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Metamath Proof Explorer

Theorem trlcocnvat

Description: Commonly used special case of trlcoat . (Contributed by NM, 1-Jul-2013)

Ref Expression
Hypotheses trlcoat.a 
|- A = ( Atoms ` K )
trlcoat.h 
|- H = ( LHyp ` K )
trlcoat.t 
|- T = ( ( LTrn ` K ) ` W )
trlcoat.r 
|- R = ( ( trL ` K ) ` W )
Assertion trlcocnvat 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( R ` ( F o. `' G ) ) e. A )

Proof

Step Hyp Ref Expression
1 trlcoat.a 
 |-  A = ( Atoms ` K )
2 trlcoat.h 
 |-  H = ( LHyp ` K )
3 trlcoat.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 trlcoat.r 
 |-  R = ( ( trL ` K ) ` W )
5 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( K e. HL /\ W e. H ) )
6 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> F e. T )
7 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> G e. T )
8 2 3 ltrncnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> `' G e. T )
9 5 7 8 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> `' G e. T )
10 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( R ` F ) =/= ( R ` G ) )
11 2 3 4 trlcnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` `' G ) = ( R ` G ) )
12 5 7 11 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( R ` `' G ) = ( R ` G ) )
13 10 12 neeqtrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( R ` F ) =/= ( R ` `' G ) )
14 1 2 3 4 trlcoat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ `' G e. T ) /\ ( R ` F ) =/= ( R ` `' G ) ) -> ( R ` ( F o. `' G ) ) e. A )
15 5 6 9 13 14 syl121anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( R ` ( F o. `' G ) ) e. A )