| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mhpaddcl.h |
⊢ 𝐻 = ( 𝐼 mHomP 𝑅 ) |
| 2 |
|
mhpaddcl.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
| 3 |
|
mhpaddcl.a |
⊢ + = ( +g ‘ 𝑃 ) |
| 4 |
|
mhpaddcl.r |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
| 5 |
|
mhpaddcl.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ) |
| 6 |
|
mhpaddcl.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 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 |
1 5
|
mhprcl |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
| 11 |
|
reldmmhp |
⊢ Rel dom mHomP |
| 12 |
11 1 5
|
elfvov1 |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
| 13 |
2
|
mplgrp |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ Grp ) → 𝑃 ∈ Grp ) |
| 14 |
12 4 13
|
syl2anc |
⊢ ( 𝜑 → 𝑃 ∈ Grp ) |
| 15 |
1 2 7 5
|
mhpmpl |
⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝑃 ) ) |
| 16 |
1 2 7 6
|
mhpmpl |
⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝑃 ) ) |
| 17 |
7 3 14 15 16
|
grpcld |
⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( Base ‘ 𝑃 ) ) |
| 18 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
| 19 |
2 7 18 3 15 16
|
mpladd |
⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) = ( 𝑋 ∘f ( +g ‘ 𝑅 ) 𝑌 ) ) |
| 20 |
19
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) supp ( 0g ‘ 𝑅 ) ) = ( ( 𝑋 ∘f ( +g ‘ 𝑅 ) 𝑌 ) supp ( 0g ‘ 𝑅 ) ) ) |
| 21 |
|
ovexd |
⊢ ( 𝜑 → ( ℕ0 ↑m 𝐼 ) ∈ V ) |
| 22 |
9 21
|
rabexd |
⊢ ( 𝜑 → { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∈ V ) |
| 23 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
| 24 |
23 8
|
grpidcl |
⊢ ( 𝑅 ∈ Grp → ( 0g ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
| 25 |
4 24
|
syl |
⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
| 26 |
2 23 7 9 15
|
mplelf |
⊢ ( 𝜑 → 𝑋 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
| 27 |
2 23 7 9 16
|
mplelf |
⊢ ( 𝜑 → 𝑌 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
| 28 |
23 18 8 4 25
|
grplidd |
⊢ ( 𝜑 → ( ( 0g ‘ 𝑅 ) ( +g ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
| 29 |
22 25 26 27 28
|
suppofssd |
⊢ ( 𝜑 → ( ( 𝑋 ∘f ( +g ‘ 𝑅 ) 𝑌 ) supp ( 0g ‘ 𝑅 ) ) ⊆ ( ( 𝑋 supp ( 0g ‘ 𝑅 ) ) ∪ ( 𝑌 supp ( 0g ‘ 𝑅 ) ) ) ) |
| 30 |
20 29
|
eqsstrd |
⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) supp ( 0g ‘ 𝑅 ) ) ⊆ ( ( 𝑋 supp ( 0g ‘ 𝑅 ) ) ∪ ( 𝑌 supp ( 0g ‘ 𝑅 ) ) ) ) |
| 31 |
1 8 9 5
|
mhpdeg |
⊢ ( 𝜑 → ( 𝑋 supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) |
| 32 |
1 8 9 6
|
mhpdeg |
⊢ ( 𝜑 → ( 𝑌 supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) |
| 33 |
31 32
|
unssd |
⊢ ( 𝜑 → ( ( 𝑋 supp ( 0g ‘ 𝑅 ) ) ∪ ( 𝑌 supp ( 0g ‘ 𝑅 ) ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) |
| 34 |
30 33
|
sstrd |
⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑔 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑔 ) = 𝑁 } ) |
| 35 |
1 2 7 8 9 10 17 34
|
ismhp2 |
⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( 𝐻 ‘ 𝑁 ) ) |