| Step |
Hyp |
Ref |
Expression |
| 1 |
|
initopropd.1 |
|- ( ph -> ( Homf ` C ) = ( Homf ` D ) ) |
| 2 |
|
initopropd.2 |
|- ( ph -> ( comf ` C ) = ( comf ` D ) ) |
| 3 |
|
initopropdlem.1 |
|- ( ph -> -. C e. _V ) |
| 4 |
|
zeroofn |
|- ZeroO Fn Cat |
| 5 |
|
ssv |
|- Cat C_ _V |
| 6 |
|
simpr |
|- ( ( ph /\ D e. Cat ) -> D e. Cat ) |
| 7 |
|
eqid |
|- ( Base ` D ) = ( Base ` D ) |
| 8 |
|
eqid |
|- ( Hom ` D ) = ( Hom ` D ) |
| 9 |
6 7 8
|
zerooval |
|- ( ( ph /\ D e. Cat ) -> ( ZeroO ` D ) = ( ( InitO ` D ) i^i ( TermO ` D ) ) ) |
| 10 |
1 2 3
|
initopropdlem |
|- ( ph -> ( InitO ` C ) = ( InitO ` D ) ) |
| 11 |
|
fvprc |
|- ( -. C e. _V -> ( InitO ` C ) = (/) ) |
| 12 |
3 11
|
syl |
|- ( ph -> ( InitO ` C ) = (/) ) |
| 13 |
10 12
|
eqtr3d |
|- ( ph -> ( InitO ` D ) = (/) ) |
| 14 |
13
|
adantr |
|- ( ( ph /\ D e. Cat ) -> ( InitO ` D ) = (/) ) |
| 15 |
1 2 3
|
termopropdlem |
|- ( ph -> ( TermO ` C ) = ( TermO ` D ) ) |
| 16 |
|
fvprc |
|- ( -. C e. _V -> ( TermO ` C ) = (/) ) |
| 17 |
3 16
|
syl |
|- ( ph -> ( TermO ` C ) = (/) ) |
| 18 |
15 17
|
eqtr3d |
|- ( ph -> ( TermO ` D ) = (/) ) |
| 19 |
18
|
adantr |
|- ( ( ph /\ D e. Cat ) -> ( TermO ` D ) = (/) ) |
| 20 |
14 19
|
ineq12d |
|- ( ( ph /\ D e. Cat ) -> ( ( InitO ` D ) i^i ( TermO ` D ) ) = ( (/) i^i (/) ) ) |
| 21 |
|
inidm |
|- ( (/) i^i (/) ) = (/) |
| 22 |
20 21
|
eqtrdi |
|- ( ( ph /\ D e. Cat ) -> ( ( InitO ` D ) i^i ( TermO ` D ) ) = (/) ) |
| 23 |
9 22
|
eqtrd |
|- ( ( ph /\ D e. Cat ) -> ( ZeroO ` D ) = (/) ) |
| 24 |
4 3 5 23
|
initopropdlemlem |
|- ( ph -> ( ZeroO ` C ) = ( ZeroO ` D ) ) |