Step |
Hyp |
Ref |
Expression |
1 |
|
mndtccat.c |
⊢ ( 𝜑 → 𝐶 = ( MndToCat ‘ 𝑀 ) ) |
2 |
|
mndtccat.m |
⊢ ( 𝜑 → 𝑀 ∈ Mnd ) |
3 |
|
oppgoppchom.d |
⊢ ( 𝜑 → 𝐷 = ( MndToCat ‘ ( oppg ‘ 𝑀 ) ) ) |
4 |
|
oppgoppchom.o |
⊢ 𝑂 = ( oppCat ‘ 𝐶 ) |
5 |
|
oppgoppchom.x |
⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐷 ) ) |
6 |
|
oppgoppchom.y |
⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝑂 ) ) |
7 |
|
eqid |
⊢ ( oppg ‘ 𝑀 ) = ( oppg ‘ 𝑀 ) |
8 |
|
eqid |
⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 ) |
9 |
7 8
|
oppgid |
⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ ( oppg ‘ 𝑀 ) ) |
10 |
9
|
a1i |
⊢ ( 𝜑 → ( 0g ‘ 𝑀 ) = ( 0g ‘ ( oppg ‘ 𝑀 ) ) ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
12 |
4 11
|
oppcbas |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝑂 ) |
13 |
12
|
eqcomi |
⊢ ( Base ‘ 𝑂 ) = ( Base ‘ 𝐶 ) |
14 |
13
|
a1i |
⊢ ( 𝜑 → ( Base ‘ 𝑂 ) = ( Base ‘ 𝐶 ) ) |
15 |
1 2
|
mndtccat |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
16 |
|
eqid |
⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 ) |
17 |
4 16
|
oppcid |
⊢ ( 𝐶 ∈ Cat → ( Id ‘ 𝑂 ) = ( Id ‘ 𝐶 ) ) |
18 |
15 17
|
syl |
⊢ ( 𝜑 → ( Id ‘ 𝑂 ) = ( Id ‘ 𝐶 ) ) |
19 |
1 2 14 6 18
|
mndtcid |
⊢ ( 𝜑 → ( ( Id ‘ 𝑂 ) ‘ 𝑌 ) = ( 0g ‘ 𝑀 ) ) |
20 |
7
|
oppgmnd |
⊢ ( 𝑀 ∈ Mnd → ( oppg ‘ 𝑀 ) ∈ Mnd ) |
21 |
2 20
|
syl |
⊢ ( 𝜑 → ( oppg ‘ 𝑀 ) ∈ Mnd ) |
22 |
|
eqidd |
⊢ ( 𝜑 → ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) ) |
23 |
|
eqidd |
⊢ ( 𝜑 → ( Id ‘ 𝐷 ) = ( Id ‘ 𝐷 ) ) |
24 |
3 21 22 5 23
|
mndtcid |
⊢ ( 𝜑 → ( ( Id ‘ 𝐷 ) ‘ 𝑋 ) = ( 0g ‘ ( oppg ‘ 𝑀 ) ) ) |
25 |
10 19 24
|
3eqtr4rd |
⊢ ( 𝜑 → ( ( Id ‘ 𝐷 ) ‘ 𝑋 ) = ( ( Id ‘ 𝑂 ) ‘ 𝑌 ) ) |