| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mndtccat.c |
|- ( ph -> C = ( MndToCat ` M ) ) |
| 2 |
|
mndtccat.m |
|- ( ph -> M e. Mnd ) |
| 3 |
|
oppgoppchom.d |
|- ( ph -> D = ( MndToCat ` ( oppG ` M ) ) ) |
| 4 |
|
oppgoppchom.o |
|- O = ( oppCat ` C ) |
| 5 |
|
oppgoppchom.x |
|- ( ph -> X e. ( Base ` D ) ) |
| 6 |
|
oppgoppchom.y |
|- ( ph -> Y e. ( 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 |
|- ( ph -> ( 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 |
|- ( ph -> ( Base ` O ) = ( Base ` C ) ) |
| 15 |
1 2
|
mndtccat |
|- ( ph -> C e. Cat ) |
| 16 |
|
eqid |
|- ( Id ` C ) = ( Id ` C ) |
| 17 |
4 16
|
oppcid |
|- ( C e. Cat -> ( Id ` O ) = ( Id ` C ) ) |
| 18 |
15 17
|
syl |
|- ( ph -> ( Id ` O ) = ( Id ` C ) ) |
| 19 |
1 2 14 6 18
|
mndtcid |
|- ( ph -> ( ( Id ` O ) ` Y ) = ( 0g ` M ) ) |
| 20 |
7
|
oppgmnd |
|- ( M e. Mnd -> ( oppG ` M ) e. Mnd ) |
| 21 |
2 20
|
syl |
|- ( ph -> ( oppG ` M ) e. Mnd ) |
| 22 |
|
eqidd |
|- ( ph -> ( Base ` D ) = ( Base ` D ) ) |
| 23 |
|
eqidd |
|- ( ph -> ( Id ` D ) = ( Id ` D ) ) |
| 24 |
3 21 22 5 23
|
mndtcid |
|- ( ph -> ( ( Id ` D ) ` X ) = ( 0g ` ( oppG ` M ) ) ) |
| 25 |
10 19 24
|
3eqtr4rd |
|- ( ph -> ( ( Id ` D ) ` X ) = ( ( Id ` O ) ` Y ) ) |