Step |
Hyp |
Ref |
Expression |
1 |
|
trlat.l |
|- .<_ = ( le ` K ) |
2 |
|
trlat.a |
|- A = ( Atoms ` K ) |
3 |
|
trlat.h |
|- H = ( LHyp ` K ) |
4 |
|
trlat.t |
|- T = ( ( LTrn ` K ) ` W ) |
5 |
|
trlat.r |
|- R = ( ( trL ` K ) ` W ) |
6 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( K e. HL /\ W e. H ) ) |
7 |
|
simp3l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> F e. T ) |
8 |
|
simp2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( P e. A /\ -. P .<_ W ) ) |
9 |
|
eqid |
|- ( join ` K ) = ( join ` K ) |
10 |
|
eqid |
|- ( meet ` K ) = ( meet ` K ) |
11 |
1 9 10 2 3 4 5
|
trlval2 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) ) |
12 |
6 7 8 11
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( R ` F ) = ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) ) |
13 |
|
simp2l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> P e. A ) |
14 |
1 2 3 4
|
ltrnat |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( F ` P ) e. A ) |
15 |
6 7 13 14
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( F ` P ) e. A ) |
16 |
|
simp3r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( F ` P ) =/= P ) |
17 |
16
|
necomd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> P =/= ( F ` P ) ) |
18 |
1 9 10 2 3
|
lhpat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) e. A /\ P =/= ( F ` P ) ) ) -> ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) e. A ) |
19 |
6 8 15 17 18
|
syl112anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) e. A ) |
20 |
12 19
|
eqeltrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A ) |