Step |
Hyp |
Ref |
Expression |
1 |
|
ismhp.h |
⊢ 𝐻 = ( 𝐼 mHomP 𝑅 ) |
2 |
|
ismhp.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
3 |
|
ismhp.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
4 |
|
ismhp.0 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
5 |
|
ismhp.d |
⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
6 |
|
ismhp.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
7 |
|
ismhp2.1 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
8 |
1 2 3 4 5 6
|
ismhp |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ) ) |
9 |
7
|
biantrurd |
⊢ ( 𝜑 → ( ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ↔ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ) ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
11 |
2 10 3 5 7
|
mplelf |
⊢ ( 𝜑 → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
12 |
11
|
ffnd |
⊢ ( 𝜑 → 𝑋 Fn 𝐷 ) |
13 |
4
|
fvexi |
⊢ 0 ∈ V |
14 |
13
|
a1i |
⊢ ( 𝜑 → 0 ∈ V ) |
15 |
|
elsuppfng |
⊢ ( ( 𝑋 Fn 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ V ) → ( 𝑑 ∈ ( 𝑋 supp 0 ) ↔ ( 𝑑 ∈ 𝐷 ∧ ( 𝑋 ‘ 𝑑 ) ≠ 0 ) ) ) |
16 |
12 7 14 15
|
syl3anc |
⊢ ( 𝜑 → ( 𝑑 ∈ ( 𝑋 supp 0 ) ↔ ( 𝑑 ∈ 𝐷 ∧ ( 𝑋 ‘ 𝑑 ) ≠ 0 ) ) ) |
17 |
|
oveq2 |
⊢ ( 𝑔 = 𝑑 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) ) |
18 |
17
|
eqeq1d |
⊢ ( 𝑔 = 𝑑 → ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 ↔ ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) |
19 |
18
|
elrab |
⊢ ( 𝑑 ∈ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ↔ ( 𝑑 ∈ 𝐷 ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) |
20 |
19
|
a1i |
⊢ ( 𝜑 → ( 𝑑 ∈ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ↔ ( 𝑑 ∈ 𝐷 ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) ) |
21 |
16 20
|
imbi12d |
⊢ ( 𝜑 → ( ( 𝑑 ∈ ( 𝑋 supp 0 ) → 𝑑 ∈ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ↔ ( ( 𝑑 ∈ 𝐷 ∧ ( 𝑋 ‘ 𝑑 ) ≠ 0 ) → ( 𝑑 ∈ 𝐷 ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) ) ) |
22 |
|
imdistan |
⊢ ( ( 𝑑 ∈ 𝐷 → ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) ↔ ( ( 𝑑 ∈ 𝐷 ∧ ( 𝑋 ‘ 𝑑 ) ≠ 0 ) → ( 𝑑 ∈ 𝐷 ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) ) |
23 |
21 22
|
bitr4di |
⊢ ( 𝜑 → ( ( 𝑑 ∈ ( 𝑋 supp 0 ) → 𝑑 ∈ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ↔ ( 𝑑 ∈ 𝐷 → ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) ) ) |
24 |
23
|
albidv |
⊢ ( 𝜑 → ( ∀ 𝑑 ( 𝑑 ∈ ( 𝑋 supp 0 ) → 𝑑 ∈ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ↔ ∀ 𝑑 ( 𝑑 ∈ 𝐷 → ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) ) ) |
25 |
|
df-ss |
⊢ ( ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ↔ ∀ 𝑑 ( 𝑑 ∈ ( 𝑋 supp 0 ) → 𝑑 ∈ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) ) |
26 |
|
df-ral |
⊢ ( ∀ 𝑑 ∈ 𝐷 ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ↔ ∀ 𝑑 ( 𝑑 ∈ 𝐷 → ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) ) |
27 |
24 25 26
|
3bitr4g |
⊢ ( 𝜑 → ( ( 𝑋 supp 0 ) ⊆ { 𝑔 ∈ 𝐷 ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ↔ ∀ 𝑑 ∈ 𝐷 ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) ) |
28 |
8 9 27
|
3bitr2d |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ↔ ∀ 𝑑 ∈ 𝐷 ( ( 𝑋 ‘ 𝑑 ) ≠ 0 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑑 ) = 𝑁 ) ) ) |