Step |
Hyp |
Ref |
Expression |
1 |
|
crngocom.1 |
⊢ 𝐺 = ( 1st ‘ 𝑅 ) |
2 |
|
crngocom.2 |
⊢ 𝐻 = ( 2nd ‘ 𝑅 ) |
3 |
|
crngocom.3 |
⊢ 𝑋 = ran 𝐺 |
4 |
1 2 3
|
iscrngo2 |
⊢ ( 𝑅 ∈ CRingOps ↔ ( 𝑅 ∈ RingOps ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝐻 𝑦 ) = ( 𝑦 𝐻 𝑥 ) ) ) |
5 |
4
|
simprbi |
⊢ ( 𝑅 ∈ CRingOps → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝐻 𝑦 ) = ( 𝑦 𝐻 𝑥 ) ) |
6 |
|
oveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐻 𝑦 ) = ( 𝐴 𝐻 𝑦 ) ) |
7 |
|
oveq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝑦 𝐻 𝑥 ) = ( 𝑦 𝐻 𝐴 ) ) |
8 |
6 7
|
eqeq12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐻 𝑦 ) = ( 𝑦 𝐻 𝑥 ) ↔ ( 𝐴 𝐻 𝑦 ) = ( 𝑦 𝐻 𝐴 ) ) ) |
9 |
|
oveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐻 𝑦 ) = ( 𝐴 𝐻 𝐵 ) ) |
10 |
|
oveq1 |
⊢ ( 𝑦 = 𝐵 → ( 𝑦 𝐻 𝐴 ) = ( 𝐵 𝐻 𝐴 ) ) |
11 |
9 10
|
eqeq12d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐻 𝑦 ) = ( 𝑦 𝐻 𝐴 ) ↔ ( 𝐴 𝐻 𝐵 ) = ( 𝐵 𝐻 𝐴 ) ) ) |
12 |
8 11
|
rspc2v |
⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝐻 𝑦 ) = ( 𝑦 𝐻 𝑥 ) → ( 𝐴 𝐻 𝐵 ) = ( 𝐵 𝐻 𝐴 ) ) ) |
13 |
5 12
|
mpan9 |
⊢ ( ( 𝑅 ∈ CRingOps ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝐻 𝐵 ) = ( 𝐵 𝐻 𝐴 ) ) |
14 |
13
|
3impb |
⊢ ( ( 𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐻 𝐵 ) = ( 𝐵 𝐻 𝐴 ) ) |