| Step |
Hyp |
Ref |
Expression |
| 1 |
|
prstcnid.c |
⊢ ( 𝜑 → 𝐶 = ( ProsetToCat ‘ 𝐾 ) ) |
| 2 |
|
prstcnid.k |
⊢ ( 𝜑 → 𝐾 ∈ Proset ) |
| 3 |
|
oduoppcbas.d |
⊢ ( 𝜑 → 𝐷 = ( ProsetToCat ‘ ( ODual ‘ 𝐾 ) ) ) |
| 4 |
|
oduoppcbas.o |
⊢ 𝑂 = ( oppCat ‘ 𝐶 ) |
| 5 |
|
eqid |
⊢ ( ODual ‘ 𝐾 ) = ( ODual ‘ 𝐾 ) |
| 6 |
5
|
oduprs |
⊢ ( 𝐾 ∈ Proset → ( ODual ‘ 𝐾 ) ∈ Proset ) |
| 7 |
2 6
|
syl |
⊢ ( 𝜑 → ( ODual ‘ 𝐾 ) ∈ Proset ) |
| 8 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
| 9 |
5 8
|
odubas |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ ( ODual ‘ 𝐾 ) ) |
| 10 |
9
|
a1i |
⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ ( ODual ‘ 𝐾 ) ) ) |
| 11 |
3 7 10
|
prstcbas |
⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ 𝐷 ) ) |
| 12 |
11
|
eqcomd |
⊢ ( 𝜑 → ( Base ‘ 𝐷 ) = ( Base ‘ 𝐾 ) ) |
| 13 |
1 2 12
|
prstcbas |
⊢ ( 𝜑 → ( Base ‘ 𝐷 ) = ( Base ‘ 𝐶 ) ) |
| 14 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
| 15 |
4 14
|
oppcbas |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝑂 ) |
| 16 |
13 15
|
eqtrdi |
⊢ ( 𝜑 → ( Base ‘ 𝐷 ) = ( Base ‘ 𝑂 ) ) |