Step |
Hyp |
Ref |
Expression |
1 |
|
chmaidscmat.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
chmaidscmat.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
3 |
|
chmaidscmat.c |
⊢ 𝐶 = ( 𝑁 CharPlyMat 𝑅 ) |
4 |
|
chmaidscmat.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
5 |
|
chmaidscmat.e |
⊢ 𝐸 = ( Base ‘ 𝑃 ) |
6 |
|
chmaidscmat.y |
⊢ 𝑌 = ( 𝑁 Mat 𝑃 ) |
7 |
|
chmaidscmat.k |
⊢ 𝐾 = ( Base ‘ 𝑌 ) |
8 |
|
chmaidscmat.m |
⊢ · = ( ·𝑠 ‘ 𝑌 ) |
9 |
|
chmaidscmat.1 |
⊢ 1 = ( 1r ‘ 𝑌 ) |
10 |
|
chmaidscmat.d |
⊢ 𝑆 = ( 𝑁 ScMat 𝑃 ) |
11 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
12 |
4
|
ply1ring |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring ) |
13 |
11 12
|
syl |
⊢ ( 𝑅 ∈ CRing → 𝑃 ∈ Ring ) |
14 |
13
|
anim2i |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( 𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ) ) |
15 |
14
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ) ) |
16 |
3 1 2 4 5
|
chpmatply1 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝐶 ‘ 𝑀 ) ∈ 𝐸 ) |
17 |
4 6
|
pmatring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑌 ∈ Ring ) |
18 |
11 17
|
sylan2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑌 ∈ Ring ) |
19 |
7 9
|
ringidcl |
⊢ ( 𝑌 ∈ Ring → 1 ∈ 𝐾 ) |
20 |
18 19
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 1 ∈ 𝐾 ) |
21 |
20
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 1 ∈ 𝐾 ) |
22 |
5 6 7 8
|
matvscl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ) ∧ ( ( 𝐶 ‘ 𝑀 ) ∈ 𝐸 ∧ 1 ∈ 𝐾 ) ) → ( ( 𝐶 ‘ 𝑀 ) · 1 ) ∈ 𝐾 ) |
23 |
15 16 21 22
|
syl12anc |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( ( 𝐶 ‘ 𝑀 ) · 1 ) ∈ 𝐾 ) |
24 |
|
risset |
⊢ ( ( 𝐶 ‘ 𝑀 ) ∈ 𝐸 ↔ ∃ 𝑐 ∈ 𝐸 𝑐 = ( 𝐶 ‘ 𝑀 ) ) |
25 |
|
oveq1 |
⊢ ( ( 𝐶 ‘ 𝑀 ) = 𝑐 → ( ( 𝐶 ‘ 𝑀 ) · 1 ) = ( 𝑐 · 1 ) ) |
26 |
25
|
eqcoms |
⊢ ( 𝑐 = ( 𝐶 ‘ 𝑀 ) → ( ( 𝐶 ‘ 𝑀 ) · 1 ) = ( 𝑐 · 1 ) ) |
27 |
26
|
a1i |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑐 ∈ 𝐸 ) → ( 𝑐 = ( 𝐶 ‘ 𝑀 ) → ( ( 𝐶 ‘ 𝑀 ) · 1 ) = ( 𝑐 · 1 ) ) ) |
28 |
27
|
reximdva |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( ∃ 𝑐 ∈ 𝐸 𝑐 = ( 𝐶 ‘ 𝑀 ) → ∃ 𝑐 ∈ 𝐸 ( ( 𝐶 ‘ 𝑀 ) · 1 ) = ( 𝑐 · 1 ) ) ) |
29 |
28
|
com12 |
⊢ ( ∃ 𝑐 ∈ 𝐸 𝑐 = ( 𝐶 ‘ 𝑀 ) → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ∃ 𝑐 ∈ 𝐸 ( ( 𝐶 ‘ 𝑀 ) · 1 ) = ( 𝑐 · 1 ) ) ) |
30 |
24 29
|
sylbi |
⊢ ( ( 𝐶 ‘ 𝑀 ) ∈ 𝐸 → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ∃ 𝑐 ∈ 𝐸 ( ( 𝐶 ‘ 𝑀 ) · 1 ) = ( 𝑐 · 1 ) ) ) |
31 |
16 30
|
mpcom |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ∃ 𝑐 ∈ 𝐸 ( ( 𝐶 ‘ 𝑀 ) · 1 ) = ( 𝑐 · 1 ) ) |
32 |
5 6 7 9 8 10
|
scmatel |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ) → ( ( ( 𝐶 ‘ 𝑀 ) · 1 ) ∈ 𝑆 ↔ ( ( ( 𝐶 ‘ 𝑀 ) · 1 ) ∈ 𝐾 ∧ ∃ 𝑐 ∈ 𝐸 ( ( 𝐶 ‘ 𝑀 ) · 1 ) = ( 𝑐 · 1 ) ) ) ) |
33 |
15 32
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( ( ( 𝐶 ‘ 𝑀 ) · 1 ) ∈ 𝑆 ↔ ( ( ( 𝐶 ‘ 𝑀 ) · 1 ) ∈ 𝐾 ∧ ∃ 𝑐 ∈ 𝐸 ( ( 𝐶 ‘ 𝑀 ) · 1 ) = ( 𝑐 · 1 ) ) ) ) |
34 |
23 31 33
|
mpbir2and |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( ( 𝐶 ‘ 𝑀 ) · 1 ) ∈ 𝑆 ) |