| 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 |
|
oppgoppcco.o |
|- ( ph -> .x. = ( comp ` D ) ) |
| 8 |
|
oppgoppcco.x |
|- ( ph -> .xb = ( comp ` O ) ) |
| 9 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
| 10 |
4 9
|
oppcbas |
|- ( Base ` C ) = ( Base ` O ) |
| 11 |
10
|
eqcomi |
|- ( Base ` O ) = ( Base ` C ) |
| 12 |
11
|
a1i |
|- ( ph -> ( Base ` O ) = ( Base ` C ) ) |
| 13 |
|
eqidd |
|- ( ph -> ( comp ` C ) = ( comp ` C ) ) |
| 14 |
1 2 12 6 6 6 13
|
mndtcco |
|- ( ph -> ( <. Y , Y >. ( comp ` C ) Y ) = ( +g ` M ) ) |
| 15 |
14
|
tposeqd |
|- ( ph -> tpos ( <. Y , Y >. ( comp ` C ) Y ) = tpos ( +g ` M ) ) |
| 16 |
|
eqid |
|- ( comp ` C ) = ( comp ` C ) |
| 17 |
11 16 4 6 6 6
|
oppccofval |
|- ( ph -> ( <. Y , Y >. ( comp ` O ) Y ) = tpos ( <. Y , Y >. ( comp ` C ) Y ) ) |
| 18 |
|
eqid |
|- ( oppG ` M ) = ( oppG ` M ) |
| 19 |
18
|
oppgmnd |
|- ( M e. Mnd -> ( oppG ` M ) e. Mnd ) |
| 20 |
2 19
|
syl |
|- ( ph -> ( oppG ` M ) e. Mnd ) |
| 21 |
|
eqidd |
|- ( ph -> ( Base ` D ) = ( Base ` D ) ) |
| 22 |
3 20 21 5 5 5 7
|
mndtcco |
|- ( ph -> ( <. X , X >. .x. X ) = ( +g ` ( oppG ` M ) ) ) |
| 23 |
|
eqid |
|- ( +g ` M ) = ( +g ` M ) |
| 24 |
|
eqid |
|- ( +g ` ( oppG ` M ) ) = ( +g ` ( oppG ` M ) ) |
| 25 |
23 18 24
|
oppgplusfval |
|- ( +g ` ( oppG ` M ) ) = tpos ( +g ` M ) |
| 26 |
22 25
|
eqtrdi |
|- ( ph -> ( <. X , X >. .x. X ) = tpos ( +g ` M ) ) |
| 27 |
15 17 26
|
3eqtr4rd |
|- ( ph -> ( <. X , X >. .x. X ) = ( <. Y , Y >. ( comp ` O ) Y ) ) |
| 28 |
8
|
oveqd |
|- ( ph -> ( <. Y , Y >. .xb Y ) = ( <. Y , Y >. ( comp ` O ) Y ) ) |
| 29 |
27 28
|
eqtr4d |
|- ( ph -> ( <. X , X >. .x. X ) = ( <. Y , Y >. .xb Y ) ) |