Step |
Hyp |
Ref |
Expression |
1 |
|
mhmcompl.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
2 |
|
mhmcompl.q |
⊢ 𝑄 = ( 𝐼 mPoly 𝑆 ) |
3 |
|
mhmcompl.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
4 |
|
mhmcompl.c |
⊢ 𝐶 = ( Base ‘ 𝑄 ) |
5 |
|
mhmcompl.h |
⊢ ( 𝜑 → 𝐻 ∈ ( 𝑅 MndHom 𝑆 ) ) |
6 |
|
mhmcompl.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
7 |
|
fvexd |
⊢ ( 𝜑 → ( Base ‘ 𝑆 ) ∈ V ) |
8 |
|
eqid |
⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
9 |
|
ovexd |
⊢ ( 𝜑 → ( ℕ0 ↑m 𝐼 ) ∈ V ) |
10 |
8 9
|
rabexd |
⊢ ( 𝜑 → { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∈ V ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
13 |
11 12
|
mhmf |
⊢ ( 𝐻 ∈ ( 𝑅 MndHom 𝑆 ) → 𝐻 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ) |
14 |
5 13
|
syl |
⊢ ( 𝜑 → 𝐻 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ) |
15 |
1 11 3 8 6
|
mplelf |
⊢ ( 𝜑 → 𝐹 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
16 |
14 15
|
fcod |
⊢ ( 𝜑 → ( 𝐻 ∘ 𝐹 ) : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑆 ) ) |
17 |
7 10 16
|
elmapdd |
⊢ ( 𝜑 → ( 𝐻 ∘ 𝐹 ) ∈ ( ( Base ‘ 𝑆 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ) |
18 |
|
eqid |
⊢ ( 𝐼 mPwSer 𝑆 ) = ( 𝐼 mPwSer 𝑆 ) |
19 |
|
eqid |
⊢ ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) |
20 |
1 3
|
mplrcl |
⊢ ( 𝐹 ∈ 𝐵 → 𝐼 ∈ V ) |
21 |
6 20
|
syl |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
22 |
18 12 8 19 21
|
psrbas |
⊢ ( 𝜑 → ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) = ( ( Base ‘ 𝑆 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ) |
23 |
17 22
|
eleqtrrd |
⊢ ( 𝜑 → ( 𝐻 ∘ 𝐹 ) ∈ ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) ) |
24 |
|
fvexd |
⊢ ( 𝜑 → ( 0g ‘ 𝑆 ) ∈ V ) |
25 |
|
mhmrcl1 |
⊢ ( 𝐻 ∈ ( 𝑅 MndHom 𝑆 ) → 𝑅 ∈ Mnd ) |
26 |
5 25
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Mnd ) |
27 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
28 |
11 27
|
mndidcl |
⊢ ( 𝑅 ∈ Mnd → ( 0g ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
29 |
26 28
|
syl |
⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
30 |
|
ssidd |
⊢ ( 𝜑 → ( Base ‘ 𝑅 ) ⊆ ( Base ‘ 𝑅 ) ) |
31 |
|
fvexd |
⊢ ( 𝜑 → ( Base ‘ 𝑅 ) ∈ V ) |
32 |
1 3 27 6 26
|
mplelsfi |
⊢ ( 𝜑 → 𝐹 finSupp ( 0g ‘ 𝑅 ) ) |
33 |
|
eqid |
⊢ ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑆 ) |
34 |
27 33
|
mhm0 |
⊢ ( 𝐻 ∈ ( 𝑅 MndHom 𝑆 ) → ( 𝐻 ‘ ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑆 ) ) |
35 |
5 34
|
syl |
⊢ ( 𝜑 → ( 𝐻 ‘ ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑆 ) ) |
36 |
24 29 15 14 30 10 31 32 35
|
fsuppcor |
⊢ ( 𝜑 → ( 𝐻 ∘ 𝐹 ) finSupp ( 0g ‘ 𝑆 ) ) |
37 |
2 18 19 33 4
|
mplelbas |
⊢ ( ( 𝐻 ∘ 𝐹 ) ∈ 𝐶 ↔ ( ( 𝐻 ∘ 𝐹 ) ∈ ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) ∧ ( 𝐻 ∘ 𝐹 ) finSupp ( 0g ‘ 𝑆 ) ) ) |
38 |
23 36 37
|
sylanbrc |
⊢ ( 𝜑 → ( 𝐻 ∘ 𝐹 ) ∈ 𝐶 ) |