Step |
Hyp |
Ref |
Expression |
1 |
|
opprbas.1 |
⊢ 𝑂 = ( oppr ‘ 𝑅 ) |
2 |
|
ringrng |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Rng ) |
3 |
1
|
opprrng |
⊢ ( 𝑅 ∈ Rng → 𝑂 ∈ Rng ) |
4 |
2 3
|
syl |
⊢ ( 𝑅 ∈ Ring → 𝑂 ∈ Rng ) |
5 |
|
oveq1 |
⊢ ( 𝑧 = ( 1r ‘ 𝑅 ) → ( 𝑧 ( .r ‘ 𝑂 ) 𝑥 ) = ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑂 ) 𝑥 ) ) |
6 |
5
|
eqeq1d |
⊢ ( 𝑧 = ( 1r ‘ 𝑅 ) → ( ( 𝑧 ( .r ‘ 𝑂 ) 𝑥 ) = 𝑥 ↔ ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑂 ) 𝑥 ) = 𝑥 ) ) |
7 |
6
|
ovanraleqv |
⊢ ( 𝑧 = ( 1r ‘ 𝑅 ) → ( ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ( ( 𝑧 ( .r ‘ 𝑂 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( .r ‘ 𝑂 ) 𝑧 ) = 𝑥 ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ( ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑂 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( .r ‘ 𝑂 ) ( 1r ‘ 𝑅 ) ) = 𝑥 ) ) ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
9 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
10 |
8 9
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
11 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
12 |
|
eqid |
⊢ ( .r ‘ 𝑂 ) = ( .r ‘ 𝑂 ) |
13 |
8 11 1 12
|
opprmul |
⊢ ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑂 ) 𝑥 ) = ( 𝑥 ( .r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) |
14 |
8 11 9
|
ringridm |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑥 ( .r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = 𝑥 ) |
15 |
13 14
|
eqtrid |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑂 ) 𝑥 ) = 𝑥 ) |
16 |
8 11 1 12
|
opprmul |
⊢ ( 𝑥 ( .r ‘ 𝑂 ) ( 1r ‘ 𝑅 ) ) = ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) 𝑥 ) |
17 |
8 11 9
|
ringlidm |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) 𝑥 ) = 𝑥 ) |
18 |
16 17
|
eqtrid |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑥 ( .r ‘ 𝑂 ) ( 1r ‘ 𝑅 ) ) = 𝑥 ) |
19 |
15 18
|
jca |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑂 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( .r ‘ 𝑂 ) ( 1r ‘ 𝑅 ) ) = 𝑥 ) ) |
20 |
19
|
ralrimiva |
⊢ ( 𝑅 ∈ Ring → ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ( ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑂 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( .r ‘ 𝑂 ) ( 1r ‘ 𝑅 ) ) = 𝑥 ) ) |
21 |
7 10 20
|
rspcedvdw |
⊢ ( 𝑅 ∈ Ring → ∃ 𝑧 ∈ ( Base ‘ 𝑅 ) ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ( ( 𝑧 ( .r ‘ 𝑂 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( .r ‘ 𝑂 ) 𝑧 ) = 𝑥 ) ) |
22 |
1 8
|
opprbas |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑂 ) |
23 |
22 12
|
isringrng |
⊢ ( 𝑂 ∈ Ring ↔ ( 𝑂 ∈ Rng ∧ ∃ 𝑧 ∈ ( Base ‘ 𝑅 ) ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ( ( 𝑧 ( .r ‘ 𝑂 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( .r ‘ 𝑂 ) 𝑧 ) = 𝑥 ) ) ) |
24 |
4 21 23
|
sylanbrc |
⊢ ( 𝑅 ∈ Ring → 𝑂 ∈ Ring ) |