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Metamath Proof Explorer

Theorem cdleme14

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, " and ... are axially perspective." We apply dalaw to cdleme13 . F and G represent f(s) and f(t) respectively. (Contributed by NM, 8-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 cdleme14 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ T ) ./\ ( F .\/ G ) ) .<_ ( ( ( T .\/ P ) ./\ ( G .\/ Q ) ) .\/ ( ( P .\/ S ) ./\ ( Q .\/ F ) ) ) )

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 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( K e. HL /\ W e. H ) )
10 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( P e. A /\ -. P .<_ W ) )
11 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> Q e. A )
12 simp23l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> P =/= Q )
13 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( S e. A /\ -. S .<_ W ) )
14 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( T e. A /\ -. T .<_ W ) )
15 simp23r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> S =/= T )
16 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> -. U .<_ ( S .\/ T ) )
17 15 16 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( S =/= T /\ -. U .<_ ( S .\/ T ) ) )
18 1 2 3 4 5 6 7 8 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 ) )
19 9 10 11 12 13 14 17 18 syl133anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ F ) ./\ ( T .\/ G ) ) .<_ ( P .\/ Q ) )
20 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> K e. HL )
21 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> S e. A )
22 simp22l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> T e. A )
23 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> P e. A )
24 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
25 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> -. S .<_ ( P .\/ Q ) )
26 1 2 3 4 5 6 7 cdleme3fa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F e. A )
27 9 10 24 13 12 25 26 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> F e. A )
28 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> -. T .<_ ( P .\/ Q ) )
29 1 2 3 4 5 6 8 cdleme3fa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ -. T .<_ ( P .\/ Q ) ) ) -> G e. A )
30 9 10 24 14 12 28 29 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> G e. A )
31 1 2 3 4 dalaw 
 |-  ( ( K e. HL /\ ( S e. A /\ T e. A /\ P e. A ) /\ ( F e. A /\ G e. A /\ Q e. A ) ) -> ( ( ( S .\/ F ) ./\ ( T .\/ G ) ) .<_ ( P .\/ Q ) -> ( ( S .\/ T ) ./\ ( F .\/ G ) ) .<_ ( ( ( T .\/ P ) ./\ ( G .\/ Q ) ) .\/ ( ( P .\/ S ) ./\ ( Q .\/ F ) ) ) ) )
32 20 21 22 23 27 30 11 31 syl133anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( ( S .\/ F ) ./\ ( T .\/ G ) ) .<_ ( P .\/ Q ) -> ( ( S .\/ T ) ./\ ( F .\/ G ) ) .<_ ( ( ( T .\/ P ) ./\ ( G .\/ Q ) ) .\/ ( ( P .\/ S ) ./\ ( Q .\/ F ) ) ) ) )
33 19 32 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ T ) ./\ ( F .\/ G ) ) .<_ ( ( ( T .\/ P ) ./\ ( G .\/ Q ) ) .\/ ( ( P .\/ S ) ./\ ( Q .\/ F ) ) ) )