Metamath Proof Explorer

Theorem ltrn11at

Description: Frequently used one-to-one property of lattice translation atoms. (Contributed by NM, 5-May-2013)

Ref Expression
Hypotheses ltrneq2.a 
|- A = ( Atoms ` K )
ltrneq2.h 
|- H = ( LHyp ` K )
ltrneq2.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion ltrn11at 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> ( F ` P ) =/= ( F ` Q ) )

Proof

Step Hyp Ref Expression
1 ltrneq2.a 
 |-  A = ( Atoms ` K )
2 ltrneq2.h 
 |-  H = ( LHyp ` K )
3 ltrneq2.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> P =/= Q )
5 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> ( K e. HL /\ W e. H ) )
6 simp2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> F e. T )
7 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> P e. A )
8 eqid 
 |-  ( Base ` K ) = ( Base ` K )
9 8 1 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
10 7 9 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> P e. ( Base ` K ) )
11 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> Q e. A )
12 8 1 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
13 11 12 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> Q e. ( Base ` K ) )
14 8 2 3 ltrn11 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) ) -> ( ( F ` P ) = ( F ` Q ) <-> P = Q ) )
15 5 6 10 13 14 syl112anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> ( ( F ` P ) = ( F ` Q ) <-> P = Q ) )
16 15 necon3bid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> ( ( F ` P ) =/= ( F ` Q ) <-> P =/= Q ) )
17 4 16 mpbird 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> ( F ` P ) =/= ( F ` Q ) )