Step |
Hyp |
Ref |
Expression |
1 |
|
mhpinvcl.h |
⊢ 𝐻 = ( 𝐼 mHomP 𝑅 ) |
2 |
|
mhpinvcl.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
3 |
|
mhpinvcl.m |
⊢ 𝑀 = ( invg ‘ 𝑃 ) |
4 |
|
mhpinvcl.r |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
5 |
|
mhpinvcl.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
6 |
|
mhpinvcl.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
8 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
9 |
|
eqid |
⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
10 |
|
reldmmhp |
⊢ Rel dom mHomP |
11 |
10 1 6
|
elfvov1 |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
12 |
2
|
mplgrp |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ Grp ) → 𝑃 ∈ Grp ) |
13 |
11 4 12
|
syl2anc |
⊢ ( 𝜑 → 𝑃 ∈ Grp ) |
14 |
1 2 7 11 4 5 6
|
mhpmpl |
⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝑃 ) ) |
15 |
7 3
|
grpinvcl |
⊢ ( ( 𝑃 ∈ Grp ∧ 𝑋 ∈ ( Base ‘ 𝑃 ) ) → ( 𝑀 ‘ 𝑋 ) ∈ ( Base ‘ 𝑃 ) ) |
16 |
13 14 15
|
syl2anc |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ ( Base ‘ 𝑃 ) ) |
17 |
|
eqid |
⊢ ( invg ‘ 𝑅 ) = ( invg ‘ 𝑅 ) |
18 |
2 7 17 3 11 4 14
|
mplneg |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) = ( ( invg ‘ 𝑅 ) ∘ 𝑋 ) ) |
19 |
18
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) supp ( 0g ‘ 𝑅 ) ) = ( ( ( invg ‘ 𝑅 ) ∘ 𝑋 ) supp ( 0g ‘ 𝑅 ) ) ) |
20 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
21 |
20 17
|
grpinvfn |
⊢ ( invg ‘ 𝑅 ) Fn ( Base ‘ 𝑅 ) |
22 |
21
|
a1i |
⊢ ( 𝜑 → ( invg ‘ 𝑅 ) Fn ( Base ‘ 𝑅 ) ) |
23 |
2 20 7 9 14
|
mplelf |
⊢ ( 𝜑 → 𝑋 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
24 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
25 |
24
|
rabex |
⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∈ V |
26 |
25
|
a1i |
⊢ ( 𝜑 → { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∈ V ) |
27 |
|
fvexd |
⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) ∈ V ) |
28 |
8 17
|
grpinvid |
⊢ ( 𝑅 ∈ Grp → ( ( invg ‘ 𝑅 ) ‘ ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
29 |
4 28
|
syl |
⊢ ( 𝜑 → ( ( invg ‘ 𝑅 ) ‘ ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
30 |
22 23 26 27 29
|
suppcoss |
⊢ ( 𝜑 → ( ( ( invg ‘ 𝑅 ) ∘ 𝑋 ) supp ( 0g ‘ 𝑅 ) ) ⊆ ( 𝑋 supp ( 0g ‘ 𝑅 ) ) ) |
31 |
19 30
|
eqsstrd |
⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) supp ( 0g ‘ 𝑅 ) ) ⊆ ( 𝑋 supp ( 0g ‘ 𝑅 ) ) ) |
32 |
1 8 9 11 4 5 6
|
mhpdeg |
⊢ ( 𝜑 → ( 𝑋 supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) |
33 |
31 32
|
sstrd |
⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) |
34 |
1 2 7 8 9 11 4 5 16 33
|
ismhp2 |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ ( 𝐻 ‘ 𝑁 ) ) |