Metamath Proof Explorer

Theorem cdlemg43

Description: Part of proof of Lemma G of Crawley p. 116, third line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013)

Ref Expression
Hypotheses cdlemg42.l 
|- .<_ = ( le ` K )
cdlemg42.j 
|- .\/ = ( join ` K )
cdlemg42.a 
|- A = ( Atoms ` K )
cdlemg42.h 
|- H = ( LHyp ` K )
cdlemg42.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg42.r 
|- R = ( ( trL ` K ) ` W )
cdlemg42.m 
|- ./\ = ( meet ` K )
Assertion cdlemg43 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( F ` ( G ` P ) ) = ( ( ( G ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( R ` G ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg42.l 
 |-  .<_ = ( le ` K )
2 cdlemg42.j 
 |-  .\/ = ( join ` K )
3 cdlemg42.a 
 |-  A = ( Atoms ` K )
4 cdlemg42.h 
 |-  H = ( LHyp ` K )
5 cdlemg42.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 cdlemg42.r 
 |-  R = ( ( trL ` K ) ` W )
7 cdlemg42.m 
 |-  ./\ = ( meet ` K )
8 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( K e. HL /\ W e. H ) )
9 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> F e. T )
10 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( P e. A /\ -. P .<_ W ) )
11 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> G e. T )
12 1 3 4 5 ltrnel 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
13 8 11 10 12 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
14 1 2 3 4 5 6 cdlemg42 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> -. ( G ` P ) .<_ ( P .\/ ( F ` P ) ) )
15 1 2 7 3 4 5 6 cdlemc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) ) /\ -. ( G ` P ) .<_ ( P .\/ ( F ` P ) ) ) -> ( F ` ( G ` P ) ) = ( ( ( G ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( ( P .\/ ( G ` P ) ) ./\ W ) ) ) )
16 8 9 10 13 14 15 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( F ` ( G ` P ) ) = ( ( ( G ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( ( P .\/ ( G ` P ) ) ./\ W ) ) ) )
17 1 2 7 3 4 5 6 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` G ) = ( ( P .\/ ( G ` P ) ) ./\ W ) )
18 8 11 10 17 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` G ) = ( ( P .\/ ( G ` P ) ) ./\ W ) )
19 18 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( F ` P ) .\/ ( R ` G ) ) = ( ( F ` P ) .\/ ( ( P .\/ ( G ` P ) ) ./\ W ) ) )
20 19 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( ( G ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( R ` G ) ) ) = ( ( ( G ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( ( P .\/ ( G ` P ) ) ./\ W ) ) ) )
21 16 20 eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( F ` ( G ` P ) ) = ( ( ( G ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( R ` G ) ) ) )