| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dihjat3.b |
|- B = ( Base ` K ) |
| 2 |
|
dihjat3.h |
|- H = ( LHyp ` K ) |
| 3 |
|
dihjat3.j |
|- .\/ = ( join ` K ) |
| 4 |
|
dihjat3.a |
|- A = ( Atoms ` K ) |
| 5 |
|
dihjat3.u |
|- U = ( ( DVecH ` K ) ` W ) |
| 6 |
|
dihjat3.s |
|- .(+) = ( LSSum ` U ) |
| 7 |
|
dihjat3.i |
|- I = ( ( DIsoH ` K ) ` W ) |
| 8 |
|
dihjat3.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
| 9 |
|
dihjat3.x |
|- ( ph -> X e. B ) |
| 10 |
|
dihjat3.p |
|- ( ph -> P e. A ) |
| 11 |
1 4
|
atbase |
|- ( P e. A -> P e. B ) |
| 12 |
10 11
|
syl |
|- ( ph -> P e. B ) |
| 13 |
|
eqid |
|- ( ( joinH ` K ) ` W ) = ( ( joinH ` K ) ` W ) |
| 14 |
1 3 2 7 13
|
djhlj |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ P e. B ) ) -> ( I ` ( X .\/ P ) ) = ( ( I ` X ) ( ( joinH ` K ) ` W ) ( I ` P ) ) ) |
| 15 |
8 9 12 14
|
syl12anc |
|- ( ph -> ( I ` ( X .\/ P ) ) = ( ( I ` X ) ( ( joinH ` K ) ` W ) ( I ` P ) ) ) |
| 16 |
|
eqid |
|- ( LSAtoms ` U ) = ( LSAtoms ` U ) |
| 17 |
1 2 7
|
dihcl |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( I ` X ) e. ran I ) |
| 18 |
8 9 17
|
syl2anc |
|- ( ph -> ( I ` X ) e. ran I ) |
| 19 |
4 2 5 7 16
|
dihatlat |
|- ( ( ( K e. HL /\ W e. H ) /\ P e. A ) -> ( I ` P ) e. ( LSAtoms ` U ) ) |
| 20 |
8 10 19
|
syl2anc |
|- ( ph -> ( I ` P ) e. ( LSAtoms ` U ) ) |
| 21 |
2 7 13 5 6 16 8 18 20
|
dihjat2 |
|- ( ph -> ( ( I ` X ) ( ( joinH ` K ) ` W ) ( I ` P ) ) = ( ( I ` X ) .(+) ( I ` P ) ) ) |
| 22 |
15 21
|
eqtrd |
|- ( ph -> ( I ` ( X .\/ P ) ) = ( ( I ` X ) .(+) ( I ` P ) ) ) |