𝑃 ) 𝑒 ) ( +g ‘ 𝑃 ) 𝑒 ) ∈ 𝐼 ) |
154 |
127 153
|
eqeltrrd |
⊢ ( ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) ∧ ( 𝑒 ∈ 𝐼 ∧ ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) ) ) → 𝑑 ∈ 𝐼 ) |
155 |
154
|
rexlimdvaa |
⊢ ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) → ( ∃ 𝑒 ∈ 𝐼 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) → 𝑑 ∈ 𝐼 ) ) |
156 |
111 155
|
syl5bi |
⊢ ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) → ( ( ( coe1 ‘ 𝑑 ) ‘ 𝑏 ) ∈ { 𝑐 ∣ ∃ 𝑒 ∈ 𝐼 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑒 ) ≤ 𝑏 ∧ 𝑐 = ( ( coe1 ‘ 𝑒 ) ‘ 𝑏 ) ) } → 𝑑 ∈ 𝐼 ) ) |
157 |
109 156
|
mpd |
⊢ ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ ( 𝑑 ∈ 𝐽 ∧ ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 ) ) → 𝑑 ∈ 𝐼 ) |
158 |
157
|
expr |
⊢ ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ 𝑑 ∈ 𝐽 ) → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) ≤ 𝑏 → 𝑑 ∈ 𝐼 ) ) |
159 |
74 158
|
sylbid |
⊢ ( ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ∧ 𝑑 ∈ 𝐽 ) → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) < ( 𝑏 + 1 ) → 𝑑 ∈ 𝐼 ) ) |
160 |
159
|
ralrimiva |
⊢ ( ( 𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) → ∀ 𝑑 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) < ( 𝑏 + 1 ) → 𝑑 ∈ 𝐼 ) ) |
161 |
160
|
3exp |
⊢ ( 𝑏 ∈ ℕ0 → ( 𝜑 → ( ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) → ∀ 𝑑 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) < ( 𝑏 + 1 ) → 𝑑 ∈ 𝐼 ) ) ) ) |
162 |
161
|
a2d |
⊢ ( 𝑏 ∈ ℕ0 → ( ( 𝜑 → ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) → ( 𝜑 → ∀ 𝑑 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑑 ) < ( 𝑏 + 1 ) → 𝑑 ∈ 𝐼 ) ) ) ) |
163 |
39 43 53 43 66 162
|
nn0ind |
⊢ ( 𝑏 ∈ ℕ0 → ( 𝜑 → ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ) |
164 |
|
rsp |
⊢ ( ∀ 𝑎 ∈ 𝐽 ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) → ( 𝑎 ∈ 𝐽 → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ) |
165 |
163 164
|
syl6com |
⊢ ( 𝜑 → ( 𝑏 ∈ ℕ0 → ( 𝑎 ∈ 𝐽 → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ) ) |
166 |
165
|
com23 |
⊢ ( 𝜑 → ( 𝑎 ∈ 𝐽 → ( 𝑏 ∈ ℕ0 → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ) ) |
167 |
166
|
imp |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐽 ) → ( 𝑏 ∈ ℕ0 → ( ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) ) |
168 |
167
|
rexlimdv |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐽 ) → ( ∃ 𝑏 ∈ ℕ0 ( ( deg1 ‘ 𝑅 ) ‘ 𝑎 ) < 𝑏 → 𝑎 ∈ 𝐼 ) ) |
169 |
35 168
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐽 ) → 𝑎 ∈ 𝐼 ) |
170 |
7 169
|
eqelssd |
⊢ ( 𝜑 → 𝐼 = 𝐽 ) |