Step |
Hyp |
Ref |
Expression |
1 |
|
mdegval.d |
⊢ 𝐷 = ( 𝐼 mDeg 𝑅 ) |
2 |
|
mdegval.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
3 |
|
mdegval.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
4 |
|
mdegval.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
5 |
|
mdegval.a |
⊢ 𝐴 = { 𝑚 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑚 “ ℕ ) ∈ Fin } |
6 |
|
mdegval.h |
⊢ 𝐻 = ( ℎ ∈ 𝐴 ↦ ( ℂfld Σg ℎ ) ) |
7 |
|
oveq12 |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( 𝑖 mPoly 𝑟 ) = ( 𝐼 mPoly 𝑅 ) ) |
8 |
7 2
|
eqtr4di |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( 𝑖 mPoly 𝑟 ) = 𝑃 ) |
9 |
8
|
fveq2d |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( Base ‘ ( 𝑖 mPoly 𝑟 ) ) = ( Base ‘ 𝑃 ) ) |
10 |
9 3
|
eqtr4di |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( Base ‘ ( 𝑖 mPoly 𝑟 ) ) = 𝐵 ) |
11 |
|
fveq2 |
⊢ ( 𝑟 = 𝑅 → ( 0g ‘ 𝑟 ) = ( 0g ‘ 𝑅 ) ) |
12 |
11 4
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ( 0g ‘ 𝑟 ) = 0 ) |
13 |
12
|
oveq2d |
⊢ ( 𝑟 = 𝑅 → ( 𝑓 supp ( 0g ‘ 𝑟 ) ) = ( 𝑓 supp 0 ) ) |
14 |
13
|
mpteq1d |
⊢ ( 𝑟 = 𝑅 → ( ℎ ∈ ( 𝑓 supp ( 0g ‘ 𝑟 ) ) ↦ ( ℂfld Σg ℎ ) ) = ( ℎ ∈ ( 𝑓 supp 0 ) ↦ ( ℂfld Σg ℎ ) ) ) |
15 |
14
|
rneqd |
⊢ ( 𝑟 = 𝑅 → ran ( ℎ ∈ ( 𝑓 supp ( 0g ‘ 𝑟 ) ) ↦ ( ℂfld Σg ℎ ) ) = ran ( ℎ ∈ ( 𝑓 supp 0 ) ↦ ( ℂfld Σg ℎ ) ) ) |
16 |
15
|
supeq1d |
⊢ ( 𝑟 = 𝑅 → sup ( ran ( ℎ ∈ ( 𝑓 supp ( 0g ‘ 𝑟 ) ) ↦ ( ℂfld Σg ℎ ) ) , ℝ* , < ) = sup ( ran ( ℎ ∈ ( 𝑓 supp 0 ) ↦ ( ℂfld Σg ℎ ) ) , ℝ* , < ) ) |
17 |
16
|
adantl |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → sup ( ran ( ℎ ∈ ( 𝑓 supp ( 0g ‘ 𝑟 ) ) ↦ ( ℂfld Σg ℎ ) ) , ℝ* , < ) = sup ( ran ( ℎ ∈ ( 𝑓 supp 0 ) ↦ ( ℂfld Σg ℎ ) ) , ℝ* , < ) ) |
18 |
10 17
|
mpteq12dv |
⊢ ( ( 𝑖 = 𝐼 ∧ 𝑟 = 𝑅 ) → ( 𝑓 ∈ ( Base ‘ ( 𝑖 mPoly 𝑟 ) ) ↦ sup ( ran ( ℎ ∈ ( 𝑓 supp ( 0g ‘ 𝑟 ) ) ↦ ( ℂfld Σg ℎ ) ) , ℝ* , < ) ) = ( 𝑓 ∈ 𝐵 ↦ sup ( ran ( ℎ ∈ ( 𝑓 supp 0 ) ↦ ( ℂfld Σg ℎ ) ) , ℝ* , < ) ) ) |
19 |
|
df-mdeg |
⊢ mDeg = ( 𝑖 ∈ V , 𝑟 ∈ V ↦ ( 𝑓 ∈ ( Base ‘ ( 𝑖 mPoly 𝑟 ) ) ↦ sup ( ran ( ℎ ∈ ( 𝑓 supp ( 0g ‘ 𝑟 ) ) ↦ ( ℂfld Σg ℎ ) ) , ℝ* , < ) ) ) |
20 |
3
|
fvexi |
⊢ 𝐵 ∈ V |
21 |
20
|
mptex |
⊢ ( 𝑓 ∈ 𝐵 ↦ sup ( ran ( ℎ ∈ ( 𝑓 supp 0 ) ↦ ( ℂfld Σg ℎ ) ) , ℝ* , < ) ) ∈ V |
22 |
18 19 21
|
ovmpoa |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 mDeg 𝑅 ) = ( 𝑓 ∈ 𝐵 ↦ sup ( ran ( ℎ ∈ ( 𝑓 supp 0 ) ↦ ( ℂfld Σg ℎ ) ) , ℝ* , < ) ) ) |
23 |
6
|
reseq1i |
⊢ ( 𝐻 ↾ ( 𝑓 supp 0 ) ) = ( ( ℎ ∈ 𝐴 ↦ ( ℂfld Σg ℎ ) ) ↾ ( 𝑓 supp 0 ) ) |
24 |
|
suppssdm |
⊢ ( 𝑓 supp 0 ) ⊆ dom 𝑓 |
25 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
26 |
|
simpr |
⊢ ( ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ∧ 𝑓 ∈ 𝐵 ) → 𝑓 ∈ 𝐵 ) |
27 |
2 25 3 5 26
|
mplelf |
⊢ ( ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ∧ 𝑓 ∈ 𝐵 ) → 𝑓 : 𝐴 ⟶ ( Base ‘ 𝑅 ) ) |
28 |
24 27
|
fssdm |
⊢ ( ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ∧ 𝑓 ∈ 𝐵 ) → ( 𝑓 supp 0 ) ⊆ 𝐴 ) |
29 |
28
|
resmptd |
⊢ ( ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ∧ 𝑓 ∈ 𝐵 ) → ( ( ℎ ∈ 𝐴 ↦ ( ℂfld Σg ℎ ) ) ↾ ( 𝑓 supp 0 ) ) = ( ℎ ∈ ( 𝑓 supp 0 ) ↦ ( ℂfld Σg ℎ ) ) ) |
30 |
23 29
|
eqtr2id |
⊢ ( ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ∧ 𝑓 ∈ 𝐵 ) → ( ℎ ∈ ( 𝑓 supp 0 ) ↦ ( ℂfld Σg ℎ ) ) = ( 𝐻 ↾ ( 𝑓 supp 0 ) ) ) |
31 |
30
|
rneqd |
⊢ ( ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ∧ 𝑓 ∈ 𝐵 ) → ran ( ℎ ∈ ( 𝑓 supp 0 ) ↦ ( ℂfld Σg ℎ ) ) = ran ( 𝐻 ↾ ( 𝑓 supp 0 ) ) ) |
32 |
|
df-ima |
⊢ ( 𝐻 “ ( 𝑓 supp 0 ) ) = ran ( 𝐻 ↾ ( 𝑓 supp 0 ) ) |
33 |
31 32
|
eqtr4di |
⊢ ( ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ∧ 𝑓 ∈ 𝐵 ) → ran ( ℎ ∈ ( 𝑓 supp 0 ) ↦ ( ℂfld Σg ℎ ) ) = ( 𝐻 “ ( 𝑓 supp 0 ) ) ) |
34 |
33
|
supeq1d |
⊢ ( ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ∧ 𝑓 ∈ 𝐵 ) → sup ( ran ( ℎ ∈ ( 𝑓 supp 0 ) ↦ ( ℂfld Σg ℎ ) ) , ℝ* , < ) = sup ( ( 𝐻 “ ( 𝑓 supp 0 ) ) , ℝ* , < ) ) |
35 |
34
|
mpteq2dva |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑓 ∈ 𝐵 ↦ sup ( ran ( ℎ ∈ ( 𝑓 supp 0 ) ↦ ( ℂfld Σg ℎ ) ) , ℝ* , < ) ) = ( 𝑓 ∈ 𝐵 ↦ sup ( ( 𝐻 “ ( 𝑓 supp 0 ) ) , ℝ* , < ) ) ) |
36 |
22 35
|
eqtrd |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 mDeg 𝑅 ) = ( 𝑓 ∈ 𝐵 ↦ sup ( ( 𝐻 “ ( 𝑓 supp 0 ) ) , ℝ* , < ) ) ) |
37 |
|
reldmmdeg |
⊢ Rel dom mDeg |
38 |
37
|
ovprc |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 mDeg 𝑅 ) = ∅ ) |
39 |
|
mpt0 |
⊢ ( 𝑓 ∈ ∅ ↦ sup ( ( 𝐻 “ ( 𝑓 supp 0 ) ) , ℝ* , < ) ) = ∅ |
40 |
38 39
|
eqtr4di |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 mDeg 𝑅 ) = ( 𝑓 ∈ ∅ ↦ sup ( ( 𝐻 “ ( 𝑓 supp 0 ) ) , ℝ* , < ) ) ) |
41 |
|
reldmmpl |
⊢ Rel dom mPoly |
42 |
41
|
ovprc |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 mPoly 𝑅 ) = ∅ ) |
43 |
2 42
|
syl5eq |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝑃 = ∅ ) |
44 |
43
|
fveq2d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( Base ‘ 𝑃 ) = ( Base ‘ ∅ ) ) |
45 |
|
base0 |
⊢ ∅ = ( Base ‘ ∅ ) |
46 |
44 3 45
|
3eqtr4g |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝐵 = ∅ ) |
47 |
46
|
mpteq1d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑓 ∈ 𝐵 ↦ sup ( ( 𝐻 “ ( 𝑓 supp 0 ) ) , ℝ* , < ) ) = ( 𝑓 ∈ ∅ ↦ sup ( ( 𝐻 “ ( 𝑓 supp 0 ) ) , ℝ* , < ) ) ) |
48 |
40 47
|
eqtr4d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 mDeg 𝑅 ) = ( 𝑓 ∈ 𝐵 ↦ sup ( ( 𝐻 “ ( 𝑓 supp 0 ) ) , ℝ* , < ) ) ) |
49 |
36 48
|
pm2.61i |
⊢ ( 𝐼 mDeg 𝑅 ) = ( 𝑓 ∈ 𝐵 ↦ sup ( ( 𝐻 “ ( 𝑓 supp 0 ) ) , ℝ* , < ) ) |
50 |
1 49
|
eqtri |
⊢ 𝐷 = ( 𝑓 ∈ 𝐵 ↦ sup ( ( 𝐻 “ ( 𝑓 supp 0 ) ) , ℝ* , < ) ) |