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Metamath Proof Explorer

Theorem cdlemg12a

Description: TODO: FIX COMMENT. (Contributed by NM, 5-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 )
cdlemg12.u 
|- U = ( ( P .\/ Q ) ./\ W )
Assertion cdlemg12a 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> ( ( P .\/ U ) ./\ ( ( G ` P ) .\/ U ) ) .<_ ( ( F ` ( G ` P ) ) .\/ U ) )

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 cdlemg12.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> K e. HL )
9 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> P e. A )
10 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> ( K e. HL /\ W e. H ) )
11 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> G e. T )
12 1 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ P e. A ) -> ( G ` P ) e. A )
13 10 11 9 12 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> ( G ` P ) e. A )
14 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> W e. H )
15 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> ( P e. A /\ -. P .<_ W ) )
16 simp22l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> Q e. A )
17 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> P =/= Q )
18 1 2 3 4 5 7 cdleme0a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
19 8 14 15 16 17 18 syl212anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> U e. A )
20 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) )
21 1 2 3 4 2llnma3r 
 |-  ( ( K e. HL /\ ( P e. A /\ ( G ` P ) e. A /\ U e. A ) /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) -> ( ( P .\/ U ) ./\ ( ( G ` P ) .\/ U ) ) = U )
22 8 9 13 19 20 21 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> ( ( P .\/ U ) ./\ ( ( G ` P ) .\/ U ) ) = U )
23 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> F e. T )
24 1 4 5 6 ltrncoat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> ( F ` ( G ` P ) ) e. A )
25 10 23 11 9 24 syl121anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> ( F ` ( G ` P ) ) e. A )
26 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ ( F ` ( G ` P ) ) e. A /\ U e. A ) -> U .<_ ( ( F ` ( G ` P ) ) .\/ U ) )
27 8 25 19 26 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> U .<_ ( ( F ` ( G ` P ) ) .\/ U ) )
28 22 27 eqbrtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ ( P .\/ U ) =/= ( ( G ` P ) .\/ U ) ) ) -> ( ( P .\/ U ) ./\ ( ( G ` P ) .\/ U ) ) .<_ ( ( F ` ( G ` P ) ) .\/ U ) )