Metamath Proof Explorer

Theorem cdleme38n

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(x) is one-to-one on P .\/ Q line. TODO: FIX COMMENT. TODO shorter if proved directly from cdleme36m and cdleme37m ? (Contributed by NM, 14-Mar-2013)

Ref Expression
Hypotheses cdleme38.l 
|- .<_ = ( le ` K )
cdleme38.j 
|- .\/ = ( join ` K )
cdleme38.m 
|- ./\ = ( meet ` K )
cdleme38.a 
|- A = ( Atoms ` K )
cdleme38.h 
|- H = ( LHyp ` K )
cdleme38.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme38.e 
|- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme38.d 
|- D = ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) )
cdleme38.v 
|- V = ( ( t .\/ E ) ./\ W )
cdleme38.x 
|- X = ( ( u .\/ D ) ./\ W )
cdleme38.f 
|- F = ( ( R .\/ V ) ./\ ( E .\/ ( ( t .\/ R ) ./\ W ) ) )
cdleme38.g 
|- G = ( ( S .\/ X ) ./\ ( D .\/ ( ( u .\/ S ) ./\ W ) ) )
Assertion cdleme38n 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> F =/= G )

Proof

Step Hyp Ref Expression
1 cdleme38.l 
 |-  .<_ = ( le ` K )
2 cdleme38.j 
 |-  .\/ = ( join ` K )
3 cdleme38.m 
 |-  ./\ = ( meet ` K )
4 cdleme38.a 
 |-  A = ( Atoms ` K )
5 cdleme38.h 
 |-  H = ( LHyp ` K )
6 cdleme38.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme38.e 
 |-  E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
8 cdleme38.d 
 |-  D = ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) )
9 cdleme38.v 
 |-  V = ( ( t .\/ E ) ./\ W )
10 cdleme38.x 
 |-  X = ( ( u .\/ D ) ./\ W )
11 cdleme38.f 
 |-  F = ( ( R .\/ V ) ./\ ( E .\/ ( ( t .\/ R ) ./\ W ) ) )
12 cdleme38.g 
 |-  G = ( ( S .\/ X ) ./\ ( D .\/ ( ( u .\/ S ) ./\ W ) ) )
13 simp313 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> R =/= S )
14 simpl1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) /\ F = G ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
15 simpl21 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) /\ F = G ) -> P =/= Q )
16 simpl22 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) /\ F = G ) -> ( R e. A /\ -. R .<_ W ) )
17 simpl23 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) /\ F = G ) -> ( S e. A /\ -. S .<_ W ) )
18 simp311 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> R .<_ ( P .\/ Q ) )
19 18 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) /\ F = G ) -> R .<_ ( P .\/ Q ) )
20 simp312 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> S .<_ ( P .\/ Q ) )
21 20 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) /\ F = G ) -> S .<_ ( P .\/ Q ) )
22 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) /\ F = G ) -> F = G )
23 19 21 22 3jca 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) /\ F = G ) -> ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ F = G ) )
24 simpl32 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) /\ F = G ) -> ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) )
25 simpl33 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) /\ F = G ) -> ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) )
26 1 2 3 4 5 6 7 8 9 10 11 12 cdleme38m 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ F = G ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> R = S )
27 14 15 16 17 23 24 25 26 syl133anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) /\ F = G ) -> R = S )
28 27 ex 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( F = G -> R = S ) )
29 28 necon3d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( R =/= S -> F =/= G ) )
30 13 29 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> F =/= G )