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