Step |
Hyp |
Ref |
Expression |
1 |
|
ltrn2eq.l |
|- .<_ = ( le ` K ) |
2 |
|
ltrn2eq.a |
|- A = ( Atoms ` K ) |
3 |
|
ltrn2eq.h |
|- H = ( LHyp ` K ) |
4 |
|
ltrn2eq.t |
|- T = ( ( LTrn ` K ) ` W ) |
5 |
|
simp3r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( F ` P ) =/= P /\ ( F ` Q ) = Q ) ) -> ( F ` Q ) = Q ) |
6 |
|
simpl1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( F ` P ) =/= P /\ ( F ` Q ) = Q ) ) /\ -. Q .<_ W ) -> ( K e. HL /\ W e. H ) ) |
7 |
|
simpl21 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( F ` P ) =/= P /\ ( F ` Q ) = Q ) ) /\ -. Q .<_ W ) -> F e. T ) |
8 |
|
simpl22 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( F ` P ) =/= P /\ ( F ` Q ) = Q ) ) /\ -. Q .<_ W ) -> ( P e. A /\ -. P .<_ W ) ) |
9 |
|
simpl23 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( F ` P ) =/= P /\ ( F ` Q ) = Q ) ) /\ -. Q .<_ W ) -> Q e. A ) |
10 |
|
simpr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( F ` P ) =/= P /\ ( F ` Q ) = Q ) ) /\ -. Q .<_ W ) -> -. Q .<_ W ) |
11 |
9 10
|
jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( F ` P ) =/= P /\ ( F ` Q ) = Q ) ) /\ -. Q .<_ W ) -> ( Q e. A /\ -. Q .<_ W ) ) |
12 |
|
simpl3l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( F ` P ) =/= P /\ ( F ` Q ) = Q ) ) /\ -. Q .<_ W ) -> ( F ` P ) =/= P ) |
13 |
1 2 3 4
|
ltrnatneq |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) =/= P ) -> ( F ` Q ) =/= Q ) |
14 |
6 7 8 11 12 13
|
syl131anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( F ` P ) =/= P /\ ( F ` Q ) = Q ) ) /\ -. Q .<_ W ) -> ( F ` Q ) =/= Q ) |
15 |
14
|
ex |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( F ` P ) =/= P /\ ( F ` Q ) = Q ) ) -> ( -. Q .<_ W -> ( F ` Q ) =/= Q ) ) |
16 |
15
|
necon4bd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( F ` P ) =/= P /\ ( F ` Q ) = Q ) ) -> ( ( F ` Q ) = Q -> Q .<_ W ) ) |
17 |
5 16
|
mpd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( F ` P ) =/= P /\ ( F ` Q ) = Q ) ) -> Q .<_ W ) |