Step |
Hyp |
Ref |
Expression |
1 |
|
sraassa.a |
⊢ 𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) |
2 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
3 |
2
|
subrgss |
⊢ ( 𝑆 ∈ ( SubRing ‘ 𝑊 ) → 𝑆 ⊆ ( Base ‘ 𝑊 ) ) |
4 |
3
|
adantl |
⊢ ( ( 𝑊 ∈ CRing ∧ 𝑆 ∈ ( SubRing ‘ 𝑊 ) ) → 𝑆 ⊆ ( Base ‘ 𝑊 ) ) |
5 |
|
eqid |
⊢ ( Cntr ‘ ( mulGrp ‘ 𝑊 ) ) = ( Cntr ‘ ( mulGrp ‘ 𝑊 ) ) |
6 |
2 5
|
crngbascntr |
⊢ ( 𝑊 ∈ CRing → ( Base ‘ 𝑊 ) = ( Cntr ‘ ( mulGrp ‘ 𝑊 ) ) ) |
7 |
6
|
adantr |
⊢ ( ( 𝑊 ∈ CRing ∧ 𝑆 ∈ ( SubRing ‘ 𝑊 ) ) → ( Base ‘ 𝑊 ) = ( Cntr ‘ ( mulGrp ‘ 𝑊 ) ) ) |
8 |
4 7
|
sseqtrd |
⊢ ( ( 𝑊 ∈ CRing ∧ 𝑆 ∈ ( SubRing ‘ 𝑊 ) ) → 𝑆 ⊆ ( Cntr ‘ ( mulGrp ‘ 𝑊 ) ) ) |
9 |
|
crngring |
⊢ ( 𝑊 ∈ CRing → 𝑊 ∈ Ring ) |
10 |
9
|
adantr |
⊢ ( ( 𝑊 ∈ CRing ∧ 𝑆 ∈ ( SubRing ‘ 𝑊 ) ) → 𝑊 ∈ Ring ) |
11 |
|
simpr |
⊢ ( ( 𝑊 ∈ CRing ∧ 𝑆 ∈ ( SubRing ‘ 𝑊 ) ) → 𝑆 ∈ ( SubRing ‘ 𝑊 ) ) |
12 |
1 5 10 11
|
sraassab |
⊢ ( ( 𝑊 ∈ CRing ∧ 𝑆 ∈ ( SubRing ‘ 𝑊 ) ) → ( 𝐴 ∈ AssAlg ↔ 𝑆 ⊆ ( Cntr ‘ ( mulGrp ‘ 𝑊 ) ) ) ) |
13 |
8 12
|
mpbird |
⊢ ( ( 𝑊 ∈ CRing ∧ 𝑆 ∈ ( SubRing ‘ 𝑊 ) ) → 𝐴 ∈ AssAlg ) |