Metamath Proof Explorer

Theorem cdleme13

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, " and are centrally perspective." F and G represent f(s) and f(t) respectively. (Contributed by NM, 7-Oct-2012)

Ref Expression
Hypotheses cdleme12.l 
|- .<_ = ( le ` K )
cdleme12.j 
|- .\/ = ( join ` K )
cdleme12.m 
|- ./\ = ( meet ` K )
cdleme12.a 
|- A = ( Atoms ` K )
cdleme12.h 
|- H = ( LHyp ` K )
cdleme12.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme12.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme12.g 
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
Assertion cdleme13 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) ) -> ( ( S .\/ F ) ./\ ( T .\/ G ) ) .<_ ( P .\/ Q ) )

Proof

Step Hyp Ref Expression
1 cdleme12.l 
 |-  .<_ = ( le ` K )
2 cdleme12.j 
 |-  .\/ = ( join ` K )
3 cdleme12.m 
 |-  ./\ = ( meet ` K )
4 cdleme12.a 
 |-  A = ( Atoms ` K )
5 cdleme12.h 
 |-  H = ( LHyp ` K )
6 cdleme12.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme12.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme12.g 
 |-  G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 1 2 3 4 5 6 7 8 cdleme12 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) ) -> ( ( S .\/ F ) ./\ ( T .\/ G ) ) = U )
10 9 6 eqtrdi 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) ) -> ( ( S .\/ F ) ./\ ( T .\/ G ) ) = ( ( P .\/ Q ) ./\ W ) )
11 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) ) -> K e. HL )
12 11 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) ) -> K e. Lat )
13 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) ) -> P e. A )
14 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) ) -> Q e. A )
15 eqid 
 |-  ( Base ` K ) = ( Base ` K )
16 15 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
17 11 13 14 16 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
18 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) ) -> W e. H )
19 15 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
20 18 19 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) ) -> W e. ( Base ` K ) )
21 15 1 3 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) )
22 12 17 20 21 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) )
23 10 22 eqbrtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( S =/= T /\ -. U .<_ ( S .\/ T ) ) ) ) -> ( ( S .\/ F ) ./\ ( T .\/ G ) ) .<_ ( P .\/ Q ) )