Step |
Hyp |
Ref |
Expression |
1 |
|
ringpropd.1 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) ) |
2 |
|
ringpropd.2 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) ) |
3 |
|
ringpropd.3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ) |
4 |
|
ringpropd.4 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ) |
5 |
1 2 3 4
|
ringpropd |
⊢ ( 𝜑 → ( 𝐾 ∈ Ring ↔ 𝐿 ∈ Ring ) ) |
6 |
|
eqid |
⊢ ( mulGrp ‘ 𝐾 ) = ( mulGrp ‘ 𝐾 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
8 |
6 7
|
mgpbas |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ ( mulGrp ‘ 𝐾 ) ) |
9 |
1 8
|
eqtrdi |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ ( mulGrp ‘ 𝐾 ) ) ) |
10 |
|
eqid |
⊢ ( mulGrp ‘ 𝐿 ) = ( mulGrp ‘ 𝐿 ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 ) |
12 |
10 11
|
mgpbas |
⊢ ( Base ‘ 𝐿 ) = ( Base ‘ ( mulGrp ‘ 𝐿 ) ) |
13 |
2 12
|
eqtrdi |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ ( mulGrp ‘ 𝐿 ) ) ) |
14 |
|
eqid |
⊢ ( .r ‘ 𝐾 ) = ( .r ‘ 𝐾 ) |
15 |
6 14
|
mgpplusg |
⊢ ( .r ‘ 𝐾 ) = ( +g ‘ ( mulGrp ‘ 𝐾 ) ) |
16 |
15
|
oveqi |
⊢ ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ ( mulGrp ‘ 𝐾 ) ) 𝑦 ) |
17 |
|
eqid |
⊢ ( .r ‘ 𝐿 ) = ( .r ‘ 𝐿 ) |
18 |
10 17
|
mgpplusg |
⊢ ( .r ‘ 𝐿 ) = ( +g ‘ ( mulGrp ‘ 𝐿 ) ) |
19 |
18
|
oveqi |
⊢ ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = ( 𝑥 ( +g ‘ ( mulGrp ‘ 𝐿 ) ) 𝑦 ) |
20 |
4 16 19
|
3eqtr3g |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ ( mulGrp ‘ 𝐾 ) ) 𝑦 ) = ( 𝑥 ( +g ‘ ( mulGrp ‘ 𝐿 ) ) 𝑦 ) ) |
21 |
9 13 20
|
cmnpropd |
⊢ ( 𝜑 → ( ( mulGrp ‘ 𝐾 ) ∈ CMnd ↔ ( mulGrp ‘ 𝐿 ) ∈ CMnd ) ) |
22 |
5 21
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝐾 ∈ Ring ∧ ( mulGrp ‘ 𝐾 ) ∈ CMnd ) ↔ ( 𝐿 ∈ Ring ∧ ( mulGrp ‘ 𝐿 ) ∈ CMnd ) ) ) |
23 |
6
|
iscrng |
⊢ ( 𝐾 ∈ CRing ↔ ( 𝐾 ∈ Ring ∧ ( mulGrp ‘ 𝐾 ) ∈ CMnd ) ) |
24 |
10
|
iscrng |
⊢ ( 𝐿 ∈ CRing ↔ ( 𝐿 ∈ Ring ∧ ( mulGrp ‘ 𝐿 ) ∈ CMnd ) ) |
25 |
22 23 24
|
3bitr4g |
⊢ ( 𝜑 → ( 𝐾 ∈ CRing ↔ 𝐿 ∈ CRing ) ) |