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Metamath Proof Explorer

Theorem cdlemg18a

Description: Show two lines are different. TODO: fix comment. (Contributed by NM, 14-May-2013)

Ref Expression
Hypotheses cdlemg12.l 
|- .<_ = ( le ` K )
cdlemg12.j 
|- .\/ = ( join ` K )
cdlemg12.m 
|- ./\ = ( meet ` K )
cdlemg12.a 
|- A = ( Atoms ` K )
cdlemg12.h 
|- H = ( LHyp ` K )
cdlemg12.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg12b.r 
|- R = ( ( trL ` K ) ` W )
Assertion cdlemg18a 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( P .\/ ( F ` Q ) ) =/= ( Q .\/ ( F ` P ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg12.l 
 |-  .<_ = ( le ` K )
2 cdlemg12.j 
 |-  .\/ = ( join ` K )
3 cdlemg12.m 
 |-  ./\ = ( meet ` K )
4 cdlemg12.a 
 |-  A = ( Atoms ` K )
5 cdlemg12.h 
 |-  H = ( LHyp ` K )
6 cdlemg12.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemg12b.r 
 |-  R = ( ( trL ` K ) ` W )
8 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) )
9 simpl1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) ) -> K e. HL )
10 simpl21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) ) -> P e. A )
11 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) ) -> ( K e. HL /\ W e. H ) )
12 simpl23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) ) -> F e. T )
13 simpl22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) ) -> Q e. A )
14 1 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ Q e. A ) -> ( F ` Q ) e. A )
15 11 12 13 14 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) ) -> ( F ` Q ) e. A )
16 1 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( F ` P ) e. A )
17 11 12 10 16 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) ) -> ( F ` P ) e. A )
18 simpl3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) ) -> P =/= Q )
19 4 5 6 ltrn11at 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> ( F ` P ) =/= ( F ` Q ) )
20 11 12 10 13 18 19 syl113anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) ) -> ( F ` P ) =/= ( F ` Q ) )
21 20 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) ) -> ( F ` Q ) =/= ( F ` P ) )
22 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) ) -> ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) )
23 2 4 hlatexch4 
 |-  ( ( ( K e. HL /\ P e. A /\ ( F ` Q ) e. A ) /\ ( Q e. A /\ ( F ` P ) e. A ) /\ ( P =/= Q /\ ( F ` Q ) =/= ( F ` P ) /\ ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) ) ) -> ( P .\/ Q ) = ( ( F ` Q ) .\/ ( F ` P ) ) )
24 9 10 15 13 17 18 21 22 23 syl323anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) ) -> ( P .\/ Q ) = ( ( F ` Q ) .\/ ( F ` P ) ) )
25 24 eqcomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) = ( P .\/ Q ) )
26 25 ex 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` Q ) ) = ( Q .\/ ( F ` P ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) = ( P .\/ Q ) ) )
27 26 necon3d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) -> ( P .\/ ( F ` Q ) ) =/= ( Q .\/ ( F ` P ) ) ) )
28 8 27 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( P .\/ ( F ` Q ) ) =/= ( Q .\/ ( F ` P ) ) )