Metamath Proof Explorer

Theorem cdlemc

Description: Lemma C in Crawley p. 113. (Contributed by NM, 26-May-2012)

Ref Expression
Hypotheses cdlemc3.l 
|- .<_ = ( le ` K )
cdlemc3.j 
|- .\/ = ( join ` K )
cdlemc3.m 
|- ./\ = ( meet ` K )
cdlemc3.a 
|- A = ( Atoms ` K )
cdlemc3.h 
|- H = ( LHyp ` K )
cdlemc3.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemc3.r 
|- R = ( ( trL ` K ) ` W )
Assertion cdlemc 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) -> ( F ` Q ) = ( ( Q .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemc3.l 
 |-  .<_ = ( le ` K )
2 cdlemc3.j 
 |-  .\/ = ( join ` K )
3 cdlemc3.m 
 |-  ./\ = ( meet ` K )
4 cdlemc3.a 
 |-  A = ( Atoms ` K )
5 cdlemc3.h 
 |-  H = ( LHyp ` K )
6 cdlemc3.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemc3.r 
 |-  R = ( ( trL ` K ) ` W )
8 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = P ) -> ( K e. HL /\ W e. H ) )
9 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = P ) -> ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
10 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = P ) -> ( F ` P ) = P )
11 1 2 3 4 5 6 7 cdlemc6 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( F ` Q ) = ( ( Q .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) )
12 8 9 10 11 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = P ) -> ( F ` Q ) = ( ( Q .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) )
13 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) =/= P ) -> ( K e. HL /\ W e. H ) )
14 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) =/= P ) -> ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
15 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) =/= P ) -> -. Q .<_ ( P .\/ ( F ` P ) ) )
16 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) =/= P ) -> ( F ` P ) =/= P )
17 1 2 3 4 5 6 7 cdlemc5 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( -. Q .<_ ( P .\/ ( F ` P ) ) /\ ( F ` P ) =/= P ) ) -> ( F ` Q ) = ( ( Q .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) )
18 13 14 15 16 17 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) =/= P ) -> ( F ` Q ) = ( ( Q .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) )
19 12 18 pm2.61dane 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) -> ( F ` Q ) = ( ( Q .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) )