Metamath Proof Explorer

Theorem ltrnatlw

Description: If the value of an atom equals the atom in a non-identity translation, the atom is under the fiducial hyperplane. (Contributed by NM, 15-May-2013)

Ref Expression
Hypotheses ltrn2eq.l 
|- .<_ = ( le ` K )
ltrn2eq.a 
|- A = ( Atoms ` K )
ltrn2eq.h 
|- H = ( LHyp ` K )
ltrn2eq.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion ltrnatlw 
|- ( ( ( 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 )

Proof

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 )