Step |
Hyp |
Ref |
Expression |
1 |
|
ringadd2.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
ringadd2.p |
⊢ + = ( +g ‘ 𝑅 ) |
3 |
|
ringadd2.t |
⊢ · = ( .r ‘ 𝑅 ) |
4 |
|
rngo2times.u |
⊢ 1 = ( 1r ‘ 𝑅 ) |
5 |
1 3 4
|
ringlidm |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵 ) → ( 1 · 𝐴 ) = 𝐴 ) |
6 |
5
|
eqcomd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵 ) → 𝐴 = ( 1 · 𝐴 ) ) |
7 |
6 6
|
oveq12d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵 ) → ( 𝐴 + 𝐴 ) = ( ( 1 · 𝐴 ) + ( 1 · 𝐴 ) ) ) |
8 |
|
simpl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵 ) → 𝑅 ∈ Ring ) |
9 |
1 4
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → 1 ∈ 𝐵 ) |
10 |
9
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵 ) → 1 ∈ 𝐵 ) |
11 |
|
simpr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵 ) → 𝐴 ∈ 𝐵 ) |
12 |
1 2 3
|
ringdir |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 1 ∈ 𝐵 ∧ 1 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ) ) → ( ( 1 + 1 ) · 𝐴 ) = ( ( 1 · 𝐴 ) + ( 1 · 𝐴 ) ) ) |
13 |
8 10 10 11 12
|
syl13anc |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵 ) → ( ( 1 + 1 ) · 𝐴 ) = ( ( 1 · 𝐴 ) + ( 1 · 𝐴 ) ) ) |
14 |
7 13
|
eqtr4d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵 ) → ( 𝐴 + 𝐴 ) = ( ( 1 + 1 ) · 𝐴 ) ) |