Step |
Hyp |
Ref |
Expression |
1 |
|
madetsmelbas.p |
⊢ 𝑃 = ( Base ‘ ( SymGrp ‘ 𝑁 ) ) |
2 |
|
madetsmelbas.s |
⊢ 𝑆 = ( pmSgn ‘ 𝑁 ) |
3 |
|
madetsmelbas.y |
⊢ 𝑌 = ( ℤRHom ‘ 𝑅 ) |
4 |
|
madetsmelbas.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
5 |
|
madetsmelbas.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
6 |
|
madetsmelbas.g |
⊢ 𝐺 = ( mulGrp ‘ 𝑅 ) |
7 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
8 |
7
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃 ) → 𝑅 ∈ Ring ) |
9 |
4 5
|
matrcl |
⊢ ( 𝑀 ∈ 𝐵 → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) ) |
10 |
9
|
simpld |
⊢ ( 𝑀 ∈ 𝐵 → 𝑁 ∈ Fin ) |
11 |
10
|
3ad2ant2 |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃 ) → 𝑁 ∈ Fin ) |
12 |
|
simp3 |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃 ) → 𝑄 ∈ 𝑃 ) |
13 |
1 2 3
|
zrhcopsgnelbas |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃 ) → ( ( 𝑌 ∘ 𝑆 ) ‘ 𝑄 ) ∈ ( Base ‘ 𝑅 ) ) |
14 |
8 11 12 13
|
syl3anc |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃 ) → ( ( 𝑌 ∘ 𝑆 ) ‘ 𝑄 ) ∈ ( Base ‘ 𝑅 ) ) |
15 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
16 |
6 15
|
mgpbas |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝐺 ) |
17 |
6
|
crngmgp |
⊢ ( 𝑅 ∈ CRing → 𝐺 ∈ CMnd ) |
18 |
17
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃 ) → 𝐺 ∈ CMnd ) |
19 |
|
simp2 |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃 ) → 𝑀 ∈ 𝐵 ) |
20 |
4 5 1
|
matepm2cl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵 ) → ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 ( 𝑄 ‘ 𝑛 ) ) ∈ ( Base ‘ 𝑅 ) ) |
21 |
8 12 19 20
|
syl3anc |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃 ) → ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 ( 𝑄 ‘ 𝑛 ) ) ∈ ( Base ‘ 𝑅 ) ) |
22 |
16 18 11 21
|
gsummptcl |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃 ) → ( 𝐺 Σg ( 𝑛 ∈ 𝑁 ↦ ( 𝑛 𝑀 ( 𝑄 ‘ 𝑛 ) ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
23 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
24 |
15 23
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( 𝑌 ∘ 𝑆 ) ‘ 𝑄 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝐺 Σg ( 𝑛 ∈ 𝑁 ↦ ( 𝑛 𝑀 ( 𝑄 ‘ 𝑛 ) ) ) ) ∈ ( Base ‘ 𝑅 ) ) → ( ( ( 𝑌 ∘ 𝑆 ) ‘ 𝑄 ) ( .r ‘ 𝑅 ) ( 𝐺 Σg ( 𝑛 ∈ 𝑁 ↦ ( 𝑛 𝑀 ( 𝑄 ‘ 𝑛 ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
25 |
8 14 22 24
|
syl3anc |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃 ) → ( ( ( 𝑌 ∘ 𝑆 ) ‘ 𝑄 ) ( .r ‘ 𝑅 ) ( 𝐺 Σg ( 𝑛 ∈ 𝑁 ↦ ( 𝑛 𝑀 ( 𝑄 ‘ 𝑛 ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) ) |