Step | Hyp | Ref
| Expression |
1 | | hbtlem.p |
. . 3
⊢ 𝑃 = (Poly1‘𝑅) |
2 | | hbtlem.u |
. . 3
⊢ 𝑈 = (LIdeal‘𝑃) |
3 | | hbtlem.s |
. . 3
⊢ 𝑆 = (ldgIdlSeq‘𝑅) |
4 | | eqid 2622 |
. . 3
⊢ (
deg1 ‘𝑅) =
( deg1 ‘𝑅) |
5 | 1, 2, 3, 4 | hbtlem1 37693 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}) |
6 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
(Base‘𝑃) =
(Base‘𝑃) |
7 | 6, 2 | lidlss 19210 |
. . . . . . . . . . 11
⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃)) |
8 | 7 | 3ad2ant2 1083 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝐼 ⊆ (Base‘𝑃)) |
9 | 8 | sselda 3603 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑃)) |
10 | | eqid 2622 |
. . . . . . . . . 10
⊢
(coe1‘𝑏) = (coe1‘𝑏) |
11 | | eqid 2622 |
. . . . . . . . . 10
⊢
(Base‘𝑅) =
(Base‘𝑅) |
12 | 10, 6, 1, 11 | coe1f 19581 |
. . . . . . . . 9
⊢ (𝑏 ∈ (Base‘𝑃) →
(coe1‘𝑏):ℕ0⟶(Base‘𝑅)) |
13 | 9, 12 | syl 17 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → (coe1‘𝑏):ℕ0⟶(Base‘𝑅)) |
14 | | simpl3 1066 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → 𝑋 ∈
ℕ0) |
15 | 13, 14 | ffvelrnd 6360 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → ((coe1‘𝑏)‘𝑋) ∈ (Base‘𝑅)) |
16 | | eleq1a 2696 |
. . . . . . 7
⊢
(((coe1‘𝑏)‘𝑋) ∈ (Base‘𝑅) → (𝑎 = ((coe1‘𝑏)‘𝑋) → 𝑎 ∈ (Base‘𝑅))) |
17 | 15, 16 | syl 17 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → (𝑎 = ((coe1‘𝑏)‘𝑋) → 𝑎 ∈ (Base‘𝑅))) |
18 | 17 | adantld 483 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → (((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅))) |
19 | 18 | rexlimdva 3031 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) →
(∃𝑏 ∈ 𝐼 ((( deg1
‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅))) |
20 | 19 | abssdv 3676 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ⊆ (Base‘𝑅)) |
21 | 1 | ply1ring 19618 |
. . . . . . . 8
⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
22 | 21 | 3ad2ant1 1082 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑃 ∈ Ring) |
23 | | simp2 1062 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝐼 ∈ 𝑈) |
24 | | eqid 2622 |
. . . . . . . 8
⊢
(0g‘𝑃) = (0g‘𝑃) |
25 | 2, 24 | lidl0cl 19212 |
. . . . . . 7
⊢ ((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0g‘𝑃) ∈ 𝐼) |
26 | 22, 23, 25 | syl2anc 693 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) →
(0g‘𝑃)
∈ 𝐼) |
27 | 4, 1, 24 | deg1z 23847 |
. . . . . . . 8
⊢ (𝑅 ∈ Ring → ((
deg1 ‘𝑅)‘(0g‘𝑃)) = -∞) |
28 | 27 | 3ad2ant1 1082 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((
deg1 ‘𝑅)‘(0g‘𝑃)) = -∞) |
29 | | nn0ssre 11296 |
. . . . . . . . . 10
⊢
ℕ0 ⊆ ℝ |
30 | | ressxr 10083 |
. . . . . . . . . 10
⊢ ℝ
⊆ ℝ* |
31 | 29, 30 | sstri 3612 |
. . . . . . . . 9
⊢
ℕ0 ⊆ ℝ* |
32 | | simp3 1063 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈
ℕ0) |
33 | 31, 32 | sseldi 3601 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈
ℝ*) |
34 | | mnfle 11969 |
. . . . . . . 8
⊢ (𝑋 ∈ ℝ*
→ -∞ ≤ 𝑋) |
35 | 33, 34 | syl 17 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → -∞
≤ 𝑋) |
36 | 28, 35 | eqbrtrd 4675 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((
deg1 ‘𝑅)‘(0g‘𝑃)) ≤ 𝑋) |
37 | | eqid 2622 |
. . . . . . . . . 10
⊢
(0g‘𝑅) = (0g‘𝑅) |
38 | 1, 24, 37 | coe1z 19633 |
. . . . . . . . 9
⊢ (𝑅 ∈ Ring →
(coe1‘(0g‘𝑃)) = (ℕ0 ×
{(0g‘𝑅)})) |
39 | 38 | 3ad2ant1 1082 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) →
(coe1‘(0g‘𝑃)) = (ℕ0 ×
{(0g‘𝑅)})) |
40 | 39 | fveq1d 6193 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) →
((coe1‘(0g‘𝑃))‘𝑋) = ((ℕ0 ×
{(0g‘𝑅)})‘𝑋)) |
41 | | fvex 6201 |
. . . . . . . . 9
⊢
(0g‘𝑅) ∈ V |
42 | 41 | fvconst2 6469 |
. . . . . . . 8
⊢ (𝑋 ∈ ℕ0
→ ((ℕ0 × {(0g‘𝑅)})‘𝑋) = (0g‘𝑅)) |
43 | 42 | 3ad2ant3 1084 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) →
((ℕ0 × {(0g‘𝑅)})‘𝑋) = (0g‘𝑅)) |
44 | 40, 43 | eqtr2d 2657 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) →
(0g‘𝑅) =
((coe1‘(0g‘𝑃))‘𝑋)) |
45 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑏 = (0g‘𝑃) → (( deg1
‘𝑅)‘𝑏) = (( deg1
‘𝑅)‘(0g‘𝑃))) |
46 | 45 | breq1d 4663 |
. . . . . . . 8
⊢ (𝑏 = (0g‘𝑃) → ((( deg1
‘𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1 ‘𝑅)‘(0g‘𝑃)) ≤ 𝑋)) |
47 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑏 = (0g‘𝑃) →
(coe1‘𝑏) =
(coe1‘(0g‘𝑃))) |
48 | 47 | fveq1d 6193 |
. . . . . . . . 9
⊢ (𝑏 = (0g‘𝑃) →
((coe1‘𝑏)‘𝑋) =
((coe1‘(0g‘𝑃))‘𝑋)) |
49 | 48 | eqeq2d 2632 |
. . . . . . . 8
⊢ (𝑏 = (0g‘𝑃) →
((0g‘𝑅) =
((coe1‘𝑏)‘𝑋) ↔ (0g‘𝑅) =
((coe1‘(0g‘𝑃))‘𝑋))) |
50 | 46, 49 | anbi12d 747 |
. . . . . . 7
⊢ (𝑏 = (0g‘𝑃) → (((( deg1
‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) =
((coe1‘𝑏)‘𝑋)) ↔ ((( deg1 ‘𝑅)‘(0g‘𝑃)) ≤ 𝑋 ∧ (0g‘𝑅) =
((coe1‘(0g‘𝑃))‘𝑋)))) |
51 | 50 | rspcev 3309 |
. . . . . 6
⊢
(((0g‘𝑃) ∈ 𝐼 ∧ ((( deg1 ‘𝑅)‘(0g‘𝑃)) ≤ 𝑋 ∧ (0g‘𝑅) =
((coe1‘(0g‘𝑃))‘𝑋))) → ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) =
((coe1‘𝑏)‘𝑋))) |
52 | 26, 36, 44, 51 | syl12anc 1324 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) →
∃𝑏 ∈ 𝐼 ((( deg1
‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) =
((coe1‘𝑏)‘𝑋))) |
53 | | eqeq1 2626 |
. . . . . . . 8
⊢ (𝑎 = (0g‘𝑅) → (𝑎 = ((coe1‘𝑏)‘𝑋) ↔ (0g‘𝑅) =
((coe1‘𝑏)‘𝑋))) |
54 | 53 | anbi2d 740 |
. . . . . . 7
⊢ (𝑎 = (0g‘𝑅) → (((( deg1
‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) =
((coe1‘𝑏)‘𝑋)))) |
55 | 54 | rexbidv 3052 |
. . . . . 6
⊢ (𝑎 = (0g‘𝑅) → (∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) =
((coe1‘𝑏)‘𝑋)))) |
56 | 41, 55 | elab 3350 |
. . . . 5
⊢
((0g‘𝑅) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ↔ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) =
((coe1‘𝑏)‘𝑋))) |
57 | 52, 56 | sylibr 224 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) →
(0g‘𝑅)
∈ {𝑎 ∣
∃𝑏 ∈ 𝐼 ((( deg1
‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}) |
58 | | ne0i 3921 |
. . . 4
⊢
((0g‘𝑅) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} → {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ≠ ∅) |
59 | 57, 58 | syl 17 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ≠ ∅) |
60 | 22 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑃 ∈ Ring) |
61 | | simpl2 1065 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝐼 ∈ 𝑈) |
62 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(algSc‘𝑃) =
(algSc‘𝑃) |
63 | 1, 62, 11, 6 | ply1sclf 19655 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑅 ∈ Ring →
(algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃)) |
64 | 63 | 3ad2ant1 1082 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) →
(algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃)) |
65 | 64 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃)) |
66 | | simprl 794 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑐 ∈ (Base‘𝑅)) |
67 | 65, 66 | ffvelrnd 6360 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → ((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃)) |
68 | | simprll 802 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋))) → 𝑓 ∈ 𝐼) |
69 | 68 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑓 ∈ 𝐼) |
70 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(.r‘𝑃) = (.r‘𝑃) |
71 | 2, 6, 70 | lidlmcl 19217 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃) ∧ 𝑓 ∈ 𝐼)) → (((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ 𝐼) |
72 | 60, 61, 67, 69, 71 | syl22anc 1327 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → (((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ 𝐼) |
73 | | simprrl 804 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋))) → 𝑔 ∈ 𝐼) |
74 | 73 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑔 ∈ 𝐼) |
75 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(+g‘𝑃) = (+g‘𝑃) |
76 | 2, 75 | lidlacl 19213 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ 𝐼 ∧ 𝑔 ∈ 𝐼)) → ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) ∈ 𝐼) |
77 | 60, 61, 72, 74, 76 | syl22anc 1327 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) ∈ 𝐼) |
78 | | simpl1 1064 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑅 ∈ Ring) |
79 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝐼 ⊆ (Base‘𝑃)) |
80 | 79, 69 | sseldd 3604 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑓 ∈ (Base‘𝑃)) |
81 | 6, 70 | ringcl 18561 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑃 ∈ Ring ∧
((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃) ∧ 𝑓 ∈ (Base‘𝑃)) → (((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ (Base‘𝑃)) |
82 | 60, 67, 80, 81 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → (((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ (Base‘𝑃)) |
83 | 79, 74 | sseldd 3604 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑔 ∈ (Base‘𝑃)) |
84 | | simpl3 1066 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑋 ∈
ℕ0) |
85 | 31, 84 | sseldi 3601 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑋 ∈
ℝ*) |
86 | 4, 1, 6 | deg1xrcl 23842 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ (Base‘𝑃) → (( deg1 ‘𝑅)‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)) ∈
ℝ*) |
87 | 82, 86 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1 ‘𝑅)‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)) ∈
ℝ*) |
88 | 4, 1, 6 | deg1xrcl 23842 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑓 ∈ (Base‘𝑃) → (( deg1
‘𝑅)‘𝑓) ∈
ℝ*) |
89 | 80, 88 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1 ‘𝑅)‘𝑓) ∈
ℝ*) |
90 | 4, 1, 11, 6, 70, 62 | deg1mul3le 23876 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑅 ∈ Ring ∧ 𝑐 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑃)) → (( deg1 ‘𝑅)‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)) ≤ (( deg1 ‘𝑅)‘𝑓)) |
91 | 78, 66, 80, 90 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1 ‘𝑅)‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)) ≤ (( deg1 ‘𝑅)‘𝑓)) |
92 | | simprlr 803 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋))) → (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) |
93 | 92 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) |
94 | 87, 89, 85, 91, 93 | xrletrd 11993 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1 ‘𝑅)‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)) ≤ 𝑋) |
95 | | simprrr 805 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋))) → (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋) |
96 | 95 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋) |
97 | 1, 4, 78, 6, 75, 82, 83, 85, 94, 96 | deg1addle2 23862 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1 ‘𝑅)‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔)) ≤ 𝑋) |
98 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(+g‘𝑅) = (+g‘𝑅) |
99 | 1, 6, 75, 98 | coe1addfv 19635 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑅 ∈ Ring ∧
(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ (Base‘𝑃) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ 𝑋 ∈ ℕ0) →
((coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))‘𝑋) =
(((coe1‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓))‘𝑋)(+g‘𝑅)((coe1‘𝑔)‘𝑋))) |
100 | 78, 82, 83, 84, 99 | syl31anc 1329 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) →
((coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))‘𝑋) =
(((coe1‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓))‘𝑋)(+g‘𝑅)((coe1‘𝑔)‘𝑋))) |
101 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(.r‘𝑅) = (.r‘𝑅) |
102 | 1, 6, 11, 62, 70, 101 | coe1sclmulfv 19653 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑅 ∈ Ring ∧ (𝑐 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑋 ∈ ℕ0) →
((coe1‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓))‘𝑋) = (𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))) |
103 | 78, 66, 80, 84, 102 | syl121anc 1331 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) →
((coe1‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓))‘𝑋) = (𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))) |
104 | 103 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) →
(((coe1‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓))‘𝑋)(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋))) |
105 | 100, 104 | eqtr2d 2657 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) =
((coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))‘𝑋)) |
106 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) → (( deg1 ‘𝑅)‘𝑏) = (( deg1 ‘𝑅)‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))) |
107 | 106 | breq1d 4663 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) → ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1 ‘𝑅)‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔)) ≤ 𝑋)) |
108 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) → (coe1‘𝑏) =
(coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))) |
109 | 108 | fveq1d 6193 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) → ((coe1‘𝑏)‘𝑋) =
((coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))‘𝑋)) |
110 | 109 | eqeq2d 2632 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) → (((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋) ↔ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) =
((coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))‘𝑋))) |
111 | 107, 110 | anbi12d 747 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) → (((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋)) ↔ ((( deg1 ‘𝑅)‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔)) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) =
((coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))‘𝑋)))) |
112 | 111 | rspcev 3309 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) ∈ 𝐼 ∧ ((( deg1 ‘𝑅)‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔)) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) =
((coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))‘𝑋))) → ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋))) |
113 | 77, 97, 105, 112 | syl12anc 1324 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋))) |
114 | | ovex 6678 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ V |
115 | | eqeq1 2626 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑎 = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) → (𝑎 = ((coe1‘𝑏)‘𝑋) ↔ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋))) |
116 | 115 | anbi2d 740 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑎 = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) → (((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋)))) |
117 | 116 | rexbidv 3052 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎 = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) → (∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋)))) |
118 | 114, 117 | elab 3350 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ↔ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋))) |
119 | 113, 118 | sylibr 224 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)))) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}) |
120 | 119 | exp45 642 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (𝑐 ∈ (Base‘𝑅) → ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) → ((𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})))) |
121 | 120 | imp 445 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) → ((𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))) |
122 | 121 | exp5c 644 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → (𝑓 ∈ 𝐼 → ((( deg1 ‘𝑅)‘𝑓) ≤ 𝑋 → (𝑔 ∈ 𝐼 → ((( deg1 ‘𝑅)‘𝑔) ≤ 𝑋 → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))))) |
123 | 122 | imp 445 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) → ((( deg1 ‘𝑅)‘𝑓) ≤ 𝑋 → (𝑔 ∈ 𝐼 → ((( deg1 ‘𝑅)‘𝑔) ≤ 𝑋 → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})))) |
124 | 123 | imp41 619 |
. . . . . . . . . . . . . 14
⊢
(((((((𝑅 ∈ Ring
∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ 𝑔 ∈ 𝐼) ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}) |
125 | | oveq2 6658 |
. . . . . . . . . . . . . . 15
⊢ (𝑒 = ((coe1‘𝑔)‘𝑋) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋))) |
126 | 125 | eleq1d 2686 |
. . . . . . . . . . . . . 14
⊢ (𝑒 = ((coe1‘𝑔)‘𝑋) → (((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ↔ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})) |
127 | 124, 126 | syl5ibrcom 237 |
. . . . . . . . . . . . 13
⊢
(((((((𝑅 ∈ Ring
∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ 𝑔 ∈ 𝐼) ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋) → (𝑒 = ((coe1‘𝑔)‘𝑋) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})) |
128 | 127 | expimpd 629 |
. . . . . . . . . . . 12
⊢
((((((𝑅 ∈ Ring
∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ 𝑔 ∈ 𝐼) → (((( deg1 ‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋)) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})) |
129 | 128 | rexlimdva 3031 |
. . . . . . . . . . 11
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) → (∃𝑔 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋)) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})) |
130 | 129 | alrimiv 1855 |
. . . . . . . . . 10
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) → ∀𝑒(∃𝑔 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋)) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})) |
131 | | eqeq1 2626 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑒 → (𝑎 = ((coe1‘𝑏)‘𝑋) ↔ 𝑒 = ((coe1‘𝑏)‘𝑋))) |
132 | 131 | anbi2d 740 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑒 → (((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋)))) |
133 | 132 | rexbidv 3052 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑒 → (∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋)))) |
134 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑔 → (( deg1 ‘𝑅)‘𝑏) = (( deg1 ‘𝑅)‘𝑔)) |
135 | 134 | breq1d 4663 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = 𝑔 → ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)) |
136 | | fveq2 6191 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = 𝑔 → (coe1‘𝑏) = (coe1‘𝑔)) |
137 | 136 | fveq1d 6193 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑔 → ((coe1‘𝑏)‘𝑋) = ((coe1‘𝑔)‘𝑋)) |
138 | 137 | eqeq2d 2632 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = 𝑔 → (𝑒 = ((coe1‘𝑏)‘𝑋) ↔ 𝑒 = ((coe1‘𝑔)‘𝑋))) |
139 | 135, 138 | anbi12d 747 |
. . . . . . . . . . . . 13
⊢ (𝑏 = 𝑔 → (((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋)) ↔ ((( deg1 ‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋)))) |
140 | 139 | cbvrexv 3172 |
. . . . . . . . . . . 12
⊢
(∃𝑏 ∈
𝐼 ((( deg1
‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑔 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋))) |
141 | 133, 140 | syl6bb 276 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑒 → (∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑔 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋)))) |
142 | 141 | ralab 3367 |
. . . . . . . . . 10
⊢
(∀𝑒 ∈
{𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ↔ ∀𝑒(∃𝑔 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋)) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})) |
143 | 130, 142 | sylibr 224 |
. . . . . . . . 9
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}) |
144 | | oveq2 6658 |
. . . . . . . . . . . 12
⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → (𝑐(.r‘𝑅)𝑑) = (𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))) |
145 | 144 | oveq1d 6665 |
. . . . . . . . . . 11
⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒)) |
146 | 145 | eleq1d 2686 |
. . . . . . . . . 10
⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → (((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ↔ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})) |
147 | 146 | ralbidv 2986 |
. . . . . . . . 9
⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → (∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ↔ ∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})) |
148 | 143, 147 | syl5ibrcom 237 |
. . . . . . . 8
⊢
(((((𝑅 ∈ Ring
∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) → (𝑑 = ((coe1‘𝑓)‘𝑋) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})) |
149 | 148 | expimpd 629 |
. . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) → (((( deg1 ‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋)) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})) |
150 | 149 | rexlimdva 3031 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → (∃𝑓 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋)) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})) |
151 | 150 | alrimiv 1855 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → ∀𝑑(∃𝑓 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋)) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})) |
152 | | eqeq1 2626 |
. . . . . . . . 9
⊢ (𝑎 = 𝑑 → (𝑎 = ((coe1‘𝑏)‘𝑋) ↔ 𝑑 = ((coe1‘𝑏)‘𝑋))) |
153 | 152 | anbi2d 740 |
. . . . . . . 8
⊢ (𝑎 = 𝑑 → (((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋)))) |
154 | 153 | rexbidv 3052 |
. . . . . . 7
⊢ (𝑎 = 𝑑 → (∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋)))) |
155 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑓 → (( deg1 ‘𝑅)‘𝑏) = (( deg1 ‘𝑅)‘𝑓)) |
156 | 155 | breq1d 4663 |
. . . . . . . . 9
⊢ (𝑏 = 𝑓 → ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋)) |
157 | | fveq2 6191 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑓 → (coe1‘𝑏) = (coe1‘𝑓)) |
158 | 157 | fveq1d 6193 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑓 → ((coe1‘𝑏)‘𝑋) = ((coe1‘𝑓)‘𝑋)) |
159 | 158 | eqeq2d 2632 |
. . . . . . . . 9
⊢ (𝑏 = 𝑓 → (𝑑 = ((coe1‘𝑏)‘𝑋) ↔ 𝑑 = ((coe1‘𝑓)‘𝑋))) |
160 | 156, 159 | anbi12d 747 |
. . . . . . . 8
⊢ (𝑏 = 𝑓 → (((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋)) ↔ ((( deg1 ‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋)))) |
161 | 160 | cbvrexv 3172 |
. . . . . . 7
⊢
(∃𝑏 ∈
𝐼 ((( deg1
‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑓 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋))) |
162 | 154, 161 | syl6bb 276 |
. . . . . 6
⊢ (𝑎 = 𝑑 → (∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑓 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋)))) |
163 | 162 | ralab 3367 |
. . . . 5
⊢
(∀𝑑 ∈
{𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ↔ ∀𝑑(∃𝑓 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋)) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})) |
164 | 151, 163 | sylibr 224 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → ∀𝑑 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}) |
165 | 164 | ralrimiva 2966 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) →
∀𝑐 ∈
(Base‘𝑅)∀𝑑 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}) |
166 | | hbtlem2.t |
. . . 4
⊢ 𝑇 = (LIdeal‘𝑅) |
167 | 166, 11, 98, 101 | islidl 19211 |
. . 3
⊢ ({𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ∈ 𝑇 ↔ ({𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ⊆ (Base‘𝑅) ∧ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ≠ ∅ ∧ ∀𝑐 ∈ (Base‘𝑅)∀𝑑 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})) |
168 | 20, 59, 165, 167 | syl3anbrc 1246 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ∈ 𝑇) |
169 | 5, 168 | eqeltrd 2701 |
1
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) ∈ 𝑇) |