Step |
Hyp |
Ref |
Expression |
1 |
|
mat2pmatscmxcl.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
mat2pmatscmxcl.k |
⊢ 𝐾 = ( Base ‘ 𝐴 ) |
3 |
|
mat2pmatscmxcl.t |
⊢ 𝑇 = ( 𝑁 matToPolyMat 𝑅 ) |
4 |
|
mat2pmatscmxcl.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
5 |
|
mat2pmatscmxcl.c |
⊢ 𝐶 = ( 𝑁 Mat 𝑃 ) |
6 |
|
mat2pmatscmxcl.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
7 |
|
mat2pmatscmxcl.m |
⊢ ∗ = ( ·𝑠 ‘ 𝐶 ) |
8 |
|
mat2pmatscmxcl.e |
⊢ ↑ = ( .g ‘ ( mulGrp ‘ 𝑃 ) ) |
9 |
|
mat2pmatscmxcl.x |
⊢ 𝑋 = ( var1 ‘ 𝑅 ) |
10 |
|
simpll |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ) ) → 𝑁 ∈ Fin ) |
11 |
4
|
ply1ring |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring ) |
12 |
11
|
ad2antlr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ) ) → 𝑃 ∈ Ring ) |
13 |
|
eqid |
⊢ ( mulGrp ‘ 𝑃 ) = ( mulGrp ‘ 𝑃 ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
15 |
4 9 13 8 14
|
ply1moncl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐿 ∈ ℕ0 ) → ( 𝐿 ↑ 𝑋 ) ∈ ( Base ‘ 𝑃 ) ) |
16 |
15
|
ad2ant2l |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ) ) → ( 𝐿 ↑ 𝑋 ) ∈ ( Base ‘ 𝑃 ) ) |
17 |
|
simpl |
⊢ ( ( 𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ) → 𝑀 ∈ 𝐾 ) |
18 |
17
|
anim2i |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ) ) → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑀 ∈ 𝐾 ) ) |
19 |
|
df-3an |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾 ) ↔ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑀 ∈ 𝐾 ) ) |
20 |
18 19
|
sylibr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ) ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾 ) ) |
21 |
3 1 2 4 5 6
|
mat2pmatbas0 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾 ) → ( 𝑇 ‘ 𝑀 ) ∈ 𝐵 ) |
22 |
20 21
|
syl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ) ) → ( 𝑇 ‘ 𝑀 ) ∈ 𝐵 ) |
23 |
14 5 6 7
|
matvscl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ) ∧ ( ( 𝐿 ↑ 𝑋 ) ∈ ( Base ‘ 𝑃 ) ∧ ( 𝑇 ‘ 𝑀 ) ∈ 𝐵 ) ) → ( ( 𝐿 ↑ 𝑋 ) ∗ ( 𝑇 ‘ 𝑀 ) ) ∈ 𝐵 ) |
24 |
10 12 16 22 23
|
syl22anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ) ) → ( ( 𝐿 ↑ 𝑋 ) ∗ ( 𝑇 ‘ 𝑀 ) ) ∈ 𝐵 ) |