Step |
Hyp |
Ref |
Expression |
1 |
|
m2cpminv0.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
m2cpminv0.i |
⊢ 𝐼 = ( 𝑁 cPolyMatToMat 𝑅 ) |
3 |
|
m2cpminv0.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
4 |
|
m2cpminv0.c |
⊢ 𝐶 = ( 𝑁 Mat 𝑃 ) |
5 |
|
m2cpminv0.0 |
⊢ 0 = ( 0g ‘ 𝐴 ) |
6 |
|
m2cpminv0.z |
⊢ 𝑍 = ( 0g ‘ 𝐶 ) |
7 |
|
eqid |
⊢ ( 𝑁 matToPolyMat 𝑅 ) = ( 𝑁 matToPolyMat 𝑅 ) |
8 |
1
|
fveq2i |
⊢ ( 0g ‘ 𝐴 ) = ( 0g ‘ ( 𝑁 Mat 𝑅 ) ) |
9 |
5 8
|
eqtri |
⊢ 0 = ( 0g ‘ ( 𝑁 Mat 𝑅 ) ) |
10 |
4
|
fveq2i |
⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ ( 𝑁 Mat 𝑃 ) ) |
11 |
6 10
|
eqtri |
⊢ 𝑍 = ( 0g ‘ ( 𝑁 Mat 𝑃 ) ) |
12 |
7 3 9 11
|
0mat2pmat |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ) → ( ( 𝑁 matToPolyMat 𝑅 ) ‘ 0 ) = 𝑍 ) |
13 |
12
|
ancoms |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( ( 𝑁 matToPolyMat 𝑅 ) ‘ 0 ) = 𝑍 ) |
14 |
13
|
eqcomd |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑍 = ( ( 𝑁 matToPolyMat 𝑅 ) ‘ 0 ) ) |
15 |
14
|
fveq2d |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝐼 ‘ 𝑍 ) = ( 𝐼 ‘ ( ( 𝑁 matToPolyMat 𝑅 ) ‘ 0 ) ) ) |
16 |
1
|
matring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐴 ∈ Ring ) |
17 |
|
eqid |
⊢ ( Base ‘ 𝐴 ) = ( Base ‘ 𝐴 ) |
18 |
17 5
|
ring0cl |
⊢ ( 𝐴 ∈ Ring → 0 ∈ ( Base ‘ 𝐴 ) ) |
19 |
16 18
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 0 ∈ ( Base ‘ 𝐴 ) ) |
20 |
2 1 17 7
|
m2cpminvid |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 0 ∈ ( Base ‘ 𝐴 ) ) → ( 𝐼 ‘ ( ( 𝑁 matToPolyMat 𝑅 ) ‘ 0 ) ) = 0 ) |
21 |
19 20
|
mpd3an3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝐼 ‘ ( ( 𝑁 matToPolyMat 𝑅 ) ‘ 0 ) ) = 0 ) |
22 |
15 21
|
eqtrd |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝐼 ‘ 𝑍 ) = 0 ) |