| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dihjatcclem.b |
|- B = ( Base ` K ) |
| 2 |
|
dihjatcclem.l |
|- .<_ = ( le ` K ) |
| 3 |
|
dihjatcclem.h |
|- H = ( LHyp ` K ) |
| 4 |
|
dihjatcclem.j |
|- .\/ = ( join ` K ) |
| 5 |
|
dihjatcclem.m |
|- ./\ = ( meet ` K ) |
| 6 |
|
dihjatcclem.a |
|- A = ( Atoms ` K ) |
| 7 |
|
dihjatcclem.u |
|- U = ( ( DVecH ` K ) ` W ) |
| 8 |
|
dihjatcclem.s |
|- .(+) = ( LSSum ` U ) |
| 9 |
|
dihjatcclem.i |
|- I = ( ( DIsoH ` K ) ` W ) |
| 10 |
|
dihjatcclem.v |
|- V = ( ( P .\/ Q ) ./\ W ) |
| 11 |
|
dihjatcclem.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
| 12 |
|
dihjatcclem.p |
|- ( ph -> ( P e. A /\ -. P .<_ W ) ) |
| 13 |
|
dihjatcclem.q |
|- ( ph -> ( Q e. A /\ -. Q .<_ W ) ) |
| 14 |
|
dihjatcc.w |
|- C = ( ( oc ` K ) ` W ) |
| 15 |
|
dihjatcc.t |
|- T = ( ( LTrn ` K ) ` W ) |
| 16 |
|
dihjatcc.r |
|- R = ( ( trL ` K ) ` W ) |
| 17 |
|
dihjatcc.e |
|- E = ( ( TEndo ` K ) ` W ) |
| 18 |
|
dihjatcc.g |
|- G = ( iota_ d e. T ( d ` C ) = P ) |
| 19 |
|
dihjatcc.dd |
|- D = ( iota_ d e. T ( d ` C ) = Q ) |
| 20 |
2 6 3 14
|
lhpocnel2 |
|- ( ( K e. HL /\ W e. H ) -> ( C e. A /\ -. C .<_ W ) ) |
| 21 |
11 20
|
syl |
|- ( ph -> ( C e. A /\ -. C .<_ W ) ) |
| 22 |
2 6 3 15 18
|
ltrniotacl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( C e. A /\ -. C .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> G e. T ) |
| 23 |
11 21 12 22
|
syl3anc |
|- ( ph -> G e. T ) |
| 24 |
2 6 3 15 19
|
ltrniotacl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( C e. A /\ -. C .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> D e. T ) |
| 25 |
11 21 13 24
|
syl3anc |
|- ( ph -> D e. T ) |
| 26 |
3 15
|
ltrncnv |
|- ( ( ( K e. HL /\ W e. H ) /\ D e. T ) -> `' D e. T ) |
| 27 |
11 25 26
|
syl2anc |
|- ( ph -> `' D e. T ) |
| 28 |
3 15
|
ltrnco |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ `' D e. T ) -> ( G o. `' D ) e. T ) |
| 29 |
11 23 27 28
|
syl3anc |
|- ( ph -> ( G o. `' D ) e. T ) |
| 30 |
2 4 5 6 3 15 16
|
trlval2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( G o. `' D ) e. T /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( R ` ( G o. `' D ) ) = ( ( Q .\/ ( ( G o. `' D ) ` Q ) ) ./\ W ) ) |
| 31 |
11 29 13 30
|
syl3anc |
|- ( ph -> ( R ` ( G o. `' D ) ) = ( ( Q .\/ ( ( G o. `' D ) ` Q ) ) ./\ W ) ) |
| 32 |
13
|
simpld |
|- ( ph -> Q e. A ) |
| 33 |
2 6 3 15
|
ltrncoval |
|- ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ `' D e. T ) /\ Q e. A ) -> ( ( G o. `' D ) ` Q ) = ( G ` ( `' D ` Q ) ) ) |
| 34 |
11 23 27 32 33
|
syl121anc |
|- ( ph -> ( ( G o. `' D ) ` Q ) = ( G ` ( `' D ` Q ) ) ) |
| 35 |
2 6 3 15 19
|
ltrniotacnvval |
|- ( ( ( K e. HL /\ W e. H ) /\ ( C e. A /\ -. C .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( `' D ` Q ) = C ) |
| 36 |
11 21 13 35
|
syl3anc |
|- ( ph -> ( `' D ` Q ) = C ) |
| 37 |
36
|
fveq2d |
|- ( ph -> ( G ` ( `' D ` Q ) ) = ( G ` C ) ) |
| 38 |
2 6 3 15 18
|
ltrniotaval |
|- ( ( ( K e. HL /\ W e. H ) /\ ( C e. A /\ -. C .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( G ` C ) = P ) |
| 39 |
11 21 12 38
|
syl3anc |
|- ( ph -> ( G ` C ) = P ) |
| 40 |
37 39
|
eqtrd |
|- ( ph -> ( G ` ( `' D ` Q ) ) = P ) |
| 41 |
34 40
|
eqtrd |
|- ( ph -> ( ( G o. `' D ) ` Q ) = P ) |
| 42 |
41
|
oveq2d |
|- ( ph -> ( Q .\/ ( ( G o. `' D ) ` Q ) ) = ( Q .\/ P ) ) |
| 43 |
11
|
simpld |
|- ( ph -> K e. HL ) |
| 44 |
12
|
simpld |
|- ( ph -> P e. A ) |
| 45 |
4 6
|
hlatjcom |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) ) |
| 46 |
43 44 32 45
|
syl3anc |
|- ( ph -> ( P .\/ Q ) = ( Q .\/ P ) ) |
| 47 |
42 46
|
eqtr4d |
|- ( ph -> ( Q .\/ ( ( G o. `' D ) ` Q ) ) = ( P .\/ Q ) ) |
| 48 |
47
|
oveq1d |
|- ( ph -> ( ( Q .\/ ( ( G o. `' D ) ` Q ) ) ./\ W ) = ( ( P .\/ Q ) ./\ W ) ) |
| 49 |
48 10
|
eqtr4di |
|- ( ph -> ( ( Q .\/ ( ( G o. `' D ) ` Q ) ) ./\ W ) = V ) |
| 50 |
31 49
|
eqtrd |
|- ( ph -> ( R ` ( G o. `' D ) ) = V ) |