Step |
Hyp |
Ref |
Expression |
1 |
|
1pmatscmul.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
2 |
|
1pmatscmul.c |
⊢ 𝐶 = ( 𝑁 Mat 𝑃 ) |
3 |
|
1pmatscmul.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
4 |
|
1pmatscmul.e |
⊢ 𝐸 = ( Base ‘ 𝑃 ) |
5 |
|
1pmatscmul.m |
⊢ ∗ = ( ·𝑠 ‘ 𝐶 ) |
6 |
|
1pmatscmul.1 |
⊢ 1 = ( 1r ‘ 𝐶 ) |
7 |
1
|
ply1ring |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring ) |
8 |
7
|
anim2i |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ) ) |
9 |
8
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸 ) → ( 𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ) ) |
10 |
|
simp3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸 ) → 𝑄 ∈ 𝐸 ) |
11 |
1 2
|
pmatring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐶 ∈ Ring ) |
12 |
11
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸 ) → 𝐶 ∈ Ring ) |
13 |
3 6
|
ringidcl |
⊢ ( 𝐶 ∈ Ring → 1 ∈ 𝐵 ) |
14 |
12 13
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸 ) → 1 ∈ 𝐵 ) |
15 |
4 2 3 5
|
matvscl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ) ∧ ( 𝑄 ∈ 𝐸 ∧ 1 ∈ 𝐵 ) ) → ( 𝑄 ∗ 1 ) ∈ 𝐵 ) |
16 |
9 10 14 15
|
syl12anc |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸 ) → ( 𝑄 ∗ 1 ) ∈ 𝐵 ) |