Proof of Theorem mdegldg
Step | Hyp | Ref
| Expression |
1 | | mdegval.d |
. . . . 5
⊢ 𝐷 = (𝐼 mDeg 𝑅) |
2 | | mdegval.p |
. . . . 5
⊢ 𝑃 = (𝐼 mPoly 𝑅) |
3 | | mdegval.b |
. . . . 5
⊢ 𝐵 = (Base‘𝑃) |
4 | | mdegval.z |
. . . . 5
⊢ 0 =
(0g‘𝑅) |
5 | | mdegval.a |
. . . . 5
⊢ 𝐴 = {𝑚 ∈ (ℕ0
↑𝑚 𝐼) ∣ (◡𝑚 “ ℕ) ∈
Fin} |
6 | | mdegval.h |
. . . . 5
⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld
Σg ℎ)) |
7 | 1, 2, 3, 4, 5, 6 | mdegval 24271 |
. . . 4
⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ*,
< )) |
8 | 7 | 3ad2ant2 1125 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ*,
< )) |
9 | 2, 3 | mplrcl 19897 |
. . . . . . . 8
⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V) |
10 | 9 | 3ad2ant2 1125 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → 𝐼 ∈ V) |
11 | 5, 6 | tdeglem1 24266 |
. . . . . . 7
⊢ (𝐼 ∈ V → 𝐻:𝐴⟶ℕ0) |
12 | 10, 11 | syl 17 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → 𝐻:𝐴⟶ℕ0) |
13 | 12 | ffund 6297 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → Fun 𝐻) |
14 | | simp2 1128 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → 𝐹 ∈ 𝐵) |
15 | | simp1 1127 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → 𝑅 ∈ Ring) |
16 | 2, 3, 4, 14, 15 | mplelsfi 19898 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → 𝐹 finSupp 0 ) |
17 | 16 | fsuppimpd 8572 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → (𝐹 supp 0 ) ∈
Fin) |
18 | | imafi 8549 |
. . . . 5
⊢ ((Fun
𝐻 ∧ (𝐹 supp 0 ) ∈ Fin) →
(𝐻 “ (𝐹 supp 0 )) ∈
Fin) |
19 | 13, 17, 18 | syl2anc 579 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → (𝐻 “ (𝐹 supp 0 )) ∈
Fin) |
20 | | simp3 1129 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → 𝐹 ≠ 𝑌) |
21 | | mdegldg.y |
. . . . . . . 8
⊢ 𝑌 = (0g‘𝑃) |
22 | | ringgrp 18950 |
. . . . . . . . 9
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) |
23 | 22 | 3ad2ant1 1124 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → 𝑅 ∈ Grp) |
24 | 2, 5, 4, 21, 10, 23 | mpl0 19849 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → 𝑌 = (𝐴 × { 0 })) |
25 | 20, 24 | neeqtrd 3038 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → 𝐹 ≠ (𝐴 × { 0 })) |
26 | | eqid 2778 |
. . . . . . . . . 10
⊢
(Base‘𝑅) =
(Base‘𝑅) |
27 | 2, 26, 3, 5, 14 | mplelf 19841 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → 𝐹:𝐴⟶(Base‘𝑅)) |
28 | 27 | ffnd 6294 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → 𝐹 Fn 𝐴) |
29 | 4 | fvexi 6462 |
. . . . . . . 8
⊢ 0 ∈
V |
30 | | ovex 6956 |
. . . . . . . . . 10
⊢
(ℕ0 ↑𝑚 𝐼) ∈ V |
31 | 5, 30 | rabex2 5053 |
. . . . . . . . 9
⊢ 𝐴 ∈ V |
32 | | fnsuppeq0 7607 |
. . . . . . . . 9
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ V ∧ 0 ∈ V) → ((𝐹 supp 0 ) = ∅ ↔ 𝐹 = (𝐴 × { 0 }))) |
33 | 31, 32 | mp3an2 1522 |
. . . . . . . 8
⊢ ((𝐹 Fn 𝐴 ∧ 0 ∈ V) → ((𝐹 supp 0 ) = ∅ ↔ 𝐹 = (𝐴 × { 0 }))) |
34 | 28, 29, 33 | sylancl 580 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → ((𝐹 supp 0 ) = ∅ ↔ 𝐹 = (𝐴 × { 0 }))) |
35 | 34 | necon3bid 3013 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → ((𝐹 supp 0 ) ≠ ∅ ↔
𝐹 ≠ (𝐴 × { 0 }))) |
36 | 25, 35 | mpbird 249 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → (𝐹 supp 0 ) ≠
∅) |
37 | 12 | ffnd 6294 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → 𝐻 Fn 𝐴) |
38 | | suppssdm 7591 |
. . . . . . . 8
⊢ (𝐹 supp 0 ) ⊆ dom 𝐹 |
39 | 38, 27 | fssdm 6309 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → (𝐹 supp 0 ) ⊆ 𝐴) |
40 | | fnimaeq0 6261 |
. . . . . . 7
⊢ ((𝐻 Fn 𝐴 ∧ (𝐹 supp 0 ) ⊆ 𝐴) → ((𝐻 “ (𝐹 supp 0 )) = ∅ ↔ (𝐹 supp 0 ) =
∅)) |
41 | 37, 39, 40 | syl2anc 579 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → ((𝐻 “ (𝐹 supp 0 )) = ∅ ↔ (𝐹 supp 0 ) =
∅)) |
42 | 41 | necon3bid 3013 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → ((𝐻 “ (𝐹 supp 0 )) ≠ ∅ ↔
(𝐹 supp 0 ) ≠
∅)) |
43 | 36, 42 | mpbird 249 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → (𝐻 “ (𝐹 supp 0 )) ≠
∅) |
44 | | imassrn 5733 |
. . . . . 6
⊢ (𝐻 “ (𝐹 supp 0 )) ⊆ ran 𝐻 |
45 | 12 | frnd 6300 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → ran 𝐻 ⊆
ℕ0) |
46 | 44, 45 | syl5ss 3832 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → (𝐻 “ (𝐹 supp 0 )) ⊆
ℕ0) |
47 | | nn0ssre 11651 |
. . . . . 6
⊢
ℕ0 ⊆ ℝ |
48 | | ressxr 10422 |
. . . . . 6
⊢ ℝ
⊆ ℝ* |
49 | 47, 48 | sstri 3830 |
. . . . 5
⊢
ℕ0 ⊆ ℝ* |
50 | 46, 49 | syl6ss 3833 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → (𝐻 “ (𝐹 supp 0 )) ⊆
ℝ*) |
51 | | xrltso 12289 |
. . . . 5
⊢ < Or
ℝ* |
52 | | fisupcl 8665 |
. . . . 5
⊢ (( <
Or ℝ* ∧ ((𝐻 “ (𝐹 supp 0 )) ∈ Fin ∧ (𝐻 “ (𝐹 supp 0 )) ≠ ∅ ∧
(𝐻 “ (𝐹 supp 0 )) ⊆
ℝ*)) → sup((𝐻 “ (𝐹 supp 0 )), ℝ*,
< ) ∈ (𝐻 “
(𝐹 supp 0 ))) |
53 | 51, 52 | mpan 680 |
. . . 4
⊢ (((𝐻 “ (𝐹 supp 0 )) ∈ Fin ∧ (𝐻 “ (𝐹 supp 0 )) ≠ ∅ ∧
(𝐻 “ (𝐹 supp 0 )) ⊆
ℝ*) → sup((𝐻 “ (𝐹 supp 0 )), ℝ*,
< ) ∈ (𝐻 “
(𝐹 supp 0 ))) |
54 | 19, 43, 50, 53 | syl3anc 1439 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → sup((𝐻 “ (𝐹 supp 0 )), ℝ*,
< ) ∈ (𝐻 “
(𝐹 supp 0 ))) |
55 | 8, 54 | eqeltrd 2859 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → (𝐷‘𝐹) ∈ (𝐻 “ (𝐹 supp 0 ))) |
56 | | fvelimab 6515 |
. . . 4
⊢ ((𝐻 Fn 𝐴 ∧ (𝐹 supp 0 ) ⊆ 𝐴) → ((𝐷‘𝐹) ∈ (𝐻 “ (𝐹 supp 0 )) ↔ ∃𝑥 ∈ (𝐹 supp 0 )(𝐻‘𝑥) = (𝐷‘𝐹))) |
57 | 37, 39, 56 | syl2anc 579 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → ((𝐷‘𝐹) ∈ (𝐻 “ (𝐹 supp 0 )) ↔ ∃𝑥 ∈ (𝐹 supp 0 )(𝐻‘𝑥) = (𝐷‘𝐹))) |
58 | | rexsupp 7596 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ V ∧ 0 ∈ V) →
(∃𝑥 ∈ (𝐹 supp 0 )(𝐻‘𝑥) = (𝐷‘𝐹) ↔ ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) ≠ 0 ∧ (𝐻‘𝑥) = (𝐷‘𝐹)))) |
59 | 31, 29, 58 | mp3an23 1526 |
. . . 4
⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ (𝐹 supp 0 )(𝐻‘𝑥) = (𝐷‘𝐹) ↔ ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) ≠ 0 ∧ (𝐻‘𝑥) = (𝐷‘𝐹)))) |
60 | 28, 59 | syl 17 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → (∃𝑥 ∈ (𝐹 supp 0 )(𝐻‘𝑥) = (𝐷‘𝐹) ↔ ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) ≠ 0 ∧ (𝐻‘𝑥) = (𝐷‘𝐹)))) |
61 | 57, 60 | bitrd 271 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → ((𝐷‘𝐹) ∈ (𝐻 “ (𝐹 supp 0 )) ↔ ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) ≠ 0 ∧ (𝐻‘𝑥) = (𝐷‘𝐹)))) |
62 | 55, 61 | mpbid 224 |
1
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) ≠ 0 ∧ (𝐻‘𝑥) = (𝐷‘𝐹))) |