Step |
Hyp |
Ref |
Expression |
1 |
|
dmatid.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
dmatid.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
3 |
|
dmatid.0 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
|
dmatid.d |
⊢ 𝐷 = ( 𝑁 DMat 𝑅 ) |
5 |
1 2 3 4
|
dmatmat |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑧 ∈ 𝐷 → 𝑧 ∈ 𝐵 ) ) |
6 |
5
|
ancoms |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ) → ( 𝑧 ∈ 𝐷 → 𝑧 ∈ 𝐵 ) ) |
7 |
6
|
ssrdv |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ) → 𝐷 ⊆ 𝐵 ) |
8 |
1 2 3 4
|
dmatid |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 1r ‘ 𝐴 ) ∈ 𝐷 ) |
9 |
8
|
ancoms |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ) → ( 1r ‘ 𝐴 ) ∈ 𝐷 ) |
10 |
9
|
ne0d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ) → 𝐷 ≠ ∅ ) |
11 |
1 2 3 4
|
dmatsubcl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ) ) → ( 𝑥 ( -g ‘ 𝐴 ) 𝑦 ) ∈ 𝐷 ) |
12 |
11
|
ancom1s |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ) ∧ ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ) ) → ( 𝑥 ( -g ‘ 𝐴 ) 𝑦 ) ∈ 𝐷 ) |
13 |
12
|
ralrimivva |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ) → ∀ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐷 ( 𝑥 ( -g ‘ 𝐴 ) 𝑦 ) ∈ 𝐷 ) |
14 |
1
|
matring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐴 ∈ Ring ) |
15 |
14
|
ancoms |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ) → 𝐴 ∈ Ring ) |
16 |
|
ringgrp |
⊢ ( 𝐴 ∈ Ring → 𝐴 ∈ Grp ) |
17 |
|
eqid |
⊢ ( -g ‘ 𝐴 ) = ( -g ‘ 𝐴 ) |
18 |
2 17
|
issubg4 |
⊢ ( 𝐴 ∈ Grp → ( 𝐷 ∈ ( SubGrp ‘ 𝐴 ) ↔ ( 𝐷 ⊆ 𝐵 ∧ 𝐷 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐷 ( 𝑥 ( -g ‘ 𝐴 ) 𝑦 ) ∈ 𝐷 ) ) ) |
19 |
15 16 18
|
3syl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ) → ( 𝐷 ∈ ( SubGrp ‘ 𝐴 ) ↔ ( 𝐷 ⊆ 𝐵 ∧ 𝐷 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐷 ( 𝑥 ( -g ‘ 𝐴 ) 𝑦 ) ∈ 𝐷 ) ) ) |
20 |
7 10 13 19
|
mpbir3and |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ) → 𝐷 ∈ ( SubGrp ‘ 𝐴 ) ) |